# Lecture Notes, Quantum Physics - UW-Madison uw. himpsel/ Lecture Notes, Quantum Physics Physics 448,...

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1

Lecture Notes, Quantum Physics

Physics 448, Prof. Franz Himpsel

Particles and Waves

Blackbody Radiation, Cosmic Background 2

Particle-Wave Duality 3

Particles 4

Waves 5

Wave Packets 6

Fourier Transform, Uncertainty Relation 7

General Quantum Mechanics

Observables, Operators 10

General properties of 12

Solutions for 1D potentials 13

Tunneling 14

Energy Levels

Hydrogen Atom 15

Spin, Pauli Matrices, Dirac Equation 17

Angular Momentum 18

Perturbations: Hyperfine, Zeeman 19

Many Electrons, Hunds Rules 20

Spectroscopies

Dipole Selection Rules 21

Photoemission, Optical, Auger 22

2

Quantum Physics

First evidence for quantum phenomena (Planck): Quantization of the electromagnetic field

energy into photons with energy h provides the exponential cutoff of the blackbody radiation

spectrum at high frequencies .

Blackbody radiation:

Spectral emittance, i.e., the power P radiated per unit area and frequency interval:

dP/d = 2hc-2 3 / (eh/kT-1) = const. (kT)3 F ( )

k = Boltzmann's constant, h = Planck's constant, dimensionless.

Frequency max of maximum emittance: max ~ T

Total emittance P, integrated over all : P ~ T4

Cosmic background radiation:

Explained as black body radiation from a hot plasma ( H+ + e- ) that existed about 3 105 years after

the Big Bang (short compared to the age of the universe: 1.5 1010 years). The temperature shift

from a 4000 K plasma to the observed 3 K microwave radiation can be viewed as Doppler shift,

making Hubble's assumption of a universe expanding with the expansion velocity proportional to

the distance from the observer. Thus, we are looking at the outermost, oldest reaches of the

universe.

Precise data on the spectral and directional distribution of the cosmic background radiation have

been obtained by the cosmic background explorer (COBE) satellite and a number of follow-up

experiments.

Physics Today, July 2000, p. 17

http://space.gsfc.nasa.gov/astro/cobe/cobe_home.html

h kT

h kT

http://space.gsfc.nasa.gov/astro/cobe/cobe_home.html

3

Particle-Wave Duality

Plane Wave: (x,t) = exp[i/ (px-Et)] = exp[i(kx-t)] = exp[i2(x/-t/T)]

This wave corresponds to a particle with energy E and momentum p.

To eliminate use the angular frequency = 2 = 2/T and the wave vector k = 2/ :

E = p = k

E = h (Planck) p = h/ (De Broglie)

Thus, E is related to the frequency , and p to the wavelength via Plancks constant h.

Since E is related to p via the particle mass m0, one can convert E to , k :

Photons: [nm] = 1.24 / E [keV] E = pc p = h/ (E = h = c/)

Electrons: [nm] = 1.23 / E [eV] E = p2/2me p = h/

Photons: k[nm1] = 5.1 ( E [keV])

Electrons: k[nm1] = 5.1 E [eV]

When do we see a particle, when a wave?

Always both: The wave function gives the probability p of finding a particle: p = ||2

When is quantum physics important?

When few particles are involved, for example when the energy of a quantum h is large compared

to the thermal energy kT (high h cutoff in the blackbody radiation).

Build - up of an

interference pattern

from single photons

4

Particle Properties of Electromagnetic Waves

The photon as a particle with energy E=h, momentum p=h/, and rest mass m0=0:

Photoemission:

+ solid e- Emax = h -

( = work function, see p. 22 for details on photoemission)

Photoelectron spectroscopy is practiced using X-ray tubes (Al K) and synchrotron radiation.

For synchrotron radiation centers see:

SRC in Madison (h=10-100eV, valence electrons): http://www.src.wisc.edu

ALS in Berkeley (h=100-1000eV, core electrons): http://www-als.lbl.gov/als/

Bremsstrahlung = Inverse Photoemission:

e- + solid hmax = E +

Inverse photoemission probes unoccupied electron states in a solid,whereas photoemission probes occupied states.

Compton Scattering:

+ e- ' + e-' ' - = C (1-cos) C = h/mec

Elastic scattering of a photon by an electron e- at rest. The electron picks up energy from the

photon. The Compton wavelength C corresponds to the wavelength of a photon with energy mec2

(for length scales compare relativity notes p. 12).

