Waves and Electromagnetism Wavefunctions Lecture notes
190
Waves and Electromagnetism Wavefunctions Lecture notes Alessandro De Angelis University of Udine, December 2012
Transcript of Waves and Electromagnetism Wavefunctions Lecture notes
Waves and Electromagnetism
Wavefunctions
Lecture notes
Alessandro De Angelis
University of Udine, December 2012
Contents
1 Differential calculus applied to vector fields 51.1 Scalar and vector fields . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 The divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 The curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Second order derivatives, and the Laplacian . . . . . . . . . . . . 91.6 The differential equation of heat flow . . . . . . . . . . . . . . . . 101.7 Differential operators in polar coordinates . . . . . . . . . . . . . 111.8 The gradient theorem . . . . . . . . . . . . . . . . . . . . . . . . 121.9 The divergence theorem (Gauss’ theorem) . . . . . . . . . . . . . 14
1.9.1 Flux of a vector field . . . . . . . . . . . . . . . . . . . . . 141.9.2 Flux through a closed surface . . . . . . . . . . . . . . . . 151.9.3 The flux from a cube . . . . . . . . . . . . . . . . . . . . . 171.9.4 Heat conduction; the diffusion equation . . . . . . . . . . 19
1.10 The curl theorem (Stokes’ theorem) . . . . . . . . . . . . . . . . 191.10.1 Circulation of a vector field . . . . . . . . . . . . . . . . . 191.10.2 The circulation around a square . . . . . . . . . . . . . . 20
1.11 Curl-free fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.12 Dirac’s delta function . . . . . . . . . . . . . . . . . . . . . . . . 23
I Waves and electromagnetism 25
2 Waves 262.1 Translations and the wave equation . . . . . . . . . . . . . . . . . 27
2.1.1 An example: mechanical waves in one dimension . . . . . 282.2 Energy transported by a wave . . . . . . . . . . . . . . . . . . . . 292.3 Sinusoidal waves and the Fourier theorem . . . . . . . . . . . . . 302.4 Amplitude, wavelength, period . . . . . . . . . . . . . . . . . . . 312.5 Transverse and longitudinal waves . . . . . . . . . . . . . . . . . 322.6 An example: sound . . . . . . . . . . . . . . . . . . . . . . . . . . 322.7 The classical Doppler effect . . . . . . . . . . . . . . . . . . . . . 33
2.7.1 Redshift of galaxies and the expansion of the Universe . . 342.7.2 The Vavilov-Cherenkov effect . . . . . . . . . . . . . . . . 37
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2.8 Composition of waves . . . . . . . . . . . . . . . . . . . . . . . . 382.8.1 Boundary conditions and steady waves . . . . . . . . . . . 382.8.2 Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.8.3 Group velocity and phase velocity . . . . . . . . . . . . . 42
2.9 Waves in three dimensions . . . . . . . . . . . . . . . . . . . . . . 442.9.1 Spherical waves . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Electromagnetism and the Maxwell’s equations 463.1 Charge density and current density . . . . . . . . . . . . . . . . . 473.2 Maxwell’s equations in differential form . . . . . . . . . . . . . . 473.3 Maxwell’s equations and continuity equation for charge . . . . . 483.4 The potentials, vector and scalar . . . . . . . . . . . . . . . . . . 493.5 Maxwell’s equations and electrostatics . . . . . . . . . . . . . . . 503.6 Maxwell’s equations and magnetostatics . . . . . . . . . . . . . . 50
4 Solutions of the Maxwell’s equations in vacuo 514.1 Maxwell’s waves and light . . . . . . . . . . . . . . . . . . . . . . 524.2 Properties of the electromagnetic waves . . . . . . . . . . . . . . 54
4.2.1 Energy transported an the electromagnetic wave . . . . . 564.2.2 The Poynting vector . . . . . . . . . . . . . . . . . . . . . 57
4.3 Photoelectric effect; the photon hypothesis . . . . . . . . . . . . . 574.4 The perception of electromagnetic waves: visible light . . . . . . 604.5 Natural units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Geometrical optics 645.1 Light propagation through different materials; transmission of
electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . 645.2 Huygens’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 The laws of reflection and refraction . . . . . . . . . . . . . . . . 65
5.3.1 The Fermat principle . . . . . . . . . . . . . . . . . . . . . 68
6 Interference and diffraction 706.1 Young’s interference experiment . . . . . . . . . . . . . . . . . . . 70
6.1.1 The two-slit case . . . . . . . . . . . . . . . . . . . . . . . 746.2 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.1 Diffraction through a single slit . . . . . . . . . . . . . . . 766.2.2 Diffraction grating . . . . . . . . . . . . . . . . . . . . . . 776.2.3 The diffraction limit . . . . . . . . . . . . . . . . . . . . . 78
7 Maxwell’s equations and Einstein’s special relativity 797.1 Classical electromagnetism is not a consistent theory . . . . . . . 797.2 Galilean transformations, relativity, and the ether . . . . . . . . . 80
7.2.1 Maxwell’s equations and the ether . . . . . . . . . . . . . 817.3 The Michelson-Morley experiment . . . . . . . . . . . . . . . . . 817.4 Einsteins’s postulates and relativity . . . . . . . . . . . . . . . . 83
7.4.1 Relativity of simultaneity . . . . . . . . . . . . . . . . . . 84
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7.4.2 Time dilation . . . . . . . . . . . . . . . . . . . . . . . . . 867.4.3 Length contraction . . . . . . . . . . . . . . . . . . . . . . 87
7.5 Invariance of the interval . . . . . . . . . . . . . . . . . . . . . . . 87
8 Lorentz transformations and the formalism of special relativity 888.1 The Lorentz transformations . . . . . . . . . . . . . . . . . . . . 88
8.1.1 A theorem by von Ignatowski . . . . . . . . . . . . . . . . 888.1.2 Transformation of velocities . . . . . . . . . . . . . . . . . 888.1.3 The relativistic Doppler effect . . . . . . . . . . . . . . . . 88
8.2 4-vectors; covariant and controvariant representation . . . . . . . 908.2.1 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . 928.2.2 Four-dimensional velocity . . . . . . . . . . . . . . . . . . 93
8.3 Spacelike and timelike events; future and past . . . . . . . . . . . 948.4 E=mc2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.5 4-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.6 The photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
9 Covariant formulation of electromagnetism 959.1 Relativistic force . . . . . . . . . . . . . . . . . . . . . . . . . . . 959.2 Electromagnetism and relativity . . . . . . . . . . . . . . . . . . 95
9.2.1 The equations for the potentials . . . . . . . . . . . . . . 959.2.2 The electromagnetic tensor . . . . . . . . . . . . . . . . . 979.2.3 Covariant expression of Maxwell’s equations . . . . . . . . 98
9.3 The transformation of the fields . . . . . . . . . . . . . . . . . . . 989.3.1 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
9.4 Covariant expression of the Lorentz force . . . . . . . . . . . . . 989.4.1 Motion of a particle under a constant force . . . . . . . . 98
9.5 Radiation from an accelerated particle . . . . . . . . . . . . . . . 98
II Wave functions; introduction to quantum physics 99
10 The crisis of classical physics 10010.1 The quantum properties of radiation . . . . . . . . . . . . . . . . 100
10.1.1 The photoelectric effect . . . . . . . . . . . . . . . . . . . 10010.1.2 The Compton effect . . . . . . . . . . . . . . . . . . . . . 10010.1.3 Blackbody radiation . . . . . . . . . . . . . . . . . . . . . 10010.1.4 Pair production . . . . . . . . . . . . . . . . . . . . . . . . 100
10.2 The wave properties of matter . . . . . . . . . . . . . . . . . . . . 10010.2.1 Diffraction of electrons . . . . . . . . . . . . . . . . . . . . 10010.2.2 de Broglie’s wavelength . . . . . . . . . . . . . . . . . . . 100
10.3 Discrete versus continuum phenomena: atomic spectra . . . . . . 100
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11 The Schrodinger equation 10111.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.2 Interpretation of the wavefunction . . . . . . . . . . . . . . . . . 10111.3 The operation of measurement; collapse of the wavefunction . . . 10111.4 Reality, ortodoxy, agnosticism . . . . . . . . . . . . . . . . . . . . 10111.5 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.6 Momentum, and operators . . . . . . . . . . . . . . . . . . . . . . 101
11.6.1 Angular momentum . . . . . . . . . . . . . . . . . . . . . 10111.7 The Hamiltonian operator . . . . . . . . . . . . . . . . . . . . . . 102
11.7.1 * A quantum view of Nother’s theorem . . . . . . . . . . 102
12 Solving the Schrodinger equation in one dimension 10312.1 Decoupling the space and time parts of the equation . . . . . . . 10312.2 Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10312.3 The infinite square well . . . . . . . . . . . . . . . . . . . . . . . 103
12.3.1 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 10312.4 Potential step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10312.5 The finite square well . . . . . . . . . . . . . . . . . . . . . . . . 10312.6 Potential barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
12.6.1 Reflection and Transmission . . . . . . . . . . . . . . . . . 10312.6.2 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
12.7 * The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 103
13 Central potentials and the Hydrogen atom 10413.1 The Schrodinger equation in 3 dimensions . . . . . . . . . . . . . 10413.2 Spherically symmetric potentials . . . . . . . . . . . . . . . . . . 10413.3 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . 10713.4 The angular part . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.4.1 Angular momentum . . . . . . . . . . . . . . . . . . . . . 10713.5 Radial probability density . . . . . . . . . . . . . . . . . . . . . . 10813.6 The H atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
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Chapter 1
Differential calculus appliedto vector fields
1.1 Scalar and vector fields
The concept of field formalizes physical properties of space. To each point ofspace (let us start with the usual 3-dimensional space) a physical quantity isassociated. Mathematically, a physical field is a function whose domain is space.
The simplest possible physical field is a scalar field. By a scalar field wemean a field which is characterized at each point by a single number – a scalar.Of course the number may change in time; we shall talk for now about whatthe field looks like at a given instant.
As an example of a scalar field, let us consider a solid block of material whichhas been heated at some places, so that the temperature of the body variesfrom point to point. Then the temperature will be a function of x, y, and z, theposition in space measured in a rectangular coordinate system. TemperatureT (x, y, z) is a scalar field.
One way of picturing scalar fields is to draw contours, which are imaginarysurfaces through points for which the field has the same value, just as contourlines on a map connect points with the same height. For a temperature fieldthe contours are called “isothermal surfaces” or isotherms. Figure 8.1 illustratesa temperature field and shows the dependence of T on x and y when z = 0.Several isotherms are drawn.
There are also vector fields. In this case, a vector is given for each point in(a region of) space. As an example, consider a rotating body: the velocity ofthe material of the body at any point is a vector which is a function of position.As a second example, consider the flow of heat in a block of material. If thetemperature in the block is high at one piece and low at another, there will be aflow of heat from the hotter places to the colder ones. The heat will be flowingin different directions in different parts of the block.
The heat flow is a directional quantity which we call ~h. Its magnitude is
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Figure 1.1: The temperature field; isotherms.
a measure of how much heat is flowing, and can be defined as the amountof thermal energy that passes, per unit time and per unit area, through aninfinitesimal surface element, perpendicular to the direction of flow. The vectorpoints in the direction of flow. In symbols: if ∆J is the thermal energy thatpasses per unit time through the surface element ∆a, then
~h = lim∆a→0
∆J
∆a~ef (1.1)
where ~ef is the versor of the heat flow. Examples of the heat flow vector arealso shown in Figure 8.1.
1.2 The gradient
For a real-valued function T (x, y, z) on <3, the gradient∇T (x, y, z) (or ~∇T (x, y, z))is a function on <3, that is, its value at a point (x, y, z) is the triple
∇T (x, y, z) =
(∂T
∂x,∂T
∂y,∂T
∂z
)=∂T
∂xi +
∂T
∂yj +
∂T
∂zk
in <3, where each of the partial derivatives is evaluated at the point (x, y, z).One can think of the symbol ∇ as being “applied” to a real-valued function Tto produce a 3-dimensional function ∇T .
Is ∇T (x, y, z) a vector? Of course it is not generally true that any threenumbers form a vector. It is true only if, when we rotate the coordinate system,
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the components of the vector transform among themselves in the correct wayfor a vector. So it is necessary to analyze how these derivatives are changed bya rotation of the coordinate system. We shall show that ∇T (x, y, z) is indeeda vector: the derivatives do transform in the correct way when the coordinatesystem is rotated. We can observe that, by the total differential theorem,
dT =∂T
∂xdx+
∂T
∂ydy +
∂T
∂zdz (1.2)
for any d~r = (dx, dy, dz). And since df is a scalar and d~r is a vector, ∇T (we
can now call it ~∇T ) must be a vector (of course a demonstration based on thetransformations of coordinates is also possible, but ennoying).
Is ~∇ a vector? Strictly speaking, no, since ∂∂x , ∂
∂y and ∂∂z are not numbers.
But it helps to think of it as a vector, as we shall see. The process of “applying”∂∂x , ∂
∂y , ∂∂z to a real-valued function T (x, y, z) can be thought of as multiplying
the quantities:(∂
∂x
)(T ) =
∂T
∂x,
(∂
∂y
)(T ) =
∂T
∂y,
(∂
∂z
)(T ) =
∂T
∂z
For this reason, ~∇ is often referred to as the “del operator”, since it “operates”on functions.
We can write the expression (1.2) as
dT = (~∇f) · (d~r) . (1.3)
~∇T represents the spatial rate of change of T . The x−component of ~∇T showshow fast T changes in the x−direction, and in the same way for the othercoordinates. What is the direction of the vector ~∇T? Equation (1.3) shows
that the rate of change of T in any direction is the component of ~∇T in thatdirection. It follows that the direction of ~∇T is that in which it has the largestpossible component- in other words, the direction in which T changes the fastest.The gradient of T has the direction of the steepest uphill slope (in T ). ~∇T isperpendicular to the isotherms.
What would it mean for the gradient to vanish? If ~∇T = 0 at (x, y, z),then dT = 0 for small displacements about the point (x, y, z). This is, then, astationary point of the function T (x, y, z). It could be a maximum (a summit),a minimum (a valley), a saddle (a pass).
1.3 The divergence
What if we make the ~∇ operator to operate on a vector field (a function of <3
into <3) instead than on a scalar field? In this case we when two possible kindsof products, the scalar product and the vector product; we shall see that bothcorrespond to interesting functions.
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Figure 1.2: Divergence.
Giving a vector function ~F defined in a domain in <3, we define as thedivergence of ~F the formal scalar product ~∇ and ~F :
~∇ · ~F =
(∂Fx∂x
+∂Fy∂y
+∂Fz∂z
), (1.4)
and it is clearly a scalar.What is its geometrical interpretation? ~∇ · ~F is a measure of how much the
vector ~F spreads out (diverges) from a point. For example, the vector functionin Figure 1.2(a) has a positive divergence (if the arrows pointed in, it would bea large negative divergence), the function in Figure 1.2(b) has zero divergence,and the function in Figure 1.2(c) again has a positive divergence.
Exercise: if the function pictured in Figure 1.2(a) is ~F(a) = x~i+ y~j, and the
function pictured in Figure 1.2(b) is ~F(b) = ~j, compute their divergence.
1.4 The curl
In a similar way, we can define the vector product of ~∇ times a vector field ~F .This is called the curl of ~F :
~∇× ~F =~ux ~uy ~uz∂/∂x ∂/∂y ∂/∂zFx Fy Fz
. (1.5)
This can be demonstrated to be a vector function.
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Figure 1.3: Curl.
Geometrical interpretation: ~∇× ~F is a measure of how much the vector ~F“curls around” the point in question. Thus the three functions in Figure 1.2 allwhen zero curl (as you can easily check for yourself), whereas the functions inFigure 1.3 when a nonzero curl, pointing in the z−direction, as the right-handrule would suggest. A whirlpool would be a region of large curl.
Exercise: if the function pictured in Figure 1.3(a) is ~F(a) = −y~i + x~j, and
the function pictured in Figure 1.3(b) is ~F(b) = x~j, compute their curl.
1.5 Second order derivatives, and the Laplacian
So far we when had only first derivatives. Why not second derivatives? We canwrite several combinations:
1. ~∇× (~∇T )
2. ~∇ · (~∇× ~F )
3. ~∇ · (~∇T )
4. ~∇(~∇ · ~F )
5. ~∇× (~∇× ~F )
You can check that these are all the legal combinations.
1. and 2. ~∇× (~∇T ) and ~∇ · (~∇× ~F ). The first two are identically zero. Letus check it for the first one: one has, by the Schwartz’s lemma:
~~∇× ~∇T =~ux ~uy ~uz∂/∂x ∂/∂y ∂/∂z∂T/∂x ∂T/∂y ∂T/∂z
= 0.
We when thus demonstrated that the curl of a gradient is zero, which is easyto remember because of the way the vectors work. It can be demonstrated that
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if the curl of a field ~F is zero, then this field is always the gradient of a scalarfield: there is some scalar field φ such that ~F = ~∇φ.
Also ~∇ · (~∇ × ~F ) = 0, and there is a similar theorem stating that if the
divergence of ~F is zero, ~F is the curl of some vector field ~A.
3. ~∇ · (~∇T ). Let us examine the third expression now. For a real-valuedfunction f(x, y, z), the Laplacian1 of T , denoted by ∇2T , is defined as
∇2T (x, y, z) = ~∇ · (~∇T ) =∂2T
∂x2+∂2T
∂y2+∂2T
∂z2. (1.6)
Sometimes the notation ∆T is used instead.The Laplacian can be thought as a scalar operator
∇2 =
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)and as such it can be applied to vector functions as well, giving as output avector:
∇2 ~A(x, y, z) =(∇2Ax(x, y, z),∇2Ay(x, y, z),∇2Az(x, y, z)
).
The explicit calculation confirms that this approach is justified.
4. ~∇(~∇· ~F ). It is a possible vector field, which may occasionally come up (forexample, see next point).
