Modern Physicslecture X
Louis de Broglie1892 - 1987
Wave Properties of Matter In 1923 Louis de Broglie postulated that perhaps matter In 1923 Louis de Broglie postulated that perhaps matter
exhibits the same “duality” that light exhibitsexhibits the same “duality” that light exhibits Perhaps all matter has both characteristics as wellPerhaps all matter has both characteristics as well Previously we saw that, for photons,Previously we saw that, for photons,
h
c
hf
c
Ep
mv
h
p
h
Which says that the wavelength of light is related to its Which says that the wavelength of light is related to its momentummomentum
Making the same comparison for matter we find…Making the same comparison for matter we find…
Quantum mechanics
Wave-particle dualityWave-particle duality Waves and particles have interchangeable propertiesWaves and particles have interchangeable properties This is an example of a system with This is an example of a system with complementary complementary
propertiesproperties
The mechanics for dealing with systems The mechanics for dealing with systems when these properties become important is when these properties become important is called “Quantum Mechanics”called “Quantum Mechanics”
The Uncertainty Principle
Measurement disturbes the system
The Uncertainty Principle Classical physicsClassical physics
Measurement uncertainty is due to limitations of the measurement Measurement uncertainty is due to limitations of the measurement apparatusapparatus
There is no limit in principle to how accurate a measurement can There is no limit in principle to how accurate a measurement can be madebe made
Quantum MechanicsQuantum Mechanics There is a fundamental limit to the accuracy of a measurement There is a fundamental limit to the accuracy of a measurement
determined by the determined by the HeisenbHeisenbeerg uncertainty principlerg uncertainty principle If a measurement of position is made with precision If a measurement of position is made with precision x and a x and a
simultaneous measurement of linear momentum is made with simultaneous measurement of linear momentum is made with precision precision ppxx, then the product of the two uncertainties can never , then the product of the two uncertainties can never be less than h/4be less than h/4
2/ xpx
Energy and time
2/ E
Uncertainty principle
The Uncertainty Principle In other words:In other words:
It is physically impossible to measure simultaneously the exact It is physically impossible to measure simultaneously the exact position and linear momentum of a particle position and linear momentum of a particle
These properties are called “complementary”These properties are called “complementary” That is only the value of one property can be known at a timeThat is only the value of one property can be known at a time Some examples of complementary properties are Some examples of complementary properties are
Which way / Interference in a double slit experimentWhich way / Interference in a double slit experiment Position / Momentum (Position / Momentum (xxp > h/4p > h/4)) Energy / Time (Energy / Time (EEt > h/4t > h/4)) Amplitude / PhaseAmplitude / Phase
Erwin Schrödinger1887 - 1961
Wave equations for probabilities In 1926 Erwin Schroedinger proposed a wave In 1926 Erwin Schroedinger proposed a wave
equation that describes how matter waves (or the equation that describes how matter waves (or the wave function) propagate in space and timewave function) propagate in space and time
The wave function contains all of the information The wave function contains all of the information that can be known about a particlethat can be known about a particle
)(
222
2
UEm
dx
d
Wave Function In quantum mechanics, matter waves are In quantum mechanics, matter waves are
described by a complex valued described by a complex valued wave functionwave function, , The absolute square gives the probability of The absolute square gives the probability of
finding the particle at some point in spacefinding the particle at some point in space
This leads to an interpretation of the double slit This leads to an interpretation of the double slit experimentexperiment
*2
Wave functions The wave function of a free particle moving The wave function of a free particle moving
along the x-axis is given byalong the x-axis is given by
This represents a snap-shot of the wave This represents a snap-shot of the wave function at a particular time function at a particular time
We cannot, however, measure We cannot, however, measure , we can , we can only measure |only measure |||22, the , the probability densityprobability density
kxAx
Ax sin2
sin
Max Born
Interpretation of the Wavefunction Max Born suggested that Max Born suggested that was the was the PROBABILITY PROBABILITY
AMPLITUDE AMPLITUDE of finding the particle per unit volumeof finding the particle per unit volume ThusThus
||||22dVdV = =dVdV ((designates complex conjugatedesignates complex conjugate)) is the probability of is the probability of
finding the particle within the volume finding the particle within the volume dVdV The quantityThe quantity | |||22is calledis called the PROBABILITY the PROBABILITY
DENSITYDENSITY Since the chance of finding the particle somewhere in Since the chance of finding the particle somewhere in
space is unity we havespace is unity we have
12
dVψdVψ*ψ
• When this condition is satisfied we say that the wavefunction is NORMALISED
0
2sin)()(
d
xAx
A particle or a wave?