Electron-Positron Pair Annihilation and Production:

e- + e+ 2

Need two photons in order to satisfy energy and momentum conservation. One photon couples to

a nucleus in pair production. Used in positron emission tomography (PET) for scanning the brain.

These processes can be treated as particle collisions where energy and momentum are conserved

(see relativity notes p.7). For a solid, only the momentum parallel to the surface is conserved.

http://www.src.wisc.edu/http://www-als.lbl.gov/als/

5

Wave Properties of Particles

Diffraction of electrons, x-rays, neutrons, atoms:

Two- or three-dimensional diffraction, depending on how many lattice planes are penetrated:

2D: Atom diffraction: Outermost atom layer only.

2D+3D: Low energy electron diffraction ( LEED, 10-100 eV ): A few lattice planes.

3D: High energy electrons (>1keV), x-rays and neutrons.

Determine the charge density with x-rays, and the spin density with neutrons.

Protein crystallography with x-rays: Wayne Hendrickson, Physics Today, Nov. 1995, p. 42.

Three-dimensional Two-dimensional

Planes of atoms diffract Rows of atoms diffract

2dplane sin = n in = out (Bragg) drow sin = n (Optical grating, LEED)

For obtaining from E , see p. 3.

Bragg reflection: Incident radiation with a continuous spectrum is monochromatized into discrete

n by reflection at a lattice plane. Monochromatic radiation is reflected only if in and out are

matched ( , 2 - scan ).

Laue pattern: Incident radiation with a continuous spectrum produces exit beams with discrete

wavelengths. The angle determines the lattice plane spacing.

Powder Diffraction: Monochromatic radiation produces ring-like diffraction patterns from

randomly-oriented crystallites.

dplane

drow

6

Wave Packets

A wave packet consists of a sum of

plane waves. It spreads (disperses)

over time if the phase velocity depends

on the wavelength . Waves with short

(high frequency ) move faster due to

their higher energy E=. This is

characteristic of a non-linear E(p)

relation, such as E=p2/m0.

Group velocity: vgroup = dE/dp = d/dk wave packet

Describes the velocity of a wave packet, and the speed of energy and information transfer. Does not

exceed c.

Phase Velocity: vphase = E / p = / k plane wave

Describes the velocity of wave fronts for a plane wave in steady state. vphase can exceed c, for

example for light in a medium with refractive index n

7

Real Space x Reciprocal Space k

Expansion of f(x) into plane waves: Definition of the Fourier transform F(k):

f(x) = (2)-1/2 F(k) exp(ikx) dk (1) (2) F(k) = (2)-1/2 f(x) exp(ikx) dx

d/dx f(x) (1) i k F(k)

f(x) dx (2) (2)1/2 F(0)

Particle Limitx small k largeDelta function* (x) (2) Constant = (2)-1/2

Wave Limitx large k smallPlane wave (2)-1/2 exp(ik0x) (1) Delta function* (k-k0)

Uncertainty

x = x0 = k = k0 =

= xn p(x) dx p(x) dx = 1 = kn P(k) dk P(k) dk = 1

Probability p(x), P(k) = ||2 = * in quantum mechanics, here p(x) ~ |f|2 , P(k) ~ |F|2

Uncertainty Relation

x k

with p = k x p

The smallest uncertainty is achieved with a Gaussian wave function f(x) = exp[- (x/)2] .

* Definition of the function : (x) f(x) dx = f(0) for all functions f(x).

(x-x0) f(x) dx = f(x0) (x) = (x) (y) (z) in three dimensions.

root mean square (rms)

8

Similarly for energy and time: t

with E = t

k, are the widths of the k, distributions that are required for a wave packet of size x, t. As

x, t become smaller, the distribution of frequencies k, in the wave packet becomes larger. This

is a general property of wave packets, which is quantified by taking the Fourier transform.

Examples:

1. Gaussian (ground state of the harmonic oscillator)

f(x) = exp[-(x/x)2] F(k) = x exp[-(kx)2]

x k 0 0

p, P = f2, F2 (quantum-mechanical x, k): x k =

p, P = f , F (standard deviation x, k): x k = 1

(half width half maximum x , k ): x k = 2 ln2 1.4

2. Finite wave train ( resolution in diffraction)

f(x) = exp[ik0x] -a x +a k0 = 2/ F(k) = (2/)1/2 sin[(kk0)a] / (kk0)

x k-a +a

0 k0

With x = a , k = /a : x k =

x k

x

The resolving power of a monochromator (Bragg crystal in 3D, grating in 2D, see p. 5) is

determined by the length of the wave train = N planes (grooves) n per plane (groove):

0 / = N n n = diffraction or

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