Ch3 Introduction to quantum mechanics

• De Broglie waves

• Davission-Germer experiment

• Uncertainty principles

• Wave function

• Barrier penetration

• Hydrogen atom

• Lasers

Some words & terms• Wave particle duality• Wavelength• Electron diffraction• De Broglie wave• Quantum• Uncertainty principle• Wave function• Schrodinger equation• Photon, electron• Sinusoidal, Exponential

• Probability density• Potential well• Barrier penetration (势垒穿透 )

• tunnel• Scanning tunnel microscope

(STM)• Operator• Eigenfuction, Eigenvalue• Laser• Spontaneous (simulated) emi

ssion

Wave-particle duality of light• The dual wave-particle nature of light is no

w a basic part of the theory of light and matter.

• Wave feature: υ, λ; • Particle feature: E, p;• Photons

– energy: E=hυ;– momentum: p=h/λ, k=2pi/λ;

De Broglie waves

• In1924, Louis de Broglie proposed that matter possesses wave as well as particle characteristics, receiving the Nobel Prize in 1929.

• A moving body behaves in certain ways as though it has a wave nature.– Photon wavelength:

– De Broglie wavelength:p

hλ

h

c

hvp

2

2

1 0

cv

mm

mv

hλ

Which property is more important?

• Problem: Find the de Broglie wavelengths of a 46g golf ball with a speed of 30m/s and an electron with a speed of 107m/s.

• For a 46g golf ball,

• For an electron,

– Radius of the hydrogen atom, 5.3x10-11m.

mmv

hmmcv 34

0 108.4 , ,

mmv

hmm 11

0 103.7 ,

The diffraction of electrons

• In 1927, Davisson and Germer showed that electron beams are diffracted when they are scattered by the regular atomic arrays of crystals.

• A beam of electrons from a heated filament (灯丝 ) is accelerated through a potential difference V. Passing through a small aperture (孔 ), the beam strikes a single crystal of nickel (镍 ). Electrons are scattered in all directions by atoms of the crystal.

The Bragg equation

• Instead of a continuous variation of scattered electron intensity with angle, distinct maximum and minimum were observed with their positions depending on the electron energy.

• The Bragg equation for maxima in the diffraction pattern is:

• In a particular case, a beam of 54eV electrons were directed perpendicularly at the nickel target and a sharp maximum in the electron distribution occurred at an angle of 50o with the original beam. Here θ=650, d=0.091nm, n=1, λ=0.166nm. 54eV0.166nm

sin2dn

Electron microscopes• The wave nature of moving electrons is the basis of

the electron microscope, the first of which was built in 1932.

• Fast electrons have wavelengths very much shorter than those of visible light. For example, an electron with 54eV (4.4x106m/s) has the wavelength of 0.166nm.

• In an electron microscope, current-carrying coils produce magnetic fields that act as lenses to focus an electron beam on a specimen. A magnification of over 1,000,000 can be reached.

The uncertainty principle• The principle describes a natural limit to the pre

cision of simultaneous measurements of the position and momentum of particles.

• If an object is said be at position of x with an uncertainty of Δx, then any simultaneous measurement of the x component of momentum must have an uncertainty Δpx consistent with

• It also implies that either wave or particle properties, but not both, may be observed in a given experiment.

hpx x

The uncertainty relations

• Heisenberg uncertainty relation for position and momentum.– It can be used to estimate the size and the

energy of the hydrogen atom in its ground state.

• Heisenberg uncertainty relation for time and energy.– It is used to study the lifetime of unstable

systems.

hpx x

hEt

Wave function• Each particle is represented by a wave functio

n, and the square of its absolute magnitude |ψ|2 evaluated at a particular place at a particular time is proportional to the probability of finding the body there at that time. Probability density.

• The probability density |ψ|2 is given by the product of ψ and its complex conjugate ψ*.

• The wave function must be single-valued, superposable, with finite partial derivatives.

Schrodinger equation

• The wave function is governed by Schrodinger equation, which is a type of equation known as eigenvalue equation. Solutions can only be obtained for certain values of energy known as energy eigenvalues.