5. ~∇× (~∇× ~F ). Let us compare this expression with the vector identity
~A× ( ~B × ~C) = ~B( ~A · ~C)− ( ~A · ~B)~C .
In order to use this formula, we should replace ~A and ~B by the operator ~∇ andput ~C = ~F . If we do that, we get
~∇× (~∇× ~F ) = ~∇(~∇ · ~F )−∇2 ~F . (1.7)
1.6 The differential equation of heat flow
Let us give an example of a law of physics written in vector notation. For heatconductors, the energy flows through the material from a surface at temperatureT2 to a surface at temperature T2 < T1. The total energy flow is proportionalto the area A of the faces, and to the temperature difference; it is also inversely
1Pierre-Simon de Laplace (1749 - 1827) was a French mathematician and astronomer whosework was pivotal to the development of mathematical astronomy and statistics. He was oneof the first scientists to postulate the existence of black holes and the notion of gravitationalcollapse.
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proportional to d, the distance between the plates (for a given temperaturedifference, the thinner the slab the greater the heat flow). Letting J be thethermal energy that passes per unit time through the slab, we write
J = k(T2 − T1)A/d .
The constant of proportionality k is called the thermal conductivity of the ma-terial.
For a small slab of area ∆a the heat flow per unit time is
∆J = k∆T∆a
∆d(1.8)
where ∆d is the thickness of the slab. ∆J/∆a is what we when defined earlier
as the magnitude of the vector ~h, whose direction is the heat flow. The heatflow will be perpendicular to the isotherms. Also, ∆T/∆d is just the rate ofchange of T with position. And since the position change is perpendicular to theisotherms, ∆T/∆d is the maximum rate of change. It is, therefore, in the limit
for ∆d→ 0, just the magnitude of ~∇T. Since the direction of ~∇T is apposite tothat of ~h, we can write the previous equation as a vector equation:
~h = −k ~∇T
(the minus sign is necessary because heat flows “downhill” in temperature.)This is the differential equation of heat conduction.
1.7 Differential operators in polar coordinates
Often – for example, in the case of central symmetries – it is convenient touse radial coordinate systems when dealing with quantities such as the gradi-ent, divergence, curl and Laplacian. We will present the expressions for theseoperators in spherical coordinates.
A point (x, y, z) can be represented in spherical (polar) coordinates (r, θ, φ),where x = r sin θ cosφ, y = r sin θ sinφ, z = r cos θ. θ (the angle down from thez axis) is called the polar angle, and φ (the angle around from the x axis) is theazimuthal angle.
At each point (r, θ, φ), ~er, ~eθ, ~eφ are unit vectors in the direction of increasingr, θ, φ, respectively (see Figure 1.4). Then the vectors ~er, ~eθ, ~eφ are orthonormal.By the right-hand rule, we see that ~eθ × ~eφ = ~er.
We can summarize the expressions for the gradient and the Laplacian appliedto a scalar field T and to a vector field ~F in spherical coordinates in the followingequations:
~∇T =∂T
∂r~er +
1
r
∂T
∂θ~eθ +
1
r sin θ
∂T
∂φ~eφ (1.9)
∇2T =1
r2
∂
∂r
(r2 ∂T
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂T
∂θ
)+
1
r2 sin2 θ
∂2T
∂φ2. (1.10)
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Figure 1.4: Spherical coordinates.
Figure 1.5: Line integral.
The derivation of the above formulas is conceptually straightforward butboring. The basic idea is to take the Cartesian equivalent of the quantity inquestion and to substitute into that formula using the appropriate coordinatetransformation.
1.8 The gradient theorem
We found previously that there were various ways of taking derivatives of fields.Some gave vector fields; some gave scalar fields. Although we developed manydifferent formulas, everything could be summarized in one rule: the operators∂/∂x, ∂/∂y, ∂/∂z are the three components of a vector operator ~∇. We wouldnow like to get some understanding of the significance of the derivatives of fields.We shall then when a better feeling for what a vector field equation means.
We when already discussed the meaning of the gradient operation (appli-
cation of ~∇ on a scalar). We take up now an integral formula involving the
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Figure 1.6: Line integral and gradient theorem.
gradient. The relation contains a very simple idea: since the gradient repre-sents the rate of change of a field quantity, if we integrate that rate of change,we should get the total change (like in the fundamental theorem of calculus).
Let us first introduce the concept of line integral for a vector field. The lineintegral from a point a to a point b of the curve is nothing but the integral ofthe dot product of the value of the function times the line element (i.e., at eachpoint the value of the function is weighted by the cosine of the angle formed bythe vector itself and by the tangent to the line):∫ b
a
~E(x, y, z) · d~l
(see Figure 1.5). For example, the work done by the force ~F from a to b along
a given path is the line integral of ~F along that path.Suppose we have a scalar function of three variables T (x, y, z). Starting at
point a, we move by a small distance d~l1 (Figure 1.6). The function T willchange by an amount
dT = ~∇T · d~l1 .
Now we move a little further, by an additional small displacement d~l2; theincremental change in T will be ~∇T ·d~l2. In this way, proceeding by infinitesimalsteps, we make a journey to point b. At each step we compute the gradient of T(at that point) and dot it into the displacement d~l: this gives us the change inT. Evidently the total change in T in going from a to b along the path selectedis ∫ b
a
~∇T · d~l = T (~b)− T (~a) .
This is called the fundamental theorem for gradients; like the “ordinary” fun-damental theorem of calculus, it says that the integral (here a line integral) ofa derivative (here the gradient) is given by the difference between the values ofthe function at the boundaries (a and b).
13
A geometrical interpretation will make use of an example. Suppose youwanted to determine the height of the Eiffel Tower. You could climb the stairs,using a ruler to measure the rise at each step, and adding them all up, or youcould place altimeters at the top and the bottom, and subtract the two readings;you should get the same answer either way.
Line integrals ordinarily depend on the path taken from a to b. But the rightside of in the gradient theorem makes no reference to the path - only to the endpoints. Evidently, gradients when the special property that their line integralsare path independent:
Corollary 1:∫ ba
(~∇T ) · d~l is independent of the path taken from a to b.
Corollary 2:∮
(~∇T ) ·d~l = 0, since the beginning and end points are identical,and hence T (b)− T (a) = 0.
1.9 The divergence theorem (Gauss’ theorem)
1.9.1 Flux of a vector field
A surface integral is an expression of the form∫S
~h · ~n da
where ~h is a vector function, and da is an infinitesimal patch of area; ~n is aunit vector with direction perpendicular to the surface. There are, of course,two directions perpendicular to any surface, so the sign of a surface integralis intrinsically ambiguous. If the surface is closed (forming a “balloon”), onefrequently puts a circle on the integral sign∮
S
~h · ~n da ;
in this case tradition dictates that “outward” is positive, but for open surfacesit is, again, arbitrary.
We will identify sometimes a flat surface with a vector perpendicular to thesurface itself, and with intensity equal to the area of the surface. The expressionof the surface integral becomes then∫
S
~h · d~a .
If ~h describes the flow of a fluid (mass per unit area per unit time), then∫S~h · d~a represents the total mass per unit time passing through the surface -
hence the alternative name, flux Φ. In the case of heat flow, we may think: ~h isthe “current density” of heat flow and the surface integral of it is the total heatcurrent directed out of the surface: that is, the thermal energy per unit time(joule per second).
14
Figure 1.7: The closed surface S defines the volume V. The unit vector ~n is theoutward normal to the surface element da, and ~h is the heat-flow vector at the
surface element.
We can generalize this idea to the case to any vector field; for instance, itmight be the electric field. We can certainly still integrate the normal componentof the electric field over an area if we wish. Although it does not appear to bethe flow of anything, we still call it the “flux”. We say flux of ~E through thesurface S the quantity
Φ =
∫S
~E · d~a .
We thus generalize the word “flux” to mean the “surface integral of the normalcomponent” of a vector.
1.9.2 Flux through a closed surface
We defined the vector ~h, which represents the heat through a unit area in aunit time. Suppose that inside a block of material we have some closed surfaceS which encloses the volume V. We would like to find out how much heat isflowing out of this volume. We can, of course, find it by calculating the totalheat flow out of the surface S (Figure 1.7).
We write da for the area of an element of the surface. The symbol stands fora two-dimensional differential. If, for instance, the area happened to be in thexy−plane, we would have da = dxdy (later we shall have integrals over volumeand for these it is convenient to consider a differential volume that is a littlecube; so when we write dV we mean dV = dxdydz)2.
The heat flow out through the surface element da is the area times thecomponent of ~h perpendicular to da. Returning to the special case of heat flow,let us take a situation in which heat is conserved. For example, imagine some
2Some texts write d2a instead of da to remind that it is kind of a second-order quantity.They also write d3V instead of dV. We will use the simpler notation, and assume that youremember that an area has two dimensions and a volume has three.
15
Figure 1.8: A volume V contained inside the surface S is divided into twopieces by a “cut” at the surface Sab. We now when the volume V1 enclosed in
the surface S1 = Sa + Sab and the volume V2 enclosed in the surfaceS2 = Sb + Sab.
material in which after an initial heating no further heat energy is generatedor absorbed. Then, if there is a net heat flow out of a closed surface, the heatcontent of the volume inside must decrease. So, in circumstances in which heatwould be conserved, we say that∮
~h · d~a = −dQdt
, (1.11)
where Q is the heat inside the surface.We shall point out an interesting fact about the flux of any vector. Imagine
that we have a closed surface S that encloses the volume V. We now separatethe volume into two parts by some kind of a “cut”, as in Figure 1.8. Now wehave two closed surfaces and volumes: the volume V1 is enclosed in the surfaceS1, which is made up of part of the original surface Sa and of the surface of thecut, Sab; the volume V2 is enclosed by S2, which is made up of the rest of theoriginal surface S, and closed off by the cut Sab.
The sum of the fluxes through S1 and S2 equals the flux through the wholesurface that we started with. The flux through the part of the surfaces Sabcommon to both S1 and S2 just exactly cancels out. For the flux of a genericvector ~C out of V1, we can write
ΦS1 =
∫Sa
~C · ~n da+
∫Sab
~C · ~n1da
and for the flux out of V2
ΦS2 =
∫Sb
~C · ~n da+
∫Sab
~C · ~n2da .
Since ~n1 = −~n2, the sum of the fluxes through S1 and S2 is just the sum of twointegrals which, taken together, give the flux through the original surface S.
16
Figure 1.9: Computation of the flux of ~C out of a small cube.
We can similarly subdivide again the volume - say by cutting V1 into twopieces. You see that the same arguments apply. So for any way of dividingthe original volume, it is generally true that the flux through the outer surface,which is the original integral, is equal to a sum of the fluxes out of all the littleinterior pieces. Then we can divide it by a large number of small cubes.
1.9.3 The flux from a cube
We now take the special case of a small cube (or parallelepiped) and find aninteresting formula for the flux out of it.
Consider a cube whose edges are lined up with the axes (Figure 1.9). Letus suppose that the coordinates of the corner nearest the origin are x, y, z.Let ∆x be the length of the cube in the x−direction, ∆y be the length in they−direction, and ∆z be the length in the z−direction (they are small quantities).
We wish to find the flux of a vector field ~C through the surface of the cube. Weshall do this by making a sum of the fluxes through each of the six faces.
First, consider the face marked 1 in the figure. The flux outward on thisface is the negative of the x−component of ~C, integrated over the area of theface. This flux is approximately
Φ1 = −Cx(1) ∆y∆z .
(since we are considering a small cube, we approximate the integral by the valueof Cx at the center of the face - which we call the point (1) - multiplied by the
17
area of the face). Similarly, the flux out of face 2 is approximately
Φ2 = Cx(2) ∆y∆z .
Cx(1) and Cx(2) are, in general, slightly different. If ∆x is small enough, wecan write
Cx(2) = Cx(1) +∂Cx∂x
∆x
(there are, of course, higher order terms, but we are considering the limit for∆x→ 0). So the flux through faces 1 and 2 is
Φ1 + Φ2 =∂Cx∂x
∆x∆y∆z .
The derivative should really be evaluated at the center of face 1; that is, at(x, y + ∆y/2, z + ∆z/2). But in the limit of an infinitesimal cube, we make anegligible error if we evaluate it at the corner (x, y, z).
Applying the same reasoning to each of the other pairs of faces, we find
Φ3 + Φ4 =∂Cy∂y
∆x∆y∆z ,
and
Φ5 + Φ6 =∂Cz∂z
∆x∆y∆z .
The total flux through all the faces is the sum of these terms. We find that∫cube
~C · d~a =
(∂Cx∂x
+∂Cy∂y
+∂Cz∂z
)∆x∆y∆z
and so we can say that for an infinitesimal cube
dΦ = (~∇ · ~C)dV .
We have shown that the outward flux from the surface of an infinitesimalcube is equal to the divergence of the vector multiplied by the volume of thecube. We now see the meaning of the divergence of a vector. The divergence ofa vector at the point P is the flux - the outgoing “flow” of ~C per unit volume,in the neighborhood of a point P .
We have connected the divergence to the flux out of an infinitesimal volume.For any finite volume we can use the fact we proved above - that the totalflux from a volume is the sum of the fluxes out of each part. We can, that is,integrate the divergence over the entire volume. This gives us the theorem thatthe integral of the normal component of any vector over any closed surface canalso be written as the integral of the divergence of the vector over the volumeenclosed by the surface. This theorem is named after Gauss3.∮
S
~C · d~a =
∫V
(~∇ · ~C) dV
3Carl Friedrich Gauss (Brunswick, 1777 - Gottingen, 1855) was a German mathematician,generally regarded as one of the greatest mathematicians of all time for his contributionsto number theory, geometry, probability theory, geodesy, planetary astronomy, the theory offunctions, and potential theory (including electromagnetism). Gauss was the only child ofpoor parents. He was rare among mathematicians in that he was a calculating prodigy.
18
Figure 1.10: Left: the circulation of ~C around the curve Γ. Right: Thecirculation around the whole loop is the sum of the circulations around the
two loops Γ1 and Γ2.
where S is any closed surface and V is the volume inside it.
1.9.4 Heat conduction; the diffusion equation
By the Gauss’ theorem, equation (1.11),∮~h · d~a = −dQdt , becomes, if we call q
the amount of heat per unit volume,
− d
dt(q∆V ) = (~∇ · ~h)∆V
and thus we can transform the integral equation (1.11) into a differential equa-tion defined locally, i.e., point by point:
−dqdt
= ~∇ · ~h .
This kind of law appears frequently in physics: it is a conservation law, or acontinuity equation.
1.10 The curl theorem (Stokes’ theorem)
1.10.1 Circulation of a vector field
The circulation of a vector field is the line integral around a closed loop.Playing the same kind of game we did with the flux, we can show that the
circulation around a loop is the sum of the circulations around two partial loops.Suppose we break up our curve of Figure 1.10(left) into two loops, by joiningtwo points (1) and (2) on the original curve by some line that cuts across asshown in Figure 1.10(right).
There are now two loops, Γ1 and Γ2; Γ1 is made up of Γa, which is that partof the original curve to the left of (1) and (2), plus Γab, the “short cut”; Γ2 ismade up of the rest of the original curve plus the short cut.
19
Figure 1.11: Some surface bounded by the loop Γ is chosen. The surface isdivided into a number of small areas, each approximately a square. The
circulation around Γ is the sum of the circulations around the little loops.
The integral along Γab will have, for the curve Γ2, the opposite sign comparedto Γ1 because the direction of travel is opposite - we must take both our lineintegrals with the same “sense” of rotation. Following the same kind of argumentwe used before, you can see that the sum of the two circulations will give justthe line integral around the original curve Γ: the parts due to Γab cancel. Thecirculation around the one part plus the circulation around the second partequals the circulation about the outer line.
We can continue the process of cutting the original loop into any numberof smaller loops. When we add the circulations of the smaller loops, there isalways a cancellation of the parts on their adjacent portions, so that the sum isequivalent to the circulation around the original single loop.
Now let us suppose that the original loop is the boundary of some surface.There are, of course, an infinite number of surfaces which all have the originalloop as the boundary. Our results will not, however, depend on which surfacewe choose. First, we break our original loop into a number of small loops thatall lie on the surface we have chosen. No matter what the shape of the surface,if we choose our small loops small enough, we can assume that each of the smallloops will enclose an area which is essentially flat. Also, we can choose our smallloops so that each is very nearly a square (Figure 1.11). Now we can calculatethe circulation around the big loop by finding the circulations around all of thelittle squares and then taking their sum.
1.10.2 The circulation around a square
How shall we find the circulation for each little square? One question is, how isthe square oriented in space? We could easily make the calculation if it had aspecial orientation. For example, if it were in one of the coordinate planes. Sincewe have not assumed anything as yet about the orientation of the coordinateaxes, we can just as well choose the axes so that the one little square we areconcentrating on at the moment lies in the xy−plane, as in Figure 1.12. If ourresult is expressed in vector notation, we can say that it will be the same no
20
Figure 1.12: Computing the circulation of C around a small square.
matter what the particular orientation of the plane.We want now to find the circulation of the field ~C around our little square. It
will be easy to do the line integral if we make the square small enough that thevector ~C does not change much along any one side of the square (the assumptionis better the smaller the square, so we are really talking about infinitesimalsquares).
Starting at the point (x, y)−the lower left corner of the figure - we go aroundin the direction indicated by the arrows. Along the first side - marked (1) - thetangential component is Cx(1) and the distance is ∆x. The first part of theintegral is Cx(1)∆x. Along the second leg, we get Cy(2)∆y. Along the third, weget −Cx(3)∆x, and along the fourth, −Cy(4)∆y. The minus signs are requiredbecause we want the tangential component in the direction of travel. The wholeline integral is then∮
~C · d~l = (Cx(1)− Cx(3))∆x+ (Cy(2)− Cy(4))∆y .
Since
Cx(3) ' Cx(1) +∂Cx∂y
∆y
and
Cy(4) ' Cy(2) +∂Cy∂x
∆x
one can write, at first order,∮~C · d~l =
(∂Cy∂x− ∂Cx
∂y
)∆x∆y .