small px small and x big
big px big and x small
Schrödinger Wave Equation The Schrödinger wave equation is one of the most The Schrödinger wave equation is one of the most
powerful techniques for solving problems in powerful techniques for solving problems in quantum physicsquantum physics
In general the equation is applied in three In general the equation is applied in three dimensions of space as well as timedimensions of space as well as time
For simplicity we will consider only the one For simplicity we will consider only the one dimensional, time independent casedimensional, time independent case
The wave equation for a wave of displacement The wave equation for a wave of displacement yy and velocity and velocity vv is given by is given by
2
2
22
2 1
t
y
vx
y
Solution to the Wave equation
We consider a trial solution by substitutingWe consider a trial solution by substituting
yy ((xx,, tt ) = ) = ((xx ) sin() sin( tt ))
into the wave equationinto the wave equation
2
2
22
2 1
t
y
vx
y
• By making this substitution we find that
ψv
ω
x
ψ2
2
2
2
• Where /v = 2/ and p = h/• Thus
/v 2(2/)
Energy and the Schrödinger Equation Consider the total energyConsider the total energy
Total energy Total energy EE = Kinetic energy + Potential Energy = Kinetic energy + Potential Energy EE = = mm vv
22/2/2 ++UU EE = = pp
/(2/(2mm )) ++UU Reorganise equation to giveReorganise equation to give
pp 22
== 22 mm ((EE - - UU )) From equation on previous slide we get From equation on previous slide we get
UEm
v
ω
22
2 2
• Going back to the wave equation we have
02
22
2
ψUEm
x
ψ
• This is the time-independent Schrödinger wave equation in one dimension
Solution to the SWE The solutions The solutions ((xx) are called the STATIONARY STATES of the ) are called the STATIONARY STATES of the
systemsystem The equation is solved by imposing BOUNDARY CONDITIONSThe equation is solved by imposing BOUNDARY CONDITIONS The imposition of these conditions leads naturally to energy levelsThe imposition of these conditions leads naturally to energy levels If we set If we set
r
e
πεU
2
04
1
We get the same results as Bohr for the energy levels of the one electron atomThe SWE gives a very general way of solving problems in quantum physics
Probability and Quantum Physics In quantum physics (or quantum mechanics) we deal with In quantum physics (or quantum mechanics) we deal with
probabilities of particles being at some point in space at probabilities of particles being at some point in space at some timesome time
We cannot specify the precise location of the particle in We cannot specify the precise location of the particle in space and timespace and time
We deal with averages of physical propertiesWe deal with averages of physical properties Particles passing through a slit will form a diffraction patternParticles passing through a slit will form a diffraction pattern Any given particle can fall at any point on the receiving Any given particle can fall at any point on the receiving
screenscreen It is only by building up a picture based on many It is only by building up a picture based on many
observations that we can produce a clear diffraction patternobservations that we can produce a clear diffraction pattern
Wave Mechanics We can solve very simple problems in quantum physics We can solve very simple problems in quantum physics
using the SWEusing the SWE This is sometimes called WAVE MECHANICSThis is sometimes called WAVE MECHANICS There are very few problems that can be solved exactlyThere are very few problems that can be solved exactly Approximation methods have to be usedApproximation methods have to be used The simplest problem that we can solve is that of a particle The simplest problem that we can solve is that of a particle
in a boxin a box This is sometimes called a particle in an infinite potential This is sometimes called a particle in an infinite potential
wellwell This problem has recently become significant as it can be This problem has recently become significant as it can be
applied to laser diodes like the ones used in CD playersapplied to laser diodes like the ones used in CD players
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