EVm

22

2

Particle in a box

• The solution to Schrodinger equation for a particle trapped in a box is just a series of standing de Broglie waves.

L

xn

Lxn

mLE

nnL

Lxx

LxxV

n

sin2

)( ,2

...3,2,1 2

,0,,

0,,0)(

222

Barrier penetration (tunneling)• In quantum mechanics, particles can sometimes be

found in or pass through regions (E<V) that are forbidden by energy conservation. This is called barrier penetration.

• Wave function in this barrier potential takes the form:– Sinusoidal oscillation in the region x<0;– Exponentials in the region 0<=x<=a,– Sinusoidal oscillation in the region x>a;

aEVmeP

)(22

a

V

Scanning tunneling microscope

• It is a remarkable application of tunneling.• The instrument mechanically tracks the electronic

wave functions outside the surface of a material, producing an extremely accurate representation of the atoms at the surface.

• When a thin metal probe is brought very close to the surface of the material. The gap between two surfaces forms the barrier. The electrons from the metal tunnel through the barrier into the probe and creates a current.

• The probe moves horizontally along the surface to maintain the same current. The map of vertical positions of the probe forms the surface structure.

probematerial

E

Hydrogen atom (I)• In a hydrogen atom, the electron’s motion is restr

icted by the inverse square electric field of nucleus. The electron is free to move in three dimensions.

• Since the electrical potential is a function of r rather than of x,y,z, Schrődinger’s equation is expressed in terms of spherical polar coordinates r,θ,φ.– r: the length of radius vector from origin O to point P;

– θ: zenith angle, angle between radius vector and +z axis,

– φ: azimuth angle, angle between the projection of the radius vector in xy plane and the +x axis,

Hydrogen atom (II) • The wave function can be divided into three

different functions which may be governed by three independent equations.

• There are three quantum numbers:– Magnetic quantum number m: the direction of the angular

momentum,

– Orbital quantum number l: the magnitude of the electron’s angular momentum,

– Principle quantum number n: the total energy of the electron,

)()()( rR

)1( llL

21

n

EEn

mLz

How atomic states are denoted?• It is customary to specify electron angular

momentum states by a letter.

• Sharp, principal, diffuse, fundamental,…• Atomic states are denoted by a combination

of the principal quantum number with the letter representing orbital angular momentum, such as 2s (n=2, l=0), 4d(n=4,l=2).

l= 0 1 2 3 4 5 6 ….s p d f g h i

Electron probability density• The probability of finding the electron in a hydrog

en atom at a distance between r and r+dr from the nucleus is:

– For different states, P(r) changes with r differently.

• Zenithal probability density has angular preference except an s state (l=0, m=0). Azimuthal probability density is a constant.

• Thus, the electron’s probability density is symmetrical about the z axis. It can be called electron cloud.

drRrdrrP 22 ||)(

Lasers• Laser is an acronym for light amplification by

simulated emission of radiation. • Three ways of radiation interacting with atomic

energies:– Spontaneous emission, high E to lower E;– Induced absorption, lower E to high E, – Induced (stimulated) emission: A passing photon with the

right energy induces the atom to emit a photon and makes a transition to the lower state, producing two photons in phase.

• Population inversion is essential for a laser, that is, a higher state has a greater population than a lower state.

He-Ne laser• A mixture of He and Ne is in the ratio of 5:1 to 1

0:1.• The purpose of the He atoms is to help achieve a

population inversion. – An electrical current excites He to 20.61eV,– Excited He atoms colliding with Ne atoms to make the

m to reach 20.66eV, a metastable state,

• The laser transition in Ne from the metastable state at 20.66eV to an excited state at 18.70eV, with the emission of a 632.8nm photon.

Usage of lasers• There are many different types of lasers, such as

the ruby laser, the carbon dioxide laser etc.• As lasers produce coherent monochromatic light

that can be confined to an intense narrow beam.• They are used to replay music in CD players, to

weld parts, to measure distance accurately and to study molecular structure. It is also used in medical treatment and surgery.