21
Neglecting second order terms, the derivative can be evaluated at (x, y). Theabove expression can be written in vector form:∮
~C · d~l =(~∇× ~C
)·∆~a .
The circulation around any loop Γ can now be easily related to the curl of thevector field. We fill in the loop with any convenient surface S, as in Figure 1.11,and add the circulations around a set of infinitesimal squares in this surface;the sum can be written as an integral.
Our result is a very useful theorem called Stokes’ theorem4∮Γ
~C · d~l =
∫S
(~∇× ~C) · d~a (1.12)
where S is any surface bounded by Γ.We must now speak about a convention of signs. The z−axis in Figure 1.12
would point toward the reader in a “usual”-that is, “right-handed”-system ofaxes. When we took our line integral with a “positive” sense of rotation, wefound that the circulation was equal to the z−component of ~∇× ~C. If we hadgone around the other way, we would have gotten the apposite sign. Now howshall we know, in general, what direction to choose for the positive directionof the “normal” component of ~∇× ~C? The “positive” normal must always berelated to the sense of rotation by the “right-hand rule”.
1.11 Curl-free fields
We would like, now, to consider some consequences of our new theorems. Takefirst the case of a vector whose curl is everywhere zero. Then Stokes’ theoremsays that the circulation around any loop is zero. Now if we choose two points(1) and (2) on a closed curve, it follows that the line integral of the tangentialcomponent from (1) to (2) is independent of which of the two possible paths istaken. We can conclude that the integral from (1) to (2) can depend only onthe location of these points - that is to say, it is some function of position only.The same logic was used where we proved that if the integral around a closedloop of some quantity is always zero, then that integral can be represented asthe difference of a function of the position of the two ends. This fact allowedus to invent the idea of a potential. We proved, furthermore, that the vectorfield was the gradient of this potential function. It follows that any vector fieldwhose curl is zero is equal to the gradient of some scalar function. That is, if~∇× ~C = 0 everywhere, there is some scalar field ψ for which ~C = ~∇ψ. We can,if we wish, describe this special kind of vector field by means of a scalar field(this shows how restrictive is the class of the conservative fields).
4Sir George Gabriel Stokes (1819 - 1903) was a mathematician and physicist. Born inIreland, he spent all of his career at University of Cambridge, where he served as the LucasianProfessor of Mathematics from 1849 until his death. Stokes made seminal contributions tofluid dynamics (including the Navier-Stokes equations), optics, and mathematical physics.
22
Let’s show something else. Suppose we have any scalar field φ. If we take itsgradient, ~∇φ, the integral of this vector around any closed loop must be zero:∮
loop
~∇φ · d~l = 0 .
Using Stokes’ theorem, we conclude that∫surface
(~∇× (~∇φ)
)· d~a = 0
over any surface; but this means that the integrand is zero:
~∇× (~∇φ) = 0 .
We obtained in a different way a result we had obtained with vector algebra.We stated, without demonstration, that a field ~C with zero curl can be
expressed as the gradient of a scaler field. Now, thanks to the Stokes’ theorem,we can demonstrate it. If the curl is zero, then the circulation along whateverloop is zero; thus the line integral from a point A to a point B is independentof the path. I can thus define, choosing arbitrarily a point A,
ϕ(B) =
∫ B
A
~C · d~l
and thus~C = ~∇ϕ .
Note that I can choose arbitrarily the point A; this reflects on the fact that,adding an arbitrary constant to ϕ, the above relation is still valid.
1.12 Dirac’s delta function
The Dirac5 delta function is used to define fields which are nonzero in a regionof measure zero, but for which the integral over space is different from zero.
In one dimension, it can be defined as a real function δ(x) being zero in allpoints apart from the origin, with the constraint∫ ∞
−∞δ(x) dx = 1
(i.e., its value must be infinite at the origin, see Figure 1.13) for a sketch.
5Paul Dirac (1902 -1984) was an English theoretical physicist who made fundamental con-tributions to the early development of both quantum mechanics and quantum electrodynamics.Among other discoveries, he formulated the Dirac equation, which describes the behaviour offermions, and predicted the existence of antimatter. Dirac shared the Nobel Prize in Physicsfor 1933 with Erwin Schrodinger, for his contributions to Quantum Mechanics.
23
Figure 1.13: A sketch of the function δ(x).
In the mathematical literature it is known as a generalized function, ordistribution. It can be obtained as the limit of a class of legitimate functions,for example, of the Gaussian functions:
δ(x) = limσ→0
1
σ√
2πe−
x2
2σ2 .
Integrals including δ(x) are perfectly legitimate, and in particular:∫ ∞−∞
δ(x− a)f(x) dx = f(a) .
It is easy to generalize the delta function to three dimensions:
δ3(~r) = δ(x)δ(y)δ(z) .
This three-dimensional delta function is zero everywhere except at (0, 0, 0),where it blows up. Its volume integral is 1.
As in the one-dimensional case, integration with δ3(~r) picks out the value ofa function f at the location of the spike (if the spike is included in the domainof integration): ∫
all space
δ(~r − ~a)f(~r) dV = f(~a) .
24
Part I
Waves andelectromagnetism
25
Chapter 2
Waves
Until 1600, scientists were dreaming to give a complete description of naturethrough the reduction to a set of elementary particles. An elementary particle isideally a point-like structure; the density of its charges (mass, electrical charge,...) can be represented by a Dirac’s delta function. Isaac Newton1 in his studiesrelated to optics used only the concept of particle to treat light’s propagation.
In the XVII century, however, some particular phenomena were observed,which could not be described by such model: their description needed the dis-placement of a delocalized perturbation. Phenomena like destructive interfer-ence (the sum of two perturbations could cancel) needed an extension of theparticle model. In order to explain such phenomena, the concept of wave wasformally introduced in 1600 in the Netherlands, where people had a long tradi-tion in the field of optics. Christiaan Huygens2 was one of the main actors ofthis intellectual revolution.
The wave is a model that describes the general propagation of a perturbation.Typical examples of phenomena usually described as waves are:
• sound waves;
• elastic waves (local deformations);
1Sir Isaac Newton (1642 - 1727) was an English physicist, mathematician, astronomer,natural philosopher, alchemist and theologian, who has been considered by many to be thegreatest and most influential scientist who ever lived. His monograph Philosophiae NaturalisPrincipia Mathematica, published in 1687, laid the foundations for most of classical mechanics.Newton built the first practical reflecting telescope and developed a theory of colour based onthe observation that a prism decomposes white light into the many colours that form the visiblespectrum. He also formulated an empirical law of cooling and studied the speed of sound. Inmathematics, Newton shares the credit with Leibniz for the development of differential andintegral calculus. Newton was also deeply involved in occult studies and interpretations ofreligion.
2Christiaan Huygens (1629 - 1695) was a prominent Dutch mathematician, astronomer,physicist and horologist. His work included early telescopic studies elucidating the nature ofthe rings of Saturn and the discovery of its moon Titan, the invention of the pendulum clockand other investigations in timekeeping, and studies of both optics and the centrifugal force.Huygens achieved note for his argument that light consists of waves.
26
Figure 2.1: A perturbation moving right with speed c.
• waves on the water surface (gravity waves);
• electromagnetic waves.
The representation of a generic wave is a function ξ(x, y, z, t).
2.1 Translations and the wave equation
If∂ξ/∂y = ∂ξ/∂z = 0 ,
then ξ(x, t) is called a plane wave; a function satisfying these conditions mightpropagate along the x axis only.
Let us consider the function of the space coordinate ξ(x, 0) at a fixed timet = 0 (we set it arbitrarily as the origin of time). Suppose that we want that aftera time t the same space function is displaced by ∆x = vt (i.e., the perturbationrepresented by that function is moving in the positive direction of the x axis ata speed v). The new function will be ξ(x− vt).
In the same way, the functional form ξ(x+ vt) will represent a perturbationmoving left with speed v.
If we set u± = (x± vt), then:∂ξ
∂x=
dξ
du±
∂u±∂x
=dξ
du±∂ξ
∂t=
dξ
du±
∂u±∂t
= ±v dξ
du±
27
Figure 2.2: Perturbation along a string.
∂2ξ
∂x2=
d
du±
(dξ
du±
)∂u±∂x
=d2ξ
du2±
∂2ξ
∂t2=
d
du±
(dξ
du±
)∂u±∂t
(±v) = v2 d2ξ
du2±
⇒ ∂2ξ
∂x2=
1
v2
∂2ξ
∂t2(2.1)
This is called the d’Alembert’s3 equation, or wave equation, in 1 dimension. Aplane wave is defined as a function satisfying the equation (2.1).
Since the wave equation is linear, the sum of two or more waves is still awave. The general solution ξ(x, t), however, is not in general a function thatmoves left or right, but a linear combination of tho functions ξ+ and ξ− movingright and left respectively:
ξ(x, t) = ξ+(x− vt) + ξ−(x+ vt) . (2.2)
2.1.1 An example: mechanical waves in one dimension
Let us consider a string of length L with a tension T . In equilibrium the stringis straight. Suppose that the string is perturbed from its rest state; a section ismoved by a small distance compared to its length, perpendicularly to the stringitself.
Let AB be a piece of the string of length dx and mass dm, witch is movedto a distance ξ from the equilibrium state. We have a force on each extreme ofthe element AB. Because of the bending of the string these two forces have thesame intensity but not the same direction. Analyzing the forces acting on theelement AB, we have:
Fx = T (cosα′ − cosα) = dmax
Fξ = T (sinα′ − sinα) = dmaξ.(2.3)
3Jean-Baptiste d’Alembert (1717 - 1783) was a French mathematician, physicist, philoso-
pher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopdie.
28
Because of the little bending, the two angles α and α′ are small, so we can write:
sinα′ − sinα ' α′ − α ' tanα′ − tanα
cosα′ − cosα ' 0 ,(2.4)
and then:
Fx = dmax ' 0
Fξ = dmaξ ' T (tanα′ − tanα).(2.5)
We see that the horizontal net force is very small compared to the verticalone. We can associate tanα to the slope of the string, i.e., to the derivativewith respect to x: ∂ξ/∂x, so we have:
Fξ = T∂
∂x
( ∂ξ∂x
)dx = T
∂2ξ
∂x2dx . (2.6)
Let µ = M/L be the linear density of the string. We have that the massof element AB is µdx; moreover the vertical acceleration is ∂2ξ/∂t2. By thesecond law of motion we can write:
µdx∂2ξ
∂t2= T
∂2ξ
∂x2dx. (2.7)
Thus,
∂2ξ
∂t2=T
µ
∂2ξ
∂x2. (2.8)
As we can see this is the d’Alembert wave equation so we now know that
the perturbabion moves through the string with velocity v =√
Tµ .
2.2 Energy transported by a wave
A wave transfers a perturbation, and thus, ultimately, energy. If u is the densityof energy per unit volume associated to the perturbation, the the intensity oftransmitted energy U per units of area and time is:
I =dU
dSdt=
dU
dSdx
dx
dt= uv, (2.9)
where I is called intensity and is measured in Wm−2. In a sinusoidal mechan-ical wave ξ0 sin(kx − ωt) (v = ω/k) a fixed point describes a simple harmonicoscillation, thus:
u =1
2kξ2
0 ∝1
2ω2ξ2
0 ⇒ I ∝ 1
2ω2ξ2
0v . (2.10)
29
2.3 Sinusoidal waves and the Fourier theorem
A periodical phenomenon is something that repeats itself equally on regularperiods of time. More precisely in mathematics a function f(t) is said to beperiodical of period T if:
f(t+ nT ) = f(t), ∀n ∈ N and ∀t. (2.11)
Well known examples of periodical functions are the trigonometric functions sinand cos.
Sinusoidal (or, which is equivalent, cosinusoidal) waves are very importantbecause of the Fourier theorem.
Figure 2.3: Example of Fourier decomposition.
The Fourier theorem states that every periodical function f(t) of periodT = 2π/ω which is finite, continuos and differentiable can be expressed by theFourier series:
f(t) = a0 +
∞∑n=1
(an cosnωt+ bn sinnωt = a0 +
∞∑n=1
cn cos(nωt+ ϕn)) (2.12)
30
where:
cn =√a2n + b2n (2.13)
tanϕn =anbn
(2.14)
a0 =1
T
∫ T
0
f(t)dt (2.15)
an =2
T
∫ T
0
f(t) cos(nωt)dtbn =2
T
∫ T
0
f(t) sin(nωt)dt (2.16)
Thus a generic wave can be written as a linear combination of sine waves.Let us analyze the proprieties of (co)sinusoidal waves:
ξ(x, t) = ξ0 cos[k(x− vt) + δ] (2.17)
To represent a (co)sinusoidal wave we can use the complex representation:
ξ(x, t) = Re[ξ0ei[k(x−vt)+δ]] (2.18)
and so we can refer to the complex wave:
ξ(x, t) = ξ0eik(x−vt) (2.19)
where ξ0 = ξ0eiδ (the amplitude absorbs the phase). As we stated before, the
real part of the complex wave is the physical wave in this context. Since thewave equation is linear, we can carry on our calculations using exponentials (theadvantage of the complex notation is that exponentials are much easier to ma-nipulate than sines and cosines), and then go back to the cosine representationwhen we want.
2.4 Amplitude, wavelength, period
We introduce some important definitions related to of a harmonic (sinusoidal)wave ξ(x, t) = ξ0 cos[k(x− vt) + δ]:
• ξ0 = max{|ξ(x, t)|} is defined as the amplitude.
• T = 2π/ω is the period, i.e., the minimum time after which the functionmakes a replica of itself (the time that a point takes to do a completeoscillation).
• ν = 1/T is the frequency, i.e., the number of periods per second; theunit of the frequency in the SI is the hertz Hz: [Hz]= [s−1].
• λ = 2π/k is the wavelength, the spatial period of the wave - i.e., theminimum distance over which the wave’s shape repeats.
• v = λ/T = λν is the velocity of the wave.
31
• k = 2π/λ is defined as the wave number.
• ω = kv = 2π/T is the angular frequency.
• δ is the phase, and it depends on the choice of the starting time.
2.5 Transverse and longitudinal waves
Let us suppose that the quantity associated to a wave ξ is a vector, for examplerepresenting an oscillation, or the electric field. We define:
• Transverse waves] the waves for which the direction of the perturbation isthe same as their direction of propagation.
• Longitudinal waves the waves for which the direction of the perturbationis perpendicular to the direction of propagation.
Figure 2.4: Transverse and longitudinal waves.
Transverse waves are said to be polarized if there is a simple law describingthe direction of oscillation ξ in the plane of oscillation, say, yz. Particular casesof polarized waves are:
• Linearly polarized waves: waves in which the direction of oscillation ξ isfixed.
• Elliptically polarized waves. These are waves that satisfy the condition:
ξy = ξ0y sin(kx− ωt) (2.20)
ξz = ξ0z cos(kx− ωt) (2.21)
ξ2y
ξ20y
+ξ2z
ξ20z
= 1 (2.22)
In particular, if ξ20y = ξ2
0x the wave is said to be circularly polarized.
2.6 An example: sound
Sound is a sequence of (approximately longitudinal) waves of pressure that prop-agates through compressible media such as air, water, or solids. Sound that is
32
Figure 2.5: A stationary source of waves and a moving source.
perceptible by humans has frequencies from about 20 Hz to 20 000 Hz; in airat standard temperature and pressure, the corresponding wavelengths of soundwaves range from 17 m to 17 mm.
Pitch is an auditory sensation in which a listener assigns musical tones to rel-ative positions on a musical scale based primarily on the frequency of vibration.Pitch is closely related to frequency, but the two are not equivalent, apart frompurely sinusoidal waves. Frequency is an objective, scientific concept, whereaspitch is subjective, and related to the sector of psychoacoustics.
The speed of sound depends on the medium the waves pass through, and is afundamental property of the material. In general, the speed of sound is propor-tional to the square root of the ratio of the elastic modulus K = −V dP/dV ofthe medium (where V is the volume and P is the pressure) to its density. Thosephysical properties and the speed of sound change with ambient conditions. Forexample, the speed of sound in gases depends on temperature. In air at NTP,the speed of sound is approximately 340 m/s. In fresh water at 20 ◦C, the speedof sound is approximately 1400 m/s. In steel, the speed of sound is about 6000m/s.
2.7 The classical Doppler effect
A peculiar effect of wave propagation is the so-called Doppler effect: the char-acteristics of a wave depend on the motion of the emitter and of the receiver. Itis a phenomenon that affects all type of waves, and it is of primary importancein astrophysics, as we shall see next.
The Doppler effect is the apparent change of frequency of the wave, whena source or an observer are in movement. We notice this effect, for example,when we are in relative movement with respect to a source of sound waves: ifwe approach the source, the frequency of the sound is higher; if we get far away,the frequency appears lower.
Let us consider a source, that emits a wave with frequency f , and an ob-
33
server. Suppose that the source and the observer move with constant relativevelocity, and that the direction of the velocity lies on the straight line joiningthe two objects. We want to analize how the Doppler effect manifests itself.We make also the semplification that one of the two objects is stationary withrespect to the medium in which the wave propagates (for example, in the caseof sound, the air of the atmosphere). We can immediately note that the phe-nomenon depends on who is moving and who is stationary with respect to themedium in which the wave propagates.
Source moving, observer at rest. Let us consider first the case in whichthe source moves with velocity vs < v in the direction of the observer. Thelength of the wave received λ′ changes. We have that
λ′ = λ− vsf.
Thus:
v
f ′=v
f− vsf
;
=⇒ f ′ = f · 1
1− vsv
;
These conditions hold only when the velocity of propagation v is larger than vs.In particular if vs � v, we can make the approximation
f ′ ' f(
1 +vsv
).
Source at rest, observer moving. Let us consider the case in which thesource lies on a fixed point and the observer moves with velocity vo in thedirection of the source. The velocity of the wave relative to the observer isv + vo, so we get
f ′ =v + voλ
=v + vov/f
= f(
1 +vov
).
