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 ψ 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, Hund’s 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 hν provides the exponential cutoff of the blackbody radiation

spectrum at high frequencies ν.

Blackbody radiation:

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

dP/dν = 2πhc-2 ⋅ ν3 / (ehν/kT-1) = const. ⋅ (kT)3 ⋅ F ( )

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

Frequency νmax of maximum emittance: νmax ~ T

Total emittance P, integrated over all ν: 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

hν kT

hν kT

3

Particle-Wave Duality

Plane Wave: ψ(x,t) = exp[i/ћ ⋅ (px-Et)] = exp[i(kx-ωt)] = exp[i2π(x/λ-t/T)]

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

To eliminate ћ use the angular frequency ω = 2πν = 2π/T and the wave vector k = 2π/λ :

E = ћω p = ћk

E = hν (Planck) p = h/λ (De Broglie)

Thus, E is related to the frequency ν, and p to the wavelength λ via Planck’s constant h.

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

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

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

Photons: k[nm−1] = 5.1 ⋅ ( E [keV])

Electrons: k[nm−1] = 5.1 ⋅ √ E [eV]

When do we see a particle, when a wave?

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

When is quantum physics important?

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

to the thermal energy kT (high hν 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=hν, momentum p=h/λ, and rest mass m0=0:

Photoemission:

γ + solid ⇒ e- Emax = hν - Φ

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

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

For synchrotron radiation centers see:

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

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

Bremsstrahlung = Inverse Photoemission:

e- + solid ⇒ γ hνmax = E + Φ

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

Compton Scattering:

γ + e- ⇒ γ' + e-' λ' - λ = λC ⋅ (1-cosθ) λC = h/mec

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

photon. The Compton wavelength λC 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+ ⇔ 2γ

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.

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⋅ sinϑ = n ⋅ λ ϑin = ϑout (Bragg) drow ⋅ sinϑ = n ⋅ λ (Optical grating, LEED)

For obtaining λ from E , see p. 3.

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

λn by reflection at a lattice plane. Monochromatic radiation is reflected only if ϑin and ϑout are

matched ( ϑ, 2ϑ - 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.

dplaneϑ

ϑ

drow

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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 λ. Waves with short

λ (high frequency ω) move faster due to

their higher energy E=ћω. This is

characteristic of a non-linear E(p)

relation, such as E=p2/m0.

Group velocity: vgroup = dE/dp = dω/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 = ω / 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<1.

Fourier Transform

The superposition of plane waves ψn into a wave packet ψ can be viewed in reverse as expansion

of ψ into plane waves ψn :

ψ(x,t) = nΣ An ⋅ exp[i(knx-ωnt)] Fourier Series (ψ periodic)

ψ(x,t) = ∫ A(k,ω) exp[i( kx - ωt )] d3k dω Fourier Integral (ψ aperiodic)

A Fourier transform is an expansion of an arbitrary function ψ(x,t) into plane waves.

In the following we restrict ourselves to functions of a single space-time coordinate f(x) or f(t) and

write F(k) or F(-ω) for the amplitude A(k,ω). F(k) is the Fourier transform of f(x), likewise F(-ω)

the Fourier transform of f(t).

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Real Space x Reciprocal Space k

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

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

d/dx f(x) (1) ⇒ i k F(k)

∫ f(x) dx ⇐ (2) (2π)1/2 F(0)

Particle Limitδx small δk largeDelta function* δ(x) (2) ⇒ Constant = (2π)-1/2

Wave Limitδx large δk smallPlane wave (2π)-1/2 exp(ik0x) ⇐ (1) Delta function* δ(k-k0)

Uncertainty

δx = √ <(x-x0)2> x0 = <x> δk = √ <(k-k0)2> k0 = <k>

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

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

Uncertainty Relation

δx ⋅ δk ≥ ½

with p = ћk δx ⋅ δp ≥ ½ ћ

The smallest uncertainty is achieved with a Gaussian wave function f(x) = exp[-½ (x/σ)2] .