We can notice that only in the first approximation the variation of frequencyis the same in both cases. The case in which both the source and the observermove is more complicated, as the case in which the direction of the movement isnot directed along the line joining them; however, there is nothing conceptuallynew.
2.7.1 Redshift of galaxies and the expansion of the Uni-verse
The Doppler effect has important applications in astrophysics. Observing theDoppler effect on the spectrum of emission of galaxies and stars in the Universe,
34
Figure 2.6: Experimental plot of the relative velocity (in km/s) of knownastrophysical objects as a function of the distance from the Earth (in Mpc; 1
pc ' 3.3 ly). The line is a fit to Hubble’s Law.
35
Figure 2.7: Redshift of emission spectrum of stars and galaxies at differentdistances. Using this shift we can calculate the relative velocity between these
objects and the Earth.
we can compute the relative velocity of these objects with respect to us, andthen the distance, thanks to the so-called Hubble’s law.
In 1929 Edwin Hubble4, studying the emission of galaxies, observed fromtheir Doppler redshift that objects in Universe move away with us with velocity
v = H0d , (2.23)
where d is the distance between the objects, and H0 is a parameter called theHubble constant (whose value is known today to be about 24km/s/Mly). Theabove relation is called Hubble’s law.
To give an isea of what H − 0 means, the speed of revolution of the Eartharound the Sun is about 30 km/s; Andromeda, the large galaxy closest to theMilky Way, has a distance d of about 2.5 Mly. However, the Hubble’s law isjust statistical and working for large distances, where gravitational attractionbecomes negligible: Andromeda is indeed approaching us.
Dimensionally we note that H0 is a frequency: H0 ' (14 × 109 years)−1.A simple interpretation of this law is that, if the Universe has always beenexpanding at a constant rate, about 14 ·109 years ago its volume was zero. Thisresult is consistent with present estimates of the age of the Universe within theso-called big bang theory.
The redshift
z =λ′
λ− 1
is also used as a metric of distance of objects.
4Edwin Hubble (1889 - 1953) was an American astronomer who played a crucial role inestablishing the field of extragalactic astronomy and is generally regarded as one of the mostimportant observational cosmologists of the 20th century.
36
Figure 2.8: The Cherenkov cone.
2.7.2 The Vavilov-Cherenkov effect
The Vavilov-Cherenkov5 (commonly just called Cherenkov) effect occurs whena wave emitter moves through a medium faster than the speed of the wave inthat medium.
In the case vs > v, as we can see in the Figure 2.8, the wave front is a cone,that takes the name of Cherenkov cone. The following relation connects theangle θ with the velocities v and vs:
cos θ = v/vs. (2.24)
To find the value of cos θ let us consider two positions of the source S1 andS2, and the corresponding points P and Q on the wave front: the wave emittedin S1 has, in P , the same phase of the one emitted in S2 in Q. For the samereason also the points S′1 and S2 have the same phase. The time that the sourcespends to go from S1 to S2 is equal to the time that the wave spends to go fromS1 to S′1. If we call a the distance S1S2 we have S1S
′1 = a cos θ. Thus we get:
a
vs=a cos θ
v
=⇒ cos θ =v
vs.
As an example, when cosmic rays interact with the atmosphere they generateshowers of particles. The charged particles radiate light, and some of themare faster than light in the atmosphere, thus generating cones of collimatedCherenkov light. This light is detected by special-purpose telescopes.
5Pavel Alekseyevich Cherenkov (1904-1990) was a Soviet physicist who shared the NobelPrize in physics in 1958 with Ilya Frank and Igor Tamm for the discovery of Cherenkovradiation, made in 1934. The discovery was made during Cherenkov’s thesis, directed by theacademician Nikolai Vavilov; when the Nobel prize was assigned, however, Vavilov was deadsince 15 years.
37
2.8 Composition of waves
Since the wave equation is linear, when we want to sum two waves we canjust sum the functions representing them. We should remember that energy isproportional to the square of the amplitude: this can create effects of positiveand negative interference that we shall discuss later.
Figure 2.9: Examples of algebraic sum of two waves.
2.8.1 Boundary conditions and steady waves
Let ξ(x, t) be the equation of a plane wave along the x axis direction. Generallywe can write
ξ(x, t) = ξ+(x− vt) + ξ−(x+ vt) (2.25)
where ξ+(x− vt) propagates in the positive direction of the x coordinate, whileξ−(x+ vt) propagates in the negative direction (Figure 2.10).
Consider now the case in which the wave is confined in a finite region, e.g., awave on a rope between two walls. When a wave ξ+ meets the wall, it changesverse of propagation, and “creates” a second wave ξ− with the same character-
Figure 2.10: Waves with opposite velocity.
38
istics but opposite velocity. If we sum the two waves we obtain a wave withconstant speed zero, also called stationary wave.
Let us analyze mathematically what happens. Since v = ω/k, and ω remainsthe same, the change of velocity from +v to −v corresponds to a change of wavenumber from +k to −k. So the two wave equations are:
ξ+(x, t) = ξ0 sin (kx− ωt) ;
ξ−(x, t) = ξ0 sin (−kx− ωt) .
Using the appropriate trigonometric formulae, the sum of the waves is:
ξ(x, t) = ξ+(x, t) + ξ−(x, t) =
= ξ0 sin (kx− ωt) + ξ0 sin (−kx− ωt) =
= 2ξ0 sin (−ωt) cos (kx) =
= −2ξ0 sin (ωt) cos (kx) .
The oscillation is maximum in the points x such that cos(kx) = 1, while isminimum in the points that satisfy the condition cos(kx) = 0. These points arecalled respectively antinodes and nodes.
Observe that only waves that have nodes on the extremities of the rope willsurvive: otherwise the amplitude of the sum of the waves in an extreme is notnull, and so also the energy (dispersed) is not null, but this contradicts theconservation of energy, because we are supposing the reflected wave to have thesame amplitude ξ0.
Imagine you are perturbing a violin string: the extremities P and Q arefixed and you generate different waves displacing the string in different points.The wave you have generated can be expressed as the sum of sinusoidal waves,from the Fourier’s theorem. All these waves have a same property: the pointsP and Q are nodes for them. After a transient, only the waves for which thepoints P and Q are nodes survive, otherwise the wave, when reflected, generatesa destructive interference. So only discrete values of λ are permitted, those forwhich the boundaries are nodal points. These are called the harmonics:
• the fundamental frequency with frequency f1 is the wave with maximumwavelenght: it is such that λ
2 = L where λ is the wavelenght and L is thestring lenght. Observe that λ = 2L;
• the second harmonic f2 is a wave with wave lenght such that 2λ2 = L, that
is λ = L;
• the third harmonic f3 is such that 3λ2 = L, that is λ = 2
3L;
• the fourth harmonic f4 has a wave lenght such that 4λ2 = L, i.e. λ = 1
2L;
• and so on . . .
The wave generated perturbing the violin string is, after a transient, the sumof these waves, as we can see in Figure 2.12. The fundamental frequency deter-mines the pitch of the note, and together with the higher harmonics determine
39
Figure 2.11: The first four harmonics.
40
Figure 2.12: A string is plucked in a certain point. This creates a wave that issum of three waves: the foundamental frequency, the second and the third
harmonics. Their sum produce a determined sound.
the timbre of the sound. We can obtain sounds with same pitch, but differenttimbre, just plucking the string in different places.
2.8.2 Beats
We will now analyze another interesting example of wave composition. Let usconsider two sinusoidal waves that propagate in the same direction, with thesame amplitude ξ0 and velocity v, but slightly different frequencies ωi.
The waves are defined by the following equations:
ξ1(x, t) = ξ0 cos (k1x− ω1t) ;
ξ2(x, t) = ξ0 cos (k2x− ω2t) .
What happens if we sum them? Let us put:
∆k = k1 − k2; 〈k〉 =k1 + k2
2;
∆ω = ω1 − ω2; 〈ω〉 =ω1 + ω2
2.
By trigonometric identities:
cos(α) + cos(β) = 2 cos
(α+ β
2
)cos
(α− β
2
),
41
Figure 2.13: Graphic representation of beats.
we obtain
f1(x, t) + f2(x, t) = ξ0 cos(k1x− ω1t) + ξ0 cos(k2x− ω2t) =
= 2ξ0 cos
(∆k
2x− ∆ω
2t
)cos(〈k〉x − 〈ω〉t). (2.26)
The resulting function is a product of two cosinusoidal functions. Figure 2.13explains the behaviour of the new wave. In this case the listener perceives anoscillating volume level; the frequency of the volume oscillation is much lowerthan the sound frequency.
This effect happens for example when two singers are not able to take thesame tonality.
2.8.3 Group velocity and phase velocity
In this section we want to analize the behaviour of a wave packet, startingwith the prerequisite that the propagation velocity of a wave with equationξ(x, t) = ξ0 cos(kx− ωt), is given by v = ω/k.
A wave packet is a short envelope of waves that travels as a unit. If thewave packet propagates in one direction (e.g. the x axis), using the Fourier’stheorem, its general form can be written as:
ξ(x, t) =
∫Ξ(k)ei(kx−ω(k)t)dk,
where Ξ(k) is a function that takes a large value in a region of area ∆k around acertain point k, and goes to zero elsewhere. For example f could be a Gaussianfunction with very low variance so that the funcion has a great peak near k. Anexample of wave packet is represented in Figure 2.14.
We want now to analize the speed of propagation of a wave packet. To dothis we observe that beats are a simple wave packet, made by two waves. So webegin studying this case, that is simpler than the general one. We know that
42
Figure 2.14: A wave packet.
Figure 2.15: In the example of beats the envelope wave (green) is given by thecosinus with longer wavelenght.
the speed of a wave whose equation is f(x, t) = ξ0 cos(kx− ωt) is given by:
v =ω
k. (2.27)
In the previous section we obtained that beats have an equation that is theproduct of two cosinusoidal functions. The speed of the envelope is given by thespeed of the factor with longer wavelenght, i.e., lower wavenumber. If we returnto (2.26), we have to compare the numbers 〈k〉 and ∆k
2 , to understand which
term has the lower wavenumber. It is easy to prove that ∆k2 ≤ 〈k〉, so we have
to consider the factor cos(
∆k2 x−
∆ω2 t). Using the (2.27) we find the velocity
venvelope =∆ω
∆k.
If we want to extend the result to general wave packets, we have to take thelimit for ∆k that goes to zero. So the velocity of the envelope, also called groupvelocity, is given by:
vg =d(ω(k))
dk. (2.28)
The phase velocity is the speed of propagation of a phase, for example of thepoint P represented in Figure 2.15. This velocity, in the case of beats, is equalto the propagation velocity of the factor with shorter wavelenght and higher
43
wavenumber, i.e., we have to look at the term cos(〈k〉x − 〈ω〉t). Rememberingthat v = ω1/k1 = ω2/k2 the phase velocity of beats is:
v =〈ω〉〈k〉
=ω1 + ω2
k1 + k2=vk1 + vk2
k1 + k2=ω1
k1=ω2
k2= v.
Note that the phase velocity is equal to the speed of the two waves that generatebeats. In general, in a dispersive medium (where v is a function of ω) the phaseand group velocities can be different.
2.9 Waves in three dimensions
Every wave we have considered so far was a planar wave moving along the x axis,whose equation is ξ(x, t) = ξ(x ± vt). A wave of this form can be decomposedin its harmonics which can be written as ξ(x, t) = ξ0 sin((kx± ωt) + φ).
To describe a wave moving in a general direction we define a new vector
~k =2π
λuv. So we have that
ξ = ξ0 sin((~k · ~r ± ωt) + φ) = ξ0 sin((kxx+ kyy + kzz ± ωt) + φ)
knowing that |~k| = ω
vwe have
∂2ξ
∂x2+∂2ξ
∂y2+∂2ξ
∂z2= ∇2ξ =
1
v2
∂2ξ
∂t2.
This last equation also has nonplanar waves as solutions.We define wavefront a surface whose phase is constant in a given moment
of time, and ray the line orthogonal to the wavefront which represents in thatpoint the direction of the wave and of the energy associated to it. If |v| is thesame in every direction we have a spherical wavefront; if |v| is the same in everydirection perpendicular to a given axis we have a cylindrical wavefront.
2.9.1 Spherical waves
A small spherical object which pulsates periodically produces a sound wave,whose wavefronts are spheres concentric to the object. These waves are anexample of spherical waves.
The equation for a single harmonic component is ξ(r, t) = A(r) sin (kr − ωt).If the medium in which the wave is travelling in is motionless and has uniformdensity then the waves propagate outwards with constant velocity, therefore thewavelength will not depend on the distance whereas, due to energy conservation,the amplitude will, decreasing as we get further from the source.
The energy flow per unit surface is I(r) = CA2(r), where C is a constant.If we have no dispersion, the power carried through a surface of radius r isconstant, and equal to
I(r)S(r) = CA2(r) · 4πr2 = cost⇒ cost
r= A(r) =
ξ0r
44
where S(r) = 4πr2 is the surface of the wavefront.Therefore the equation of a spherical wave in a nondispersive medium is
ξ(r, t) =ξ0r
sin(kr − ωt).
45
Chapter 3
Electromagnetism and theMaxwell’s equations
The Maxwell’s equations1∮S
~E · d~S =q
ε0(3.1)∮
C
~E · d~l = − d
dt
∫S(C)
~B · d~S (3.2)∮S
~B · d~S = 0 (3.3)∮C
~B · d~l = µ0I + ε0µ0d
dt
∫S(C)
~E · d~S , (3.4)
together with the equation describing the motion of a particle of electrical chargeq in an electromagnetic field
~F = q(~E + ~v × ~B) (3.5)
(Lorentz2 force), provide a complete description of electromagnetic field and ofits dynamical effects.
We want to write Maxwell’s equations in a local form, and to transform theintegro-differential equations above into purely differential equations.
1James Clerk Maxwell (1831 - 1879) was a Scottish physicist. His most prominent achieve-ment was formulating classical electromagnetic theory. Maxwell’s equations, published in1865, demonstrate that electricity, magnetism and light are all manifestations of the samephenomenon, namely the electromagnetic field. Maxwell also contributed to the Maxwell-Boltzmann distribution, which gives the statistical distribution of velocities in a classical per-fect gas in equilibrium. Einstein kept a photograph of Maxwell on his study wall, alongsidepictures of Faraday and Newton.
2Hendrik Lorentz (1853 - 1928) was a Dutch physicist who gave important contributionsto electromagnetism. He also derived the equations subsequently used by Albert Einsteinto describe the transformation of space and time coordinates in different inertial referenceframes. He was awarded the 1902 Nobel Prize in Physics.
46
3.1 Charge density and current density
We first write in terms of local variables the electric charge and the electriccurrent.
The charge density ρ(t, ~r) is defined as the charge per unit volume in a point~r at a time t:
q =
∫V
ρ(t, ~r) dV . (3.6)
The current density~(t, ~r) is defined as the intensity of electrical current perunit surface:
I =
∫S
~(t, ~r) · d~S . (3.7)
3.2 Maxwell’s equations in differential form
Let us examine Equation (3.1). The charge q is contained in the volume volumeV , thus we can write
q =
∫V
(S)ρ dV .
By Gauss’ theorem, ∮S
~E · d~S =
∫V (S)
(~∇ · ~E) dV
and thus~∇ · ~E =
ρ
ε0. (3.8)
In the same way we get from Equation (3.3), by applying Gauss’ theorem,∮~B · d~S = 0 =⇒ ~∇ · ~B = 0 . (3.9)
Equations ∮C(S)
~E · d~l = − d
dt
∫S
~B · d~S∮C(S)
~B · d~l = µ0I + ε0µ0d
dt
∫S
~E · d~S
become respectively, by the application of Stokes’ theorem, and of Equation(3.7), ∫
S
(~∇× ~E) · d~S = − d
dt
∫S
~B · d~S∫S
(~∇× ~B) · d~S = µ0
∫S
~ · d~S + ε0µ0d
dt
∫S
~E · d~S
47
and thus
~∇× ~E = −∂~B
∂t
~∇× ~B = ε0µ0∂~E∂t
+ µ0~ .
These are called respectively the law of Faraday3-Lenz4 law and the Ampere5-Maxwell law.
Thus Maxwell’s equations can be written in differential form as:
~∇ · ~E =ρ
ε0
~∇× ~E = −∂~B
∂t~∇ · ~B = 0
~∇× ~B = ε0µ0∂~E∂t
+ µ0~
These equations allow calculating the electric and magnetic fields from thecharge distribution ρ(t, ~r) and the current density ~(t, ~r).
3.3 Maxwell’s equations and continuity equationfor charge
If we take the divergence of both sides of the Ampere-Maxwell law
~∇× ~B = ε0µ0∂~E∂t
+ µ0~
we obtain
~∇ · (~∇× ~B) = ε0µ0~∇ · ∂
~E∂t
+ µ0~∇ ·~ .
But ~∇· (~∇× ~B) = 0, and we can exchange the derivatives with respect to spaceand time:
ε0µ0∂
∂t(~∇ · ~E) + µ0
~∇ ·~ = 0 .
Gauss’ law tells us that ~∇ · ~E = ρ/ε0; thus
~∇ ·~ +∂ρ
∂t= 0 . (3.10)
3Michael Faraday (1791 - 1867) was an English scientist who contributed to the fields ofelectromagnetism and electrochemistry. His main discoveries include those of electromagneticinduction, diamagnetism and electrolysis. Although Faraday received little formal educationhe was one of the most influential scientists in history.
4Heinrich Lenz (1804 -1865) was a Russian physicist of Baltic ethnicity. He is most notedfor formulating Lenz’s law in electrodynamics in 1833.
5Andre-Marie Ampere (1775 - 1836) was a French physicist and mathematician who isgenerally regarded as one of the main founders of the science of electrodynamics.
48
The equation above is a continuity equation for charge. If there is a netelectric current is flowing out of a region, then the charge in that region mustbe decreasing by the same amount. Charge is conserved.
3.4 The potentials, vector and scalar
The physics in the Maxwell equations depends only on the magnetic and electricfields. We shall tray to express these fields in terms of some generating fields,called potentials, which have no immediate relation to physics.
The divergence of ~B is zero, and this means that we can always represent ~Bas the curi of another vector field. Conversely, the divergence of a curl is alwayszero. Thus we can always relate the magnetic field to a field we will call ~A by
~B = ~∇× ~A .
The field ~A is called the vector potential.We remind that the scalar potential φ, such that
~E = −~∇φ
in electrostatics, was not completely specified by its definition. If we have foundφ, we can always find another potential φ′ that is equivalent from the point ofview of physics by adding a constant:
φ′ = φ+ C .
The new potential φ′ gives the same electric fields, since the gradient of a con-stant is zero; φ′ and φ represent the same physics.
Similarly, we can have different vector potentials ~A which give the samemagnetic fields. Again, because ~B is obtained from ~A by differentiation, addinga constant to ~A does not change anything physical. But we can add to ~A anyfield which is the gradient of some scalar field, without changing the physics:
~B = ~∇× ~A =⇒ ~∇× ( ~A+ ~∇ψ) = ~∇× ~A+ ~∇× ~∇ψ = ~∇× ~A = ~B .
It is usually convenient to take some of the freedom out of ~A by arbitrarilyplacing some other condition on it (in much the same way that we found itconvenient - often - to choose to make the potential zero at large distances).
We can, for instance, restrict ~A by choosing arbitrarily what the divergenceof ~A must be. We can always do that without affecting ~B. This is becausealthough ~A and ~A′ have the same curi, and give the same ~B, they do not needto have the same divergence. By a suitable choice of ψ we can make ~∇× ~A′ anywell-behaved function we wish.
What should we choose for ~∇ · ~A? The choice should be made to get thegreatest mathematical convenience and will depend on the problem we are doing.For magnetostatics, we make the simple choice
~∇ · ~A = 0 (3.11)
49
(later, when we take up electrodynamics, we shall make a different choice). Ourcomplete definition of A is then, for the moment,
~∇× ~A = ~B and ∇ · A = 0 .
3.5 Maxwell’s equations and electrostatics
Two of the Maxwell’s laws are the traditional Ampere-Maxwell and Faraday-Lenz laws.
Can we obtain the two other laws of electrostatics and magnetostatics : theCoulomb’s law, stating that the electric field by a point charge q is
~E =
(1
4πε0
)~urr2
(3.12)
and the Biot-Savart6 law, stating that the elementary magnetic field generatedby a current I over an element of conductor d~l at a radius ~r is
d ~B =(µ0
4π
)Id~l × ~urr2
. (3.13)
from the Maxwell equations? It turns out that this is indeed possible.We do the demonstration for the Coulomb’s law. The differential form of the
Maxwell’s law is equivalent to its integral form (3.1). Due to symmetry reasons,the field must be directed radially. Then if we choose as a surface a sphere ofradius r, equation (3.1) becomes
|~E| 4πr2 =q
ε0,
which demonstrates the statement.
3.6 Maxwell’s equations and magnetostatics
For the demonstration of the Biot-Savart law from Maxwell’s equations, seethe Feynman’s lectures on Physics, Volume 2, Chapter 14 (it is not part of theprogram of the exam).
6Jean-Baptiste Biot (1774 - 1862) was a French physicist, astronomer, and mathemati-cian who established the reality of meteorites, made an early balloon flight, and studied thepolarization of light. Felix Savart (1791 - 1841), professor at College de France, was the co-originator of the Biot-Savart Law, along with Biot. Together, they worked on the theory ofmagnetism and electrical currents. Their law was developed about 1820.
50
Chapter 4
Solutions of the Maxwell’sequations in vacuo
Maxwell’s equations are four partial-derivatives first order equations; two ofthem are coupled in the unknown fields ~E and ~B. We shall see now that inregions of space and time where no charges or currents are present, these can bedecoupled in two equations, one for ~E and one for ~B. The resulting equationsare wave equations; we shall discuss them, together with their implications.
Let us go back to the Maxwell’s equations in differential form:
• Coulomb’s law:~∇ · ~E =
ρ
ε0
• Faraday-Lenz’s law:
~∇× ~E = −∂~B
∂t
• the equation stating that there are no magnetic charges (monopoles):
~∇ · ~B = 0
• Ampere-Maxwell’s law:
~∇× ~B = ε0µ0∂~E∂t
+ µ0~ .
Now we consider a region without charges and we are going to analyze thesituation. We can write the Maxwell’s equations in vacuo as:
~∇ · ~E = 0 (4.1)
~∇× ~E = −∂~B
∂t(4.2)
51
~∇ · ~B = 0 (4.3)
~∇× ~B = ε0µ0∂E
∂t(4.4)
A trivial solution would be that the electromagnetic field is everywhere zero.Let us examine the characteristics of nontrivial solutions.
We suppose that there would be field generators outside the no-charges re-gion. Consequently the region inherits information about the charges in theUniverse. Equation 4.2 can be written as:
~∇×(~∇× ~E
)= − ∂
∂t
(~∇× ~B
)= −ε0µ0
∂2~E∂t2
thanks to equation 4.4
On the other hand
−µ0ε0∂2~E∂t2
= ~∇×(~∇× ~E
)= ~∇
(~∇ · ~E
)︸ ︷︷ ︸
=0, eq. 4.1
−∇2~E = −∇2~E
and thus we obtain:∂2~E∂t2
=1
ε0µ0∇2~E . (4.5)
We observe that the electric field in the empty space satisfies the d’Alembert’sequation: it is thus a wave.
Similarly to what was done for the electric field we can write
−µ0ε0∂2 ~B
∂t2= ε0µ0
∂
∂t
(~∇× ~E
)︸ ︷︷ ︸eq.4.2
= ~∇×(~∇× ~B
)= ~∇
(~∇ · ~B
)︸ ︷︷ ︸
=0,eq. 4.3
−∇2 ~B = −∇2 ~B
and thus we obtain∂2 ~B
∂t2=
1
ε0µ0∇2 ~B . (4.6)
This is the wave equation for the magnetic field in empty space; we note thatit has the same form as that of the electric field.
4.1 Maxwell’s waves and light
Since1
4πε0' 9 · 109 N ·m2
C
µ0
4π= 10−7 T ·m
A
52
we can write the velocity of this wave:
v2 =1
ε0µ0=
14πε0µ0
4π
' 9 · 109
10−7' 9 · 1016 m2
s2.
Then:
v =
√1
ε0µ0' 3 · 108 m/s (4.7)
that is consistent with the value of the speed of light. It becomes then naturalto assume that light is an electromagnetic wave.
Hertz1 presented the decisive experimental confirmation that light is a Max-well’s wave in 1888, by generating light through an oscillating electromagneticfield. He wrote: “The connection between light and electricity is now estab-lished ... In every flame, in every luminous particle, we see an electrical process... Thus, the domain of electricity extends over the whole of nature. It evenaffects ourselves intimately: we perceive that we possess ... an electrical organ -the eye.” By 1900, then, three great branches of physics, electricity, magnetism,and optics, had merged into a single unified theory (and it was soon apparentthat visible light represents only a tiny window in the vast spectrum of electro-magnetic radiation, from radio though microwaves, infrared and ultraviolet, toX-rays and gamma rays.)
The Ampere-Maxwell equation (4.4) becomes, if we do this assumption,
~∇× ~B = ε0µ0∂~E∂t
=1
c2∂~E∂t
.
Thus the speed of light enters in the fondamental laws of nature.This fact has an important consequence: in classical mechanics the Maxwell’s
equations are not the same for different reference frames in relative motion withconstant velocity. Indeed, in classical physics, all the laws depend on acceler-ation, which is invariant between the inertial reference frames (the coordinatesare transformed from one reference frame to another by means of the Galilei2
transformations); velocities cannot enter in a fundamental law valid in all iner-tial frames, since they are not an invariant quantity.
Since the Maxwell’s equations violate the Galilean relativity, if we assumethat they are correct, there could be two possibilities (not mutually exclusive):
• Relativity is not a legitimate assumption: there is a preferred referenceframe, in which electromagnetism can be described (this has been called
1Heinrich Hertz (1857 - 1894) was a German physicist who clarified and expanded JamesClerk Maxwell’s electromagnetic theory of light.
2Galileo Galilei (1564 - 1642) was an Italian physicist, mathematician, astronomer, andphilosopher who played a major role in the scientific revolution. His achievements includeimprovements to the telescope and consequent astronomical observations and support forCopernicanism. His contributions to observational astronomy include the discovery of thephases of Venus, of the four largest satellites of Jupiter (named the Galilean moons in hishonour), and the observation and analysis of sunspots. Galilei also worked in military scienceand technology.
53
the aether). For example, this would be the case if electromagnetic fieldswould be the perturbation of a static medium.
• The relativistic transformations of Galilei do not work.
We shall see that the answer to this question brought to us one of the mostfascinating theories of the XX century - Einstein’s3 special relativity.
4.2 Properties of the electromagnetic waves
Now we study the caracteristics of elettromagnetic waves in vacuo. We supposeto be very far away from the electromagnetic charges, so the equation of thewave depends on a single variable. Then ~E(x, y, z, t) ≡ ~E(z, t) and
∂~E∂x
=∂~E∂y
=∂ ~B
∂x=∂ ~B
∂y= 0
(the waves must be plane waves). Thus the Maxwell’s equations become:
∂−→Ez∂z
= 0 (4.8)
uz ×∂~E∂z
= −∂~B
∂t. (4.9)
Indeed
~∇× ~E = ux
(∂−→Ez∂y− ∂−→Ey∂z
)− uy
(∂−→Ez∂x− ∂−→Ex∂z
)+ uz
(∂−→Ey∂x− ∂−→Ex∂y
)=
= uz ×∂~E∂z
= −∂~B
∂t.
Analogously for the magnetic field:
∂−→Bz∂z
= 0 (4.10)
3Albert Einstein (1879 - 1955) was a German-born physicist who deeply changed the rep-resentation of the Universe by the human species. While best known for his mass-energyequivalence formula E = mc2 (published in 1905), he received the 1921 Nobel Prize in Physicsfor his discovery of the law of the photoelectric effect (also in 1905), which was pivotal in es-tablishing quantum theory within physics. Near the beginning of his career, Einstein thoughtthat Newtonian mechanics could not reconcile the laws dynamics with the laws of the electro-magnetic field. This led to the development of his special theory of relativity (again in 1905).He realized, however, that the principle of relativity could also be extended to gravitationalfields, and with his subsequent theory of gravitation in 1916, he published a paper on the gen-eral theory of relativity. He was visiting the United States when Adolf Hitler came to powerin 1933, and did not go back to Germany, where he was a professor in Berlin. He settled inthe U.S., becoming a citizen in 1940. On the eve of World War II, he helped the set up ofthe Manhattan Project, and ultimately the construction of the atomic bomb. Later, however,he highlighted the danger of nuclear weapons. Einstein was affiliated with the Institute forAdvanced Study in Princeton until his death.
54
uz ×∂ ~B
∂z=
1
c2∂~E∂t
. (4.11)
We observe that
0 = uz ·
(uz ×
∂ ~B
∂z
)=
1
c2∂−→Ez∂t
=⇒ ∂−→Ez∂t
= 0
0 = uz ·
(uz ×
∂~E∂z
)= −∂
−→Bz∂t
=⇒ ∂−→Bz∂t
= 0
and thus we can conclude that Ez and Bz are constants, and in particular theydo not depend on the variables x, y, z, t. If we postulate that the Universe hasfinite energy, we conclude that Ez = Bz = 0. Recall that the energy densityassociated with the presence of electric and magnetic field are respectively
u~E =ε0E2
2u ~B =
B2
2µ0. (4.12)
Unless the Universe has infinite energy, Ez = Bz = 0: the electromagnetic wavemust be transverse.
We rewrite all the equations obtained to understand the relations betweenthe electric field and the magnetic field in an electromagnetic wave.
Ez = 0 (4.13)
Bz = 0 (4.14)
∂Ex∂z
= −∂By∂t
(4.15)
∂Ey∂z
=∂Bx∂t
(4.16)
∂Ex∂t
= c2∂By∂z
(4.17)
∂Ey∂t
= −c2 ∂Bx∂z
(4.18)
If we set υ = z − ct we have that ∂υ∂z = 1 and ∂υ
∂t = −c. We can obtain fromequation 4.15
∂By∂t
= −∂Ex∂z
= −∂Ex∂υ
∂υ
∂z= −∂Ex
∂υ
⇒ By =
∫∂By∂t
dt = −∫∂Ex∂υ
dt =1
c
∫∂Ex∂υ
dυ
⇒ By =Exc
+ constant
The constant must be equal to zero, not to have infinite energy; in the end wehave then
By =Exc. (4.19)
55
Analogously from Equation (4.16)
Bx = −Eyc. (4.20)
In conclusion we have obtained that
| ~B| = |~E|c
and
~B · ~E =
(−EycEx +
ExcEy + 0
)= 0.
In other words, the electric wave is perpendicular and proportional to the mag-netic wave.
We also observe from (4.19) and (4.20) that
~E × ~B = EB ~uz : (4.21)
i.e., the vector product ~E × ~B gives the direction of propagation of the wave.
4.2.1 Energy transported an the electromagnetic wave
We found the following equations:
∂2~~E∂t2
= c2∂2~~E∂x2
c2 =1
ε0µ0
∂2 ~~B
∂t2= c2
∂2 ~~B
∂x2| ~~B| = |
~~E|c
The energy densities associated with the electric and magnetic fields arerespectively
u~E =ε0E2
2u ~B =
B2
2µ0
so that the total energy density is
u =ε0E2
2+B2
2µ0.
Considering that B = E/c and ε0µ0 = 1/c2,we have that in an electromag-netic wave the magnetic and electric components of the total energy are equal:
u ~B =B2
2µ0=E2
2µ0c2= u~E
so we can write the total energy density as
u = 2u~E = ε0E2.
56
4.2.2 The Poynting vector
Let u be the energy density per volume unit, the energy carried by a wave,perpendicularly to its direction, per time and surface unit is
I =dU
dSdt=
dU
dSdx
dx
dt= uv.
We call I the intensity and it is measured in W/m2.So far we assumed that the origin of the energy flux is coincident to the
wave source, ignoring possible dissipative effects. The energy carried by anelectromagnetic wave in vacuum per time and surface unit is
S = uc = ε0E2c = ε0EBc2 =1
µ0EB
By the properties of ~E and ~B in an electromagnetic wave we have that
~S =1
µ0
~~E × ~B
has the same direction of the wave and its magnitude is equal to the power persurface unit. This vector is called the Poynting’s vector4.
A sine wave of the form E = E0 cos(kz − ωt), B = B0 cos(kz − ωt) has anaverage power of EB/2µ0.
The electromagnetic wave also carries momentum. Based on considerationssupported by Einstein’s relativity theory (see later), the momentum carried per
surface unit is related to energy as ~S/c.
4.3 Photoelectric effect; the photon hypothesis
When light interacts with material objects, it often reveals a particle-like nature,with behaving like a group of point-like objects interacting with individual atomsor molecules.
In the phenomenon called photoelectric effect, light is able to knock electronsout of the surface of a metal, creating an electric current. The phenomenon wasfirst observed by Hertz in 1887. The characteristics of such an emission cannotbe explained unless one assumes (Einstein suggested this explanation in oneof his famous articles published in 1905, and he was awarded the Nobel prizefor this) that light interacts like a group of wave packets (we shall call themphotons) interacting as particles.
Let us analyse the photoelectric effect using the apparatus in figure 4.1, inwhich light of frequency ν is directed onto a target and ejects electrons from it.
A potential difference V is maintained between the target and the collectorto sweep up these electrons, called photoelectrons. This collection produces a
4John Poynting (1852 - 1914) was an English physicist, college professor in Birmingham.
57
Figure 4.1: Sketch of the apparatus used to study the photoelectric effect.
Figure 4.2: The photoelectric effect.
58
photoelectric current I that is measured with meter A. We adjust the poten-tial difference V by moving the sliding contact so that the collector is slightlynegative with respect to the target. This potential difference acts to slow downthe ejected electrons. We then vary V until it reaches a certain value, called thestopping potential Vstop, at which point the reading of meter A has just droppedto zero. When V = Vstop, the most energetic ejected electrons are turned backjust before reaching the collector.
ν0
dKmaxdν = h
ν
Kmax
Figure 4.3: The kinetic energyKmax, as a function of the
frequency.
Then Kmax, the kinetic energy ofthese most energetic electrons, is
Kmax = eVstop
where e is the elementary charge. Mea-surements show that for light of a givenfrequency, Kmax does not depend on theintensity of the light source.
Now let us vary the frequency ν of theincident light and measure the associatedstopping potential Vstop. Figure 4.3 isa plot of Vstop versus f . Note that thephotoelectric effect does not occur if thefrequency is smaller than a certain cutofffrequency ν0 or, equivalently, if the wave-length is greater than the corresponding
cutoff wavelength λ0 = cν0
. This is true no matter how intense the incident lightis.
This observation is in sharp contrast with a classical view of reality, whichwould make us expect to see electrons ejected for every frequency, given a brightenough light. This is not what happens. For light below the cutoff frequency ν0,the photoelectric effect does not occur, no matter how bright the light sourceis.
The existence of a cutoff frequency is, however, just what we should expectif the energy is transferred via photons. The electrons within the target are heldthere by electric forces. To just escape from the target, an electron must pickup a certain minimum energy Φ, where Φ is a property of the target materialcalled its work function. If the energy hν transferred to an electron by a photonexceeds the work function of the material (if hν > Φ), the electron can escapethe target. If the energy transferred does not exceed the work function (that is,if hν < Φ), the electron cannot escape. This is what figure 4.3 shows.
Einstein summed up the results of such photoelectric experiments in theequation
hν = Kmax + Φ (photoelectric equation) . (4.22)
This is a statement of the conservation of energy for a single photon absorptionby a target with work function Φ. Energy equal to the photon’s energy hν istransferred to a single electron in the material of the target. If the electronis to escape from the target, it must pick up energy at least equal to Φ. Any
59
additional energy (hν − Φ) that the electron acquires from the photon appearsas kinetic energy K of the electron. In the most favourable circumstance, theelectron can escape through the surface without losing any of this kinetic energyin the process; it then appears outside the target with the maximum possiblekinetic energy Kmax.
Therefore we can conclude that the a light wave of frequency ν is made ofquanta of energy
hν =h
2πω = ~ω
where h is the Planck’s constant, which has value
h ' 6.63× 10−34 J s = 4.14× 10−15 eV s .
4.4 The perception of electromagnetic waves: vis-ible light
A typical human eye will respond to wavelengths from about 750 nm (the wave-length of red) to 390 nm (the wavelength of purple). This range is similar to theSun’s radiation spectrum.
If we take a green laser (of wavelength 0.5 µm, that is 5.00× 10−7 m), wehave
ν =3× 108 m/s
5.00× 10−7 m= 0.6× 1015 Hz.
Therefore the energy associated to the green light is
E = hν = 6× 10−34 · 0.6× 1015 ' 3× 10−19 J ' 2 eV.
4.5 Natural units
The International system of units (SI) can be constructed on the basis of fourfundamental units: of length (the meter m), of time (the second s), of mass (thekilogram kg), of charge (the coulomb C)5.
These units are inappropriate for the world of fundamental physics: theradius of a nucleus is of the order of 10−15 m, also called one femtometer (fm)or one fermi; the mass me of an electron is of the order of 10−30 kg; the chargeof an electron is (in absolute value) of the order of 10−19 C. By using such unitswe would carry along a lot of exponents! Thus in particle physics we better useunits like the electron charge for the electrostatic charge, and the electronvolteV and its multiples (keV, MeV, GeV, TeV) for the energy.
Length 1 fm 10−15 mMass 1 MeV/c2 1.78× 10−30 kgCharge |e| 1.60× 10−19 C
5For reasons related only to metrology, i.e., of reproducibility and accuracy of the definition,in the standard SI the unit of electrical current, the ampere A, is used instead of the coulomb;the two definitions are however conceptually equivalent.
60
Note the unit of mass, in which the relation E = mc2 establishing equivalenceof mass and energy (we shall discuss this relationship later) is used implicitly:what one is doing here is to use 1 eV ' 1.60× 10−19 J as the new fundamentalunit of energy.
With these new units, the mass of a proton is about 0.938 GeV/c2, and themass of the electron is about 0.511 MeV/c2. The fundamental energy level ofHydrogen is about −13.6 eV.
In addition to this, nature is providing us with two constants which areparticularly appropriate for the world of fundamental physics: the speed of lightc ' 3.00×108 m/s = 3.00×1023 fm/s, and the Planck’s constant ~ ' 1.05×10−34
J s ' 6.58 × 10−16 eV s. It seems then natural to express speeds in terms ofc, and angular moments in terms of ~. When we do this we switch to theso-called Natural Units (NU). The minimal set of natural units (not includingelectromagnetism) could then be
Speed 1 c 3.00× 108 m/sAngular momentum 1 ~ 1.05× 1034 J sEnergy 1 eV 1.60× 10−19 J
In such a system, ~ = c = 1.After these conventions, just one unit can be used to describe the mechanical
Universe: we choose energy, and thus we can express al mechanical quantitiesin terms of eV and of its multiples. It is immediate to express momenta andmasses directly in terms of energy; related to lengths and energies, we can usethe fact that
~c ' 1.97× 10−13MeV m = 3.15× 10−26J m
~ ' 6.58× 10−22MeV s = 1.05× 10−34J s
(the first relation can also be written as ~c ' 0.197 GeV fm. By choosing naturalunits, all factors of ~ and c may be omitted from equations, which leads toconsiderable simplifications. For example, the relativistic energy relation
E2 = p2c2 +m2c4
becomesE2 = p2 +m2 .
To express 1 m and 1 s in NU, we can just write
1m = 1m~c ' 5.10× 1012MeV−1
1s = 1s~ ' 1.52× 1021 .MeV−1
Both length and time are thus, in natural units, expressed as inverse of energy.The first relation can also be written as 1fm ' 5.10GeV−1: note that when youwhen a quantity expressed in MeV−1, in order to express it in GeV−1, you mustmultiply (and not divide) by a factor of 1000.
61
mks NUQuantity p q r nAction (~) 1 2 -1 0Velocity (c) 0 1 -1 0Mass 1 0 0 0Length 0 1 0 -1Time 0 0 1 -1Momentum 1 1 -1 1Energy 1 2 -2 1
Table 4.1: Dimensions of different physical quantities in the SI and NU.
Let us now find a general rule to transform quantities expressed in naturalunits into SI units, and vice-versa.
To express a quantity in NU back to SI we first restore the ~ and c factorsby dimensional arguments and then use the conversion factors ~ and c (or ~c)to evaluate the result. The dimension of c is [c] = [m/s]; the dimension of ~ is[kgm2s−1].
The vice-versa (from SI to NU) also easy. A quantity with metre-kilogram-second (mks) dimensions MpLqT r (where M represents the mass, L the lengthand T the time) has the NU dimensions [Ep−q−r], where E represents energy.Since ~ and c do not appear in NU, this is the only relevant dimension, anddimensional checks and estimates are very simple. The quantity Q in the SI canbe expressed in NU as
QNU = QSI
(5.62× 1029 MeV
kg
)p(5.10× 1012 MeV−1
m
)q×
(1.52× 1021 MeV−1
s
)rMeVp−q−r
The NU and SI dimensions are listed for some important quantities in Table4.1.
Finally, let us discuss how to treat electromagnetism. To do so, we mustintroduce a new unit, charge for example. We can redefine the unit charge byobserving that
e2
4πε0
has the dimension of [J m], and thus is a pure number in NU. By dividing by~c one has:
e2
4πε0~c' 1
137.
Imposing that the electric permeability of vacuum ε0 = 1 (thus automaticallyµ0 = 1 for the magnetic permeability of vacuum, since from Maxwell’s equations
62
ε0µ0 = 1/c2) we obtain the new definition of charge, and with such a definition:
α =e2
4π' 1
137.
This is called the Lorentz-Heaviside convention. Elementary charge in NU be-comes then a pure number:
e ' 0.303 .
63
Chapter 5
Geometrical optics
In cases where the wavelength is small compared to other length scales in a phys-ical system, light waves can be modeled by light rays, moving on straight-linetrajectories representing the direction of the propagation. This is the domainof the so-called geometrical optics.
5.1 Light propagation through different materi-als; transmission of electromagnetic waves
If light travels through a transparent material, rather than in vacuum, theelectromagnetic equations still hold, substituting effective constants ε and µfor ε0 and µ0, where ε = εrε0 and µ = µrµ0 (µr and εr are respectively therelative magnetic constant and relative dielectric constant). µr ' 1 for non-ferromagnetic materials, and εr ≥ 1; for example εr ' 1 + 6 × 10−4 in air atNTP, εr ' 80 in water (which has a highly polar molecule), and εr ' 5 toεr ' 10 in common glass.
So, in transparent (non-polarizable) materials, light travels with speed:
v = c′ =1√εµ
=c
√εrµr
' c√εr
=:c
n
where n ' √εr ≥ 1 is the ratio between speed of light in vacuum and speed oflight through the material, and it is called refractive index.
5.2 Huygens’ principle
In the XVII century Huygens formulated a conjecture about the propagation ofwaves that only later, around 1882, Kirchhoff1 demonstrated for electromagneticwaves on the basis of Maxwell’s equations, introducing some improvements.
1Gustav Kirchhoff (1824 - 1887) was a German physicist who contributed to the funda-mental understanding of electrical circuits, spectroscopy, and the emission of black-body (hecoined the term “black body”) radiation by heated objects.
64
Figure 5.1: Huygens’ principle.
The Huygens’ principle states that all points on a given wave front can betaken as point sources for the production of spherical secondary waves, calledwavelets, which propagate in the forward direction with the speed characteristicof waves in that medium.
The corrections introduced by Kirchhoff were that the amplitude varies ac-cording to a decreasing function of the angle θ with respect to the direction ofpropagation (f(θ) ' (1 + cos θ)/2), and that the phase of the emitter is antic-ipated by π/2 with respect to the phase of the wavefront. In many problems,however, these two points can be neglected.
5.3 The laws of reflection and refraction
A Huygens’ construction can be used to derive the laws of reflection and refrac-tion of light between two optical media with different indices of refraction; theseare called Snelll’s laws2.
Assume a wave with fronts separated by a wavelength λ1 traveling withspeed v1 in an optically clear medium incident on the boundary with a secondoptically clear medium.
Theorem 1. When a wave meets an obstacle on his path, the angle of incidenceθ1 is equal to the angle of reflection θ2.
2Willebrord Snell, or Snellius (1580 - 1626), was a Dutch astronomer and mathematician.His name has been attached to the laws of reflection and of refraction of light.
65
To demonstrate it, we can look at the Figure 5.3: when the wave fronttouches the surface of reflection, every point behaves like a source of a newwave, that propagates form the point from which the principal wave came. Theresult of these propagations is a wave that appears like the original wave, justreflected by an angle equal to the angle θ1.
A second application of the rule that Huygens postulated is the demonstra-tion of the phenomenon of refraction.
Theorem 2. If a wave passes from a medium with rifraction index n1 to onewith index n2, then the direction of the light obeys the following equation
n1 sin θ1 = n2 sin θ2, (5.1)
where θ1 and θ2 are the angles formed by the direction of the wave with thenormal of the surface, before and after the change of medium of propagationrespectively (vi = c/ni is the speed of light in the two media).
Let us look at Figure 5.3. If we call T the time at which the wavefrontbegins to cross the second medium, and T + ∆t the time at which a second rayhits the second medium, then in a time of ∆t the wave formed by the waveletshas moved inside the second medium of a distance of v2∆t. We obtain thatAC sin θ1 = v1∆t, and AC sin θ2 = v2∆t. From that we obtain
sin θ1
sin θ2=v1
v2=n2
n1. (5.2)
Note that if n1 > n2, an angle of incidency θc exists, called the critical angle,for which the wave does not propagate in the second medium. This is becausethe equation (5.2) becomes:
θ2 = arcsin
(n1
n2sin θ1
).
Thus refraction (transmission) is possible only if
−1 <n1
n2sin θ1 < 1;
arcsin
(−n2
n1
)< θ1 < arcsin
(n2
n1
).
This principle is used in optical fibres: the waves in a tiny tube cannot exit fromthe fibres because of the elevate refraction index.
We also observe that in the transmission between two media, frequency ofwaves does not change, consistent with Huygens’ principle: in fact the waveletshave all the same frequency as the frequency of the principal wave.
66
Figure 5.2: The phenomenon of reflection.
67
Figure 5.3: The phenomenon of refraction.
5.3.1 The Fermat principle
The Fermat3 principle is equivalent to the Huygens’ principle, and states thata wave that propagates in more than one medium, covers the path that requestthe minimum time.
To better understand this principle we can look at the following example.When we are in a point S on a beach and we want to reach a point P in thesea, the fastest path for us is not along the straight line that connects S andP : it is better for us cover a longer distance on the sands, where our speed isfaster.
We try now to demonstrate the law of refraction using this principle. As wecan see in Figure 5.4 we have to move from S to P . We call C the point along
3Pierre de Fermat (1607 - 1665) was a French lawyer in Toulouse, and an amateur math-ematician who is given credit for early developments that led to infinitesimal calculus. Hemade notable contributions to analytic geometry, probability, and optics, and in particular tonumber theory.
68
Figure 5.4: The Fermat’s Principle.
our path in which the medium changes. The total time is given by
t =
√a2 + x2
v1+
√b2 + (c− x)2
v2;
dt
dx=
x
v1
√a2 + x2
− c− xv2
√b2 + (c− x)2
=
=sin θ1
v1− sin θ2
v2.
Imposing the derivative of the function to equal 0, we get
sin θ1
sin θ2=v1
v2=n2
n1.
that is the law of refraction.
69
Chapter 6
Interference and diffraction
Light waves interfere with each other much like mechanical waves do: inter-ference associated with light waves arises when the electromagnetic fields thatconstitute the individual waves combine.
Since light frequency is very high compared to the sensitivity of the humaneye (and of most instruments: the typical frequency of 1015 Hz is very difficultto reach), for interference between two sources of light to be observed, there aretwo conditions which must be met: the sources must be coherent (they mustmaintain a constant phase with respect to each other), and the waves must haveidentical wavelengths.
6.1 Young’s interference experiment
In 1801, Thomas Young1 experimentally proved that light is a wave, contrary towhat most other scientists then thought. He did so by demonstrating that lightundergoes interference, as do water waves, sound waves, and waves of all othertypes. In addition, he was able to measure the average wavelength of sunlight;his value, 570 nm, is impressively close to the modern accepted value of 555 nm.We shall here examine Young’s experiment as an example of the interference oflight waves.
Figure (6.1) shows the basic arrangement of Young’s experiment. Light froma distant monochromatic source of wavelength λ illuminates slit S0 in screen A.The emerging light then spreads via diffraction to illuminate two slits S1 andS2 in screen B (this is done to guarantee coherence of the two sources).
The snapshot of Figure (6.1) depicts the interference of the overlappingwaves. We observe the interference on a viewing screen C intercepting the light.Where it does so, points of interference maxima form visible bright rows-calledbright bands, bright fringes, or (loosely speaking) maxima - that extend across
1Thomas Young (1773 - 1829) was an English medical doctor and polymath. He madenotable scientific contributions to the fields of vision, light, solid mechanics, energy, physiology,language, musical harmony, and in the decipherment of Egyptian hieroglyphs.
70
Figure 6.1: Arrangement of Young’s interference experiment.
the screen. Dark regions - called dark bands, dark fringes, or (loosely speaking)minima - result from fully destructive interference and are visible between ad-jacent pairs of bright fringes (maxima and minima more properly refer to thecenter of a band.) The pattern of bright and dark fringes on the screen is calledan interference pattern. Figure (6.2) is a photograph of part of the interferencepattern that would be seen by an observer standing to the left of screen C at adistance L� d, where d is the distance between S1 and S2.
Where are the dark fringes and the bright fringes located? Let us examinethe difference in optical path between the two rays in Figure (6.3).
When one wave travels an integer number of wavelengths farther than theother, the waves arrive in phase, and a bright fringe occurs. This condition isverified for
δ = r2 − r1 ' d sin θ = mλ
(this assumes the paths are parallel; they are not exactly parallel, but the aboveis a very good approximation since L � d). The absolute value of the integerm is called the order of the maximum.
When destructive interference occurs, a dark fringe is observed. This needsa path difference of an odd half wavelength
δ = r2 − r1 ' d sin θ =
(m+
1
2
)λ .
Let us take a coordinate z on the screen, starting from z = 0 at θ = 0. Then,
71
Figure 6.2: Interference pattern.
Figure 6.3: Difference of path length.
72
Figure 6.4: N-slits interference.
z = L tan θ, and, for small θ, z ' L sin θ. One has thus
zbright ' λL
dm (6.1)
zdark ' λL
d
(m+
1
2
)(6.2)
(with m = 0,±1,±2, ...). Young’s setup is thus effective to measure the wave-length of light, which is amplified by a factor L/d.
What happens in between the maxima and the minima? Let us solve thegeneral problem of the interference of N equally spaced sources in phase, eachof amplitude A (Figure (6.4). Let A(θ) be the common amplitude of each wave(which can be a function of θ due to the attenuation).
The total field Etot as a function of θ is
Etot =
N∑n=1
En =
N∑n=1
A(θ)ei(krn−ωt) = A(θ)ei(kr1−ωt)N∑n=1
eik(n−1)d sin θ . (6.3)
With ζ = eikd sin θ, the sum is a sum of the geometrical series 1 + ζ + ζ2 + ...+
73
ζN−1 = (ζN − 1)/(ζ − 1). Thus
Etot = A(θ)ei(kr1−ωt)eikNd sin θ − 1
eikd sin θ − 1
= A(θ)ei(kr1−ωt)eik(N/2)d sin θ
eik(1/2)d sin θ
eik(N/2)d sin θ − e−ik(N/2)d sin θ
eik(1/2)d sin θ − e−ik(1/2)d sin θ
= A(θ)ei(kr1−ωt)eik((N−1)/2)d sin θ sin (N/2)kd sin θ
sin (1/2)kd sin θ.
The amplitude is thus
Atot(θ) = A(0)sin (N/2)kd sin θ
sin (1/2)kd sin θ≡ A(0)
sin (Nα/2)
sin (α/2)
with
α ≡ kd sin θ =2πd sin θ
λ.
The amplitude at θ = 0 is
Atot(0) = limθ→0
Atot(θ) = NA(0) .
and thusAtot(θ)
Atot(0)=A(θ)
A(0)
sin (Nα/2)
N sin (α/2).
When we go to intensities,
Itot(θ)
Itot(0)=
(A(θ)
A(0)
sin (Nα/2)
N sin (α/2)
)2
.
Even for large angles, the effect of A(θ) is to simply act as an envelopefunction of the oscillating sine functions. We can always bring A(θ) back in ifwe want to, but the more interesting behavior of Atot(θ)) is the oscillatory part.We are generally concerned with the locations of the maxima and minima ofthe oscillations and not with the actual value of the amplitude. We can pose,for the moment, Atot(θ) ' Atot(0).
What does the Itot(θ)/Itot(0) ratio look like as a function of θ? The plot forN = 4 is shown in Figure (6.5). If we are actually talking about small angles,then we have α = kd sin θ ' kdz/D.
6.1.1 The two-slit case
In particular, for the two-slit case,
Itot(θ)
Itot(0)='
(sin (2α/2)
N sin (α/2)
)2
= cos2(α/2) . (6.4)
74
Figure 6.5: 4-slits interference.
6.2 Diffraction
Any wave passing through an opening experiences diffraction. Diffraction meansthat the wave spreads out on the other side of the opening rather than theopening casting a sharp shadow.
Diffraction is most noticeable when the opening is about the same size asthe wavelength of the wave. If light passes through a narrow slit, it produces acharacteristic pattern of light and dark areas called a diffraction pattern, whichcan be explained by the interference of light traveling different optical paths.
Light passing a sharp edge also exhibits a diffraction pattern. Huygens’Principle describes this spreading out, and a Huygens’ construction can be usedto quantify the diffraction phenomenon. For example, Figure (6.6) shows co-herent light incident on an opening, which has dimensions comparable to thewavelength of the light. Rather than casting a sharp shadow, light spreads outon the other side of the opening. We can describe this spreading out by using aHuygens’ construction and assuming that spherical wavelets are emitted at sev-eral points inside the opening. The resulting light waves on the right side of theopening undergo interference and produce a characteristic diffraction pattern.
Light waves can also go around the edges of barriers. In this case, the lightfar from the edge of the barrier continues to travel like the light waves shown inFigure (6.6). The light near the edge of the barrier seems to bend around thebarrier and is described by the sources of wavelets near the edge.
75
Figure 6.6: Diffraction through a single slit.
Figure 6.7: Diffraction through a single slit.
6.2.1 Diffraction through a single slit
Huygens’ principle requires that the waves spread out after they pass throughslits. This spreading out of light from its initial line of travel is called diffraction;in general, diffraction occurs when waves pass through small openings, aroundobstacles or by sharp edges.
Fraunhofer diffraction occurs when the rays leave the diffracting object inparallel directions (screen very far from the slit).
We shall now discuss what happens when a plane wave impinges on justone wide slit with width a instead of a number of infinitesimally thin ones (seeFigure 6.7).
By Huygens’ principle, we can consider the wide slit to consist of an infinitenumber of line sources (or point sources, if we ignore the direction perpendicularto the page) next to each other, each creating a cylindrical wave. In other words,the diffraction pattern from one continuous wide slit is equivalent to the large-N
76
limit of the N−slit result.We shall perform this calculation by an integral over all the phases from
the possible paths from different parts of the wide slit. Let the slit run fromy = −a/2 to y = a/2, and let B(θ)dy be the amplitude that would be presentat a location θ on the screen if only an infinitesimal slit of width dy was open.So B(θ) is the amplitude (on the screen) per unit length (in the slit): B(θ)dy isthe analog of the A(θ) in the case of interference. If we measure the pathlengthsrelative to the midpoint of the slit, then the path that starts at position y isshorter by y sin θ. It therefore has a relative phase of e−ky sin θ.
Integrating over all the paths that emerge from the different values of y(through slits of width dy) gives the total wave at position θ on the screen as(up to an overall phase from the y = 0 point, and ignoring the temporal part ofthe phase)
Etot(θ) =
∫ +a/2
−a/2dy B(θ) e−iky sin θ .
This is the continuous version of the discrete sum in the case of the multi-slitinterference. B(θ) falls off like 1 =
√r; however, we shall assume that θ is small,
which means that we can assume cosθ ' 1.
Etot(θ) ' B(0)
∫ +a/2
−a/2dy e−iky sin θ =
B(0)
−ik sin θ
(e−ik(a/2) sin θ − eik(a/2) sin θ
)= B(0)
−2i sin(ka sin θ
2
)−ik sin θ
= B(0)asin(
12ka sin θ
)12ka sin θ
.
There is no phase here, so this itself is the amplitude Atot(θ). Taking theusual limit at θ = 0, we obtain Atot(0) = B(0)a. Therefore, Atot(θ)/Atot(0) =sin(β/2)/(β/2), where β = ka sin θ. Since the intensity is proportional to thesquare of the amplitude, we arrive at the equation
Itot(θ)
Itot(0)=
(sin(β/2)
β/2
)2
with β = ka sin θ = 2πa sin θ/λ.From Figure (6.8), we see that most of the intensity of the diffraction pattern
is contained within the main bump where β < 2π. Numerically, you can shownthat the fraction of the total area that lies under the main bump is about 90%.So it makes sense to say that the angular width of the pattern is given by
sin θ ' λ
a.
6.2.2 Diffraction grating
The diffracting grating consists of many equally spaced parallel slits. A typicalgrating contains several thousand lines per centimeter.
77
Figure 6.8: The function sin2(β/2)/(β/2)2.
The intensity of the pattern on the screen is the result of the combined effectsof interference and diffraction.
6.2.3 The diffraction limit
Making a perfect lens that produces flawless images has been a dream of lensmakers for centuries. This is however impossible: whenever an object is imagedby an optical system, such as the lens of a camera, or our eyes, fine features arepermanently lost in the image.
Roughly, the diffraction limit is given by the fact that a lens of aperture ahas an intrinsic diffraction of a light of wavelength λ such that the minimumuncertainty δθ on the measurement of angles is, as seen in the previous Section,
δθ ' kλa
where k depends on the geometry of the lens and on the definition of uncertainty,and it is of the order of unity.
For example, the human eye has an opening of a diameter of about 1mm.Thus the minimum angle we can appreciate (let us assume a wavelength of 500nm) is about 500 nm/1mm = 0.5 mrad. The Moon is about d ' 4 × 108 kmaway from us: then, when we look to the Moon, one pixel of our view has – atbest: the atmosphere introduces additional distortions – a typical size of 200km. A perfect 10-m telescope could register pixels of 20 m: size matters inastronomy and optics in general. In any case, we could not take from the Eartha picture of the astronauts on the Moon.
78
Consequences of the invariance of the speed of light: a reminder
• From the principle of relativity it follows that the velocity of propagation of electromagnetic interactions is the same in all inertial systems of reference
• We assume that the velocity of propagation of interactions is a universal constant, and equal to c– Relativity of simultaneity
– Time dilation
– Length contraction
(time and space intervals are not absolute)
More on length contraction
• The observers on the car, whose frame of reference we have chosen to call S’, measure the length of their car to be L’. They verify this by watching the front end of the car and then the rear end of the car pass the center of the platform, and measuring the amount of time t’ that passes between those two events. According to the observers on the car, the car is at rest and the subway platform is moving at a velocity -V relative to the car.
L’ = V t’
t '= t L =L'
• To an observer standing at the center of the platform when the subway car goes by, the car has a length L that can be calculated by measuring the time that passes between the time when the front end of the car reaches the center of the platform and the time when the rear end of the car reaches the center of the platform: L=V t
• The interval t is a proper time interval for the observer standing on the platform, because the two events whose times are being measured both happen at the center of the platform. Therefore the time intervals are related by
Examples of time dilation
and length contraction:
The muon experiment
Think of the muon’s point of view…
The propagation of signal
• Let’s look at two inertial reference systems K and K' with coordinate axes XYZ and X’ Y’ Z’, where the system K' moves relative to K along the X(X') axis.
• Suppose signals start out from some point A on the X' axis in two opposite directions. Since the velocity of propagation of a signal in the K' system, as in all inertial systems, is equal (for both directions) to c, signals will reach points B and C, equidistant from A, at the same time (in the K' system)
• The same two events (arrival of the signal at B and C) can by no means be simultaneous for an observer in the K system. In fact, the velocity of a signal relative to the K system has, according to the principle of relativity, the same value c, and since the point B moves (relative to the K system) toward the source of its signal, while the point C moves in the direction away from the signal (sent from A to C), in the K system the signal will reach point B earlier than point C.
• We shall frequently use the concept of event. An event is described by the place where it occurred and the time when it occurred. Thus an event occurring in a certain material particle is defined by the three coordinates of that particle
• We designate the time in the systems K and K' by t and t'
Intervals
Transformation of intervals
As already shown, if ds = 0 in one inertial system, then ds' = 0 in any other system. On the other hand, ds and ds' are infinitesimals of the same order. From these two conditions it follows that ds2 and ds'2 must be proportional to each other:
Timelike events
Timelike events (2)
Spacelike events
Future and past• Let us take some event O as our origin of time and space coordinates. Let us now consider what relation other events bear to the given event O. For visualization, we shall consider only one space dimension and the time, marking them on two axes.
• Uniform rectilinear motion of a particle, passing through x = 0 at t = 0, is represented by a straight line going through O and inclined to the t axis at an angle whose tangent is the velocity of the particle. Since the maximum possible velocity is c, there is a maximum angle which this line can subtend with the t axis.
• The two lines represent the propagation of signals (with the velocity of light) in opposite directions passing through O (i.e. going through x=0 at t=0). All lines representing the motion of particles can lie only in the regions aOc and dOb. On the lines ab and cd, x = ± ct.
• First consider events whose world points lie within the region aOc. For all the points of this region c2t2-x2 >0. The interval between any event in this region and the event O is timelike. In this region t>0, i.e. all the events in this region occur "after" the event O. But two events which are separated by a timelike interval cannot occur simultaneously in any reference system. Consequently it is impossible to find a reference system in which any of the events in region aOe occurred "before" the event O, i.e. at time t<0. Thus all the events in region aOc are future events relative to O in all reference systems. This region can be called the absolute future relative to O.
Future and past(2)
• In the same way, all events in the region bOd are in the absolute past relative to O; i.e. events in this region occur before the event O in all systems of reference.
• Next consider regions dOa and eOb. The interval between any event in this region and the event O is spacelike. These events occur at different points in space in every reference system. Therefore these regions can be said to be absolutely remote relative to O.
• Note that if we consider all three space coordinates instead of just one, then instead of the two intersecting lines of Fig. 2 we would have a “cone" x2 + y2 + z2 – c2t2 = 0 in the 4-dimensional coordinate system x, y, z. t, the axis of the cone coinciding with the t axis. (This cone is called the light cone.) The regions of absolute future and absolute past are then represented by the two interior portions of this cone.
• Two events can be related causally to each other only if the interval between them is timelike; this follows immediately from the fact that no interaction can propagate with a velocity greater than the velocity of light. As we have just seen, it is precisely for these events that the concepts "earlier" and "later" have an absolute significance, which is a necessary condition for the concepts of cause and effect to have meaning.
The Lorentz transformations
The Lorentz transformations (2)
The Lorentz transformations (3)
The Lorentz transformations (4)
Length contraction, again
Transformation of velocities
Exercises at home
• Are Lorentz transformations commutative?
• Derive time dilation from the Lorentz transformations
Doppler effect in classical physics: General solution
• Numerator – Receiver (observer)– Toward +– Away –
• Denominator – Source– Toward –– Away +
Doppler effect from Lorentz transformations
• Now, the signal transmitted by one observer, and received by another, is a light wave. It makes a difference compared to the situation in Doppler effect with sound waves: there is no medium, and the velocity of the wave (i.e., of light) is the same for both observers. And the the “transmitter” and “receiver” move relative to each other with such a speed that relativistic effects have to be taken into account.
• We will consider the following situation: the “transmitter” is in the frame O that moves away with speed -u (meaning: to the left) from the frame O’ in which the “receiver” located. At a moment the “transmitter” starts to broadcast a light wave.
•
•
• ’
•
t0
t0
'=t'(c + u)
t0
'=c
'=
t'(c + u)
t0
'=
t0
t '
1
1+ u /ct'
t0
=
'=1 u2 /c 2
1+ u /c
=(1 u /c)(1+ u /c)
(1+ u /c)(1+ u /c)
=1 u /c
1+ u /c
4-ve
ctor
s
4-tensors
Unit tensor and metric tensor
The completely antisymmetric tensor of rank 4
Covariant derivatives
xμ μ ;xμ
μ
μ
μ =1
c 2
2
t 22 =
4-dimensional velocity
Relativistic dynamics: mass and energy
E = h
( (
E=mc2
Relativistic momentum
Relativistic momentum
In addition, it is proportional to the space part of the 4-velocity
Energy-momentum vector (4-momentum)
pμ = m uμ = mc (1 , v/c) is instead a 4-vector
pμpμ = m2c2 = E2/c2 – p2
The photon
Relativistic force
Vector potential and scalar potential
By defining Aμ = ( /c , A) and jμ = (c , j) the above equations become
Aμ = 0 jμ
The continuity equation tells that ; thus jμ is a 4-vector.
Also Aμ is then a 4-vector.
•
j +t= 0 μ jμ = 0
Electromagnetism and relativity
The electromagnetic tensor
E i=
0Ai+
iA0 F 0i= E i
Bi=
ˆ 1 ˆ 2 ˆ 3 1 2 3
A1 A2 A3
Covariant expression of Maxwell’s equations
• Non ce ne sarebbe bisogno, ma…
μF μ = μ
μA Aμ( ) = μ
μ( )A μAμ( )
Ma μAμ =t
+
A = 0
μF μ = j
Invariants constructed from the electromagnetic tensor
Lorentz force: covariant expression
Lorentz force: covariant expression
Motion of a particle under constant force
qeEu =dp0
dsdp0
dt = ddt
m
1 v 2
= Fv
Decay of an unstable particle
Elastic collision (1 + 2 1’+2’)
Elastic collision: target at rest
Compton scattering• If the incident particle 1 has 0 mass (photon)
cos =E '1 E1 + m(E '1 E1)
E '1 E1
=1+ m1E1
1E '1
=1 m1
h '
1
h
=1
mc
h'( )
1
1
The fall of Classical Physics
2
Classical physics: Fundamental Models
Particle Model (particles, bodies)
Motion in 3 dimension; for each time t, position and speed
are known (they are well-defined numbers, regardless we
know them). Mass is known.
Systems and rigid objects
Extension of particle model
Wave Model (light, sound, …)
Generalization of the particle model: energy is transported,
which can be spread (de-localized)
Interference
2
3
Classical physics at the end of XIX Century
Scientists are convinced that the particle and wave model can describe the
evolution of the Universe, when folded with
Newton’s laws (dynamics)
Description of forces
Maxwell’s equations
Law of gravity.
…
We live in a 3-d world, and motion happens in an absolute time. Time and
space (distances) intervals are absolute.
The Universe is homogeneous and isotropical; time is homogeneous.
Relativity
The physics entities can be described either in the particle or in the wave
model.
Natura non facit saltus (the variables involved in the description are
continuous).
4
Something is wrong
Relativity, continuity, wave/particle (I)
Maxwell equations are
not relativistically
covariant!
Moreover, a series of
experiments seems to
indicate that the
speed of light is
constant (Michelson-
Morley, …) A speed!
3
5
Something is wrong
Relativity, continuity, wave/particle (IIa)
In the beginning of the XX
century, it was known that
atoms were made of a
heavy nucleus, with
positive charge, and by
light negative electrons
Electrostatics like gravity:
planetary model
All orbits allowed
But: electrons, being
accelerated, should radiate
and eventually fall into the
nucleus
6
Something is wrong
Relativity, continuity, wave/particle (IIb)
If atoms emit energy in the form of photons due to
level transitions, and if color is a measure of energy,
they should emit at all wavelengths – but they don’t
4
7
Something is wrong
Relativity, continuity, wave/particle (III)
Radiation has a particle-like behaviour, sometimes
Particles display a wave-like behaviour, sometimes
=> In summary, something wrong involving the
foundations:
Relativity
Continuity
Wave/Particle duality
8
Need for a new physics
A reformulation of physics was needed
This is fascinating!!! Involved philosophy, logics, contacts
with civilizations far away from us…
A charming story in the evolution of mankind
But… just a moment… I leaved up to now with classical
physics, and nothing bad happened to me!
Because classical physics fails at very small scales, comparable with
the atom’s dimensions, 10-10 m, or at speeds comparable with the
speed of light, c ~ 3 108 m/s
Under usual conditions, classical physics makes a good job.
Warning: What follows is logically correct, although
sometimes historically inappropriate.
5
9
I
Light behaves like a particle,
sometimes
10
Photoelectric Effect Features
and Photon Model explanation
The experimental results contradict all four classical
predictions
Einstein interpretation: All electromagnetic radiation can be
considered a stream of quanta, called photons
A photon of incident light gives all its energy hƒ to a single
electron in the metal
h is called the Planck constant, and plays a
fundamental role in Quantum Physics
6
11
The Compton Effect
Compton dealt with Einstein’s idea of
photon momentum
Einstein: a photon with energy E carries a
momentum of E/c = hƒ / c
According to the classical theory,
electromagnetic waves of frequency ƒo
incident on electrons should scatter,
keeping the same frequency – they
scatter the electron as well…
12
Compton’s experiment showed that, at
any given angle, a different frequency of
radiation is observed
The graphs show the scattered x-ray for
various angles
Again, treating the photon as a particle of
energy hf explains the phenomenon. The
shifted peak, ‘> 0, is caused by the
scattering of free electrons
This is called the Compton shift equation
7
13
Compton Effect, Explanation
The results could be explained, again, by treating the
photons as point-like particles having
energy hƒ
momentum hƒ / c
Assume the energy and momentum of the isolated
system of the colliding photon-electron are conserved
Adopted a particle model for a well-known wave
The unshifted wavelength, o, is caused by x-rays
scattered from the electrons that are tightly bound to
the target atoms
The shifted peak, ', is caused by x-rays scattered
from free electrons in the target
14
Every object at T > 0 radiates electromagnetically, and
absorbes radiation as well
Stefan-Boltzmann law:
Blackbody: the
perfect absorber/emitter
Blackbody radiation
“Black” body
Classical interpretation: atoms in the object vibrate; since <E> ~
kT, the hotter the object, the more energetic the vibration, the
higher the frequency
The nature of the radiation leaving the cavity through the
hole depends only on the temperature of the cavity walls
8
15
Experimental findings
& classical calculation
Wien’s law: the emission
peaks at
Example: for Sun T ~ 6000K
But the classical calculation
(Rayleigh-Jeans) gives a
completely different result…
Ultraviolet catastrophe
16
Experimental findings
& classical calculation
Classical calculation (Raileigh-
Jeans): the blackbody is a set of
oscillators which can absorb any
frequency, and in level transition
emit/absorb quanta of energy:
No maximum; a ultraviolet
catastrophe should absorb all
energy Experiment
9
17
Planck’s hypothesis
Only the oscillation modes for which
E = hf
are allowed…
18
Interpretation
The classical calculation is accurate for large
wavelengths, and is the limit for h -> 0
Elementary oscillators can have only
quantized energies, which satisfy
E=nhf (h is an universal constant, n is
an integer –quantum- number)
Transitions are accompanied by the
emission of quanta of energy (photons)
n
4
3
2
1
E
4hf
3hf
2hf
hf
10
19
Which lamp emits e.m. radiation ?
1) A
2) B
3) A & B
4) None
20
Particle-like behavior of light:
now smoking guns…
The reaction
has been
recorded
millions of
times…
11
21
Bremsstrahlung
"Bremsstrahlung" means in German
"braking radiation“; it is the radiation
emitted when electrons are decelerated or
"braked" when they are fired at a metal
target. Accelerated charges give off
electromagnetic radiation, and when the
energy of the bombarding electrons is
high enough, that radiation is in the x-ray
region of the electromagnetic spectrum. It
is characterized by a continuous
distribution of radiation which becomes
more intense and shifts toward higher
frequencies when the energy of the
bombarding electrons is increased.
22
Summary
The wave model cannot explain the behavior of light
in certain conditions
Photoelectric effect
Compton effect
Blackbody radiation
Gamma conversion/Bremsstrahlung
Light behaves like a particle, and has to be
considered in some conditions as made by single
particles (photons) each with energy
h ~ 6.6 10-34 Js is called the Planck’s constant
12
23
II
Particles behave like waves,
sometimes
24
Should, symmetrically, particles display
radiation-like properties?
The key is a diffraction experiment: do particles show
interference?
A small cloud of Ne atoms was cooled down to T~0. It
was then released and fell with zero initial velocity onto
a plate pierced with two parallel slits of width 2 μm,
separated by a distance of d=6 μm. The plate was
located H=3.5 cm below the center of the laser trap.
The atoms were detected when they reached a screen
located D=85 cm below the plane of the two slits. This
screen registered the impacts of the atoms: each dot
represents a single impact. The distance between two
maxima, y, is 1mm.
The diffraction pattern is consistent with the diffraction
of waves with
13
25
Diffraction of electrons
Davisson & Germer 1925:
Electrons display diffraction patterns !!!
26
de Broglie’s wavelength
What is the wavelength associated to a particle?
de Broglie’s wavelength:
Explains quantitatively the diffraction by Davisson and Germer……
Note the symmetry
What is the wavelength of an electron moving at 107 m/s ?
(smaller than an atomic length; note the dependence on m)
14
27
Atomic spectra
Why atoms emit according to a discrete energy spectrum?
Something must
be there...
Balmer
28
Electrons in atoms: a semiclassical model
Similar to waves on a cord, let’s imagine that
the only possible stable waves are stationary…
2 r = n n=1,2,3,…
=> Angular momentum is quantized (Bohr
postulated it…)
15
29
v
r
m
F
NB:
• In SI, ke = (1/4 0) ~ 9 x 109 SI units
• Total energy < 0 (bound state)
• <Ek> = -<Ep/2> (true in general for bound states, virial theorem)
Only special values are possible for the radius !
Hydrogen (Z=1)
30
Energy levels
The radius can only assume
values
The smallest radius (Bohr’s radius) is
Radius and energy are related:
And thus energy is quantized:
16
31
Transitions
An electron, passing from an orbit of energy Ei
to an orbit with Ef < E
i, emits energy [a photon
such that f = (Ei-E
f)/h]
32
Level transitions and energy quanta
We obtain Balmer’s relation!
17
33
Limitations
Semiclassical models wave-particle duality can explain
phenomena, but the thing is still insatisfactory,
When do particles behave as particles, when do they behave
as waves?
Why is the atom stable, contrary to Maxwell’s equations?
We need to rewrite the fundamental models, rebuilding
the foundations of physics…
34
Wavefunction
Change the basic model!
We can describe the position of a particle
through a wavefunction (r,t). This can account
for the concepts of wave and particle (extension
and simplification).
Can we simply use the D’Alembert waves, real
waves? No…
18
35
Wavefunction - II
We want a new kind of “waves” which can account for particles, old waves,
and obey to F=ma.
And they should reproduce the characteristics of “real” particles: a
particle can display interference corresponding to a size of 10-7 m, but
have a radius smaller than 10-10 m
Waves of what, then? No more of energy,
but of probability
The square of the wavefunction is the intensity, and it gives the
probability to find the particle in a given time in a given place.
Waves such that F=ma? We’ll see that they cannot be a function in R,
but that C is the minimum space needed for the model.
36
SUMMARY
Close to the beginning of the XX century, people thought that
physics was understood. Two models (waves, particles). But:
Quantization at atomic level became experimentally evident
Particle-like behavior of radiation: radiation can be considered in some
conditions as a set of particles (photons) each with energy
Wave-like property of particles: particles behave in certain condistions as
waves with wavenumber
Role of Planck’s constant, h ~ 6.6 10-34 Js
Concepts of wave and particle need to be unified: wavefunction
(r,t).
19
37
L’equazione di Schroedinger
38
Proprieta’ della funzione d’onda
20
39
L’equazione di S.
1
L1 – The Schroedinger
equation
• A particle of mass m on the x-axis is subject to
a force F(x, t)
• The program of classical mechanics is to
determine the position of the particle at any
given time: x(t). Once we know that, we can
figure out the velocity v = dx/dt, the momentum
p = mv, the kinetic energy T = (1/2)mv2, or any
other dynamical variable of interest.
• How to determine x(t) ? Newton's second law:
F = ma.
– For conservative systems - the only kind that occur
at microscopic level - the force can be expressed as
the derivative of a potential energy function, F = -dV/
dx, and Newton's law reads m d2x/dt2 = -dV/ dx
– This, together with appropriate initial conditions
(typically the position and velocity at t = 0),
determines x(t).
2
• A particle of mass m, moving along the
x axis, is subject to a force
F(x, t) = -dV/ dx
• Quantum mechanics approaches this
same problem quite differently. In this
case what we're looking for is the wave
function, (x, t), of the particle, and we
get it by solving the Schroedinger
equation:
• In 3 dimensions,
~ 10-34 Js
3
The statistical interpretation
• What is this "wave function", and what can it tell you?
After all, a particle, by its nature, is localized at a point,
whereas the wave function is spread out in space (it's
a function of x, for any given time t). How can such an
object be said to describe the state of a particle?
• Born's statistical interpretation:
Quite likely to find the particle near A, and relatively
unlikely near B.
• The statistical interpretation introduces a kind of
indeterminacy into quantum mechanics, for even if you
know everything the theory has to tell you about the
particle (its wave function), you cannot predict with
certainty the outcome of a simple experiment to
measure its position
– all quantum mechanics gives is statistical information
about the possible results
• This indeterminacy has been profoundly disturbing
4
Realism, ortodoxy,
agnosticism - 1
• Suppose I measure the position of the particle,
and I find C. Question: Where was the particle
just before I made the measurement?
There seem to be three plausible answers to
this question…
1. The realist position: The particle was at C. This
seems a sensible response, and it is the one
Einstein advocated. However, if this is true QM
is an incomplete theory, since the particle
really was at C, and yet QM was unable to tell
us so. The position of the particle was never
indeterminate, but was merely unknown to the
experimenter. Evidently is not the whole
story: some additional information (a hidden
variable) is needed to provide a complete
description of the particle
5
Realism, ortodoxy,
agnosticism - 2
• Suppose I measure the position of the particle,
and I find C. Question: Where was the particle
just before I made the measurement?
2. The orthodox position: The particle wasn't
really anywhere. It was the act of
measurement that forced the particle to "take a
stand“. Observations not only disturb what is to
be measured, they produce it .... We compel
the particle to assume a definite position. This
view (the so-called Copenhagen interpretation)
is associated with Bohr and his followers.
Among physicists it has always been the most
widely accepted position. Note, however, that
if it is correct there is something very peculiar
about the act of measurement - something that
over half a century of debate has done
precious little to illuminate.
6
Realism, ortodoxy,
agnosticism - 3
• Suppose I measure the position of the particle,
and I find C. Question: Where was the particle
just before I made the measurement?
3. The agnostic position: Refuse to answer. This
is not as silly as it sounds - after all, what
sense can there be in making assertions about
the status of a particle before a measurement,
when the only way of knowing whether you
were right is precisely to conduct a
measurement, in which case what you get is
no longer "before the measurement"? It is
metaphysics to worry about something that
cannot, by its nature, be tested. One should
not think about the problem of whether
something one cannot know anything about
exists
7
Realism, ortodoxy,
or agnosticism?
• Suppose I measure the position of the particle, and I find C. Question: Where was the particle just before I made the measurement?
• Until recently, all three positions had their
partisans. But in 1964 John Bell demonstrated
that it makes an observable difference if the
particle had a precise (though unknown)
position prior to the measurement. Bell's
theorem made it an experimental question
whether 1 or 2 is correct. The experiments
have confirmed the orthodox interpretation: a
particle does not have a precise position prior
to measurement; it is the measurement that
insists on one particular number, and in a
sense creates the specific result, statistically
guided by the wave function.
• Still some agnosticism is tolerated…
8
Collapse of
the
wavefunction
• Suppose I measure the position of the particle, and I find C. Question: Where will be the particle immediately after?
• Of course in C. How does the
orthodox interpretation explain
that the second measurement
is bound to give the value C?
Evidently the first
measurement radically alters
the wave function, so that it is
now sharply peaked about C.
The wave function collapses
upon measurement (but soon
spreads out again, following
the Schroedinger equation, so
the second measurement
must be made quickly). There
are, then, two entirely distinct
kinds of physical processes:
"ordinary", in which evolves
under the Schroedinger
equation, and
"measurements", in which
suddenly collapses.
9
Normalization
• | (x, t)|2 is the probability density for finding
the particle at point x at time t.
The integral of | (x, t)|2 over space must be 1
(the particle has to be somewhere).
• The wave function is supposed to be
determined by the Schroedinger equation--we
can't impose an extraneous condition on
without checking that the two are consistent.
• Fortunately, the Schroedinger equation is
linear: if is a solution, so too is A , where
A is any (complex) constant. What we must
do, then, is pick this undetermined
multiplicative factor so that The integral of |
(x, t)|2 over space must be 1 This process is
called normalizing the wave function.
• Physically realizable states correspond to the
"square-integrable" solutions to Schroedinger's
equation.
10
Will a normalized function stay as such?
11
Expectation values
• For a particle in state , the expectation value
of x is
• It does not mean that if you measure the
position of one particle over and over again,
this is the average of the results
– On the contrary, the first measurement (whose
outcome is indeterminate) will collapse the wave
function to a spike at the value obtained, and the
subsequent measurements (if they're performed
quickly) will simply repeat that same result.
• Rather, <x> is the average of measurements
performed on particles all in the state , which
means that either you must find some way of
returning the particle to its original state after
each measurement, or you prepare an
ensemble of particles, each in the same state
, and measure the positions of all of them:
<x> is the average of them.
12
Momentum
13
More on operators
• One could also simply observe that
Schroedinger’s equations works as if
(exercise: apply on the plane wave). In
3 dimensions,
Compound operators
• Kinetic energy is
14
Angular momentum
15
Exercise
• A particle is represented at t=0 by the wavefunction
(x, 0) = A(a2-x2) |x| < a (a>0).
= 0 elsewhere
a Determine the normalization constant A
b, c What is the expectation value for x and for p at t=0?
16
Exercise (cont.)
• A particle is represented at t=0 by the wavefunction
(x, 0) = A(a2-x2) |x| < a (a>0).
= 0 elsewhere
d, e Compute <x2>, <p2>
f, g Compute the uncertainty on x, p
h Verify the uncertainty principle in this case
17
L2 – The time-independent
Schroedinger equation
• Supponiamo che il potenziale U sia
indipendente dal tempo
18
19
, soluzione della prima equazione (eq.agli autovalori detta
equazione di S. stazionaria), e’ detta autofunzione
20
1. They are stationary states. Although
the wave function itself
does (obviously) depend on t, the
probability density does not - the time
dependence cancels out. The same
thing happens in calculating the
expectation value of any dynamical
variable
3 comments on the stationary solutions: 1
21
2. They are states of definite energy. In
mechanics, the total energy is called the
Hamiltonian:
H(x, p) = p2/2m + V(x).
The corresponding Hamiltonian operator,
obtained by the substitution p -> p operator
Note: it is true in general that, if is
eigenfunction of an operator, the measurement
gives as a result certainly the eigenvalue
3 comments on the stationary solutions: 2
22
3. They are a basis. The general solution
is a linear combination of separable
solutions. The time-independent
Schroedinger equation might yield an
infinite collection of solutions, each with
its associated value of the separation
constant; thus there is a different wave
function for each allowed energy.
The S. equation is linear: a linear
combination of solutions is itself a
solution.
It so happens that every solution to the
(time-dependent) S. equation can be
written as a linear combination of
stationary solutions.
To really play the game, mow we must
input some values for V
3 comments on the stationary solutions: 3
23
Free particle (V=0, everywhere)
24
The infinite square well
25
Infinite square well, 2
26
Infinite square well, 3
But…
27
Infinite square well, 4
11/6/12
1
Intermezzo: Heisenberg principle (theorem)
10
11
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2
12
The uncertainty principle (qualitative)
Imagine that you're holding one end of a long rope, and you generate a
wave by shaking it up and down rhythmically.
Where is that wave? Nowhere, precisely - spread out over 50 m or so.
What is its wavelength? It looks like ~6 m
By contrast, if you gave the rope a sudden jerk you'd get a relatively narrow
bump traveling down the line. This time the first question (Where precisely is
the wave?) is a sensible one, and the second (What is its wavelength?)
seems difficult - it isn't even vaguely periodic, so how can you assign a
wavelength to it?
=> Uncertainty is a characteristic of the wave representation
x(m)
x(m)
13
The more precise a wave's x is, the less precise is , and vice versa. A
theorem in Fourier analysis makes this rigorous…
This applies to any wave, and in particular to the QM wave function. is
related to p by the de Broglie formula
Thus a spread in corresponds to a spread in p, and our observation says
that the more precisely determined a particle's position is, the less precisely
is p
This is Heisenberg's famous uncertainty principle. (we'll prove it later, but I
want to anticipate it now)
The uncertainty principle (qualitative, II)
x(m)
x(m)
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General rules for the linear piece-wise potential
Divide the interval in N regions with constant
potential Vi
For each region, solve the Schroedinger equation
Real exponentials for E <Vi; imaginary otherwise
Impose boundary/initial conditions
Impose continuity conditions
14
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18
Finite Square-Well Potential
The finite square-well potential is
The Schrödinger equation outside the finite well in regions I and III is
or using
yields . Considering that the wave function must be zero at
infinity, the solutions for this equation are
19
Inside the square well, where the potential V is zero, the wave equation
becomes where
Instead of a sinusoidal solution we have
The boundary conditions require that
and the wave function must be smooth where the regions meet.
Note that the
wave function is
nonzero outside
of the box.
Finite Square-Well Solution
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20
Penetration Depth
The penetration depth is the distance outside the potential well where
the probability significantly decreases. It is given by
It should not be surprising to find that the penetration distance that
violates classical physics is proportional to Planck’s constant.
E t (V0 E)m x
p= (V0 E)
m
2m(V0 E) 2m(V0 E) 2
21
Barriers and Tunneling
Consider a particle of energy E approaching a potential barrier of height V0 and
the potential everywhere else is zero.
We will first consider the case when the energy is greater than the potential
barrier.
In regions I and III the wave numbers are:
In the barrier region we have
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22
Reflection and Transmission
The wave function will consist of an incident wave, a reflected wave, and a
transmitted wave.
The potentials and the Schrödinger wave equation for the three regions are
as follows:
The corresponding solutions are:
As the wave moves from left to right, we can simplify the wave functions to:
23
Probability of Reflection and Transmission
The probability of the particles being reflected R or transmitted T is:
Because the particles must be either reflected or transmitted we have:
R + T = 1.
By applying the boundary conditions x ± , x = 0, and x = L, we arrive
at the transmission probability:
Notice that there is a situation in which the transmission probability is 1.
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24
Tunneling
Now we consider the situation where classically the particle does not have enough
energy to surmount the potential barrier, E < V0.
The quantum mechanical result is one of the most remarkable features of modern
physics, and there is ample experimental proof of its existence. There is a small,
but finite, probability that the particle can penetrate the barrier and even emerge
on the other side.
The wave function in region II becomes
The transmission probability that
describes the phenomenon of tunneling is
25
Uncertainty Explanation
Consider when L >> 1 then the transmission probability becomes:
This violation allowed by the uncertainty principle is equal to the
negative kinetic energy required! The particle is allowed by quantum
mechanics and the uncertainty principle to penetrate into a classically
forbidden region. The minimum such kinetic energy is:
1
QM in 3 dimensions
2
V(r): separation of variables
3
ˆ =2
,2 ,
ˆ 2 Y = l l 1Y
ˆ 2 = l l 1
ˆ
2.
4
The angular equation
5
The angular equation -
ˆ = i ,
ˆ = m ˆ = m
ˆ
.
6
The angular equation -
Pl are the Legendre polynomials, defined by the Rodrigues formula:
7
8
ˆ = l(l +1)
9
Spherical harmonics
= (-1)m for m>=0 and =1 for m<0. The Y are orthogonal, so
10
The radial equation
11
The H atom
u(r) = rR(r)
12
Asymptotic behavior
u(r) = rR(r)
13
The radial solution
u(r) = rR(r)
n > l
is the q-th Laguerre polynomial.
Remember: only the solutions for n>l are valid functions
14
The radial
solution: energy
15
Ground state
16
n > 1
17
n > 1 (continued)
Since they are eigenvectors for different eigenvalues
18
Example
19
Graphs
20
Angular momentum
21
22