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

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

root mean square (rms)

8

Similarly for energy and time: δω ⋅ δt ≥ ½

with E = ћω δΕ ⋅ δt ≥ ½ ћ

δk, δω are the widths of the k, ω distributions that are required for a wave packet of size δx, δt. As

δx, δt become smaller, the distribution of frequencies δk, δω 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[-½(x/σx)2] F(k) = σx ⋅ exp[-½(kσx)2]

x k 0 0

p, P = f2, F2 (quantum-mechanical δx, δk): δx ⋅ δk = ½

p, P = f , F (standard deviation σx, σk): σx ⋅ σk = 1

(half width half maximum γx , γk ): γx

⋅ γk = 2 ln2 ≈ 1.4

2. Finite wave train (λ resolution in diffraction)

f(x) = exp[ik0x] -a ≤ x ≤ +a k0 = 2π/λ F(k) = (2/π)1/2 sin[(k−k0)a] / (k−k0)

x k-a +a

0 k0

With ∆x = a , ∆k = π/a : ∆x ⋅ ∆k = π

γx γk

δδx

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 ⋅ λ per plane (groove):

λ0 / δλ = N ⋅ n n = diffraction order

3. Lorentzian (damped oscillator, mean free path)

This describes the decay time versus the spectral linewidth. The spatial analog is the mean free path

versus the momentum broadening. For quantum mechanical uncertainty one has to take the square

of ψ ~ f and of its Fourier transform Ψ ~ F. This gives a Lorentzian for |F|2 :

|f(t)|2 = exp[-t/τ] τ = ½τ′

t

f(t) = exp[-iω0t - t/τ' ] for t ≥ 0

With ∆t = τ , ∆ω = Γ = 1/τ (full width half m

Lifetime τ versus Linewidth:

wit

Mean free path λ versus k-Broadening:

wi

For the examples 2 and 3 the quantum-mecha

<k2> = ∫ k2 |F(k)|2 dk diverges. The abrupt step

τ0

1 12 2|F(ω)|2 = ⋅ Γ = 1/τ

9

2π (ω−ω0) + (Γ/2)

ω

F(ω) = i (2π)−1/2 ⋅ [ω−ω0 + i/τ']−1

aximum): ∆t ⋅ ∆ω = 1

∆ω = 1 / τ

h E=ћω ∆E = ћ / τ

∆k = 1 / λ

th p=ћk ∆p = ћ / λ

nical definition of δk and δω does not exist because

in f causes too many Fourier components at high k.

ω0

Γ

Course of Action in Quantum Mechanics

Energy E(p) E = -p2/2m + V

⇓Substitution E = iћ ∂/∂t , p = -iћ ∂/∂x Relativity Notes, p. 11

⇓

Schrödinger Equation iћ ∂ψ/∂t = [ -ћ2∇2/2m + V ] ψ […] = H = Hamiltonian

⇓Wave Function ψ

⇓Normalization ∫ ψ*ψ dx = 1 Divide ψ by [∫ ψ*ψ dx]½

⇓Probability Density ψ*ψ Probability of finding a particle

⇓Observable <A> = ∫ ψ*A ψ dx

⇓Uncertainty δA 2 δA = √ <A2> − <A>2

⇓Uncertainty Relation: δA ⋅

Exact Observables ⇒

A ψn = an ⋅ ψn A = Oper

H ψn = En ⋅ ψn H = Ham

Complete Set of Quantum

To fully characterize a state

eigenfunctions (for example

information that can be meas

Orthonormality: ∫ ψ*n(x)

Completeness: Σn ψ n(x)

Expansion of ψ: ψ(x) = ∫

Expectation valueof an operator A

= √ < (A − <A>) > root mean square (rms)

10

δB ≥ [A,B] = A⋅B − B⋅A

Eigenvalues ⇒ Quantum Numbers:

ator, an = Eigenvalues, ψn = Eigenfunctions, δA = 0

iltonian, ψ(r,t) = ψn(r) ⋅ exp[−i(En/ћ) ⋅ t]

Numbers and Basis Functions:

one has to find the largest set of operators A,B,... that have common

E, L2, Lz). The eigenvalues an contain the maximum amount of

ured. The operators A,B,... commute pairwise: [A,B] = 0 .

⋅ ψ m(x) dx = δnm

⋅ ψ*n(x′) = δ(x-x′) See p. 7 for the definition of the δ−function.

δ(x-x′) ψ(x′) dx′ = Σn ψn(x) ⋅ ∫ ψn*(x′) ⋅ ψ (x′) dx′ = Σn an ψn(x)an

< [A,B] > 2i

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Dirac Brackets:

Definitions: | n > = ψn(x) < n | m > = ∫ ψ*n(x) ⋅ ψ m(x) dx (scalar)

< n | = ψn*(x) | m > < n | = ψm(x) ⋅ ψ*n(x′) (operator)

< n | A | m > = ∫ ψ*n(x) ⋅ Α ψ m(x) dx (scalar)

Orthogonality: < n | m > = δnm

Completeness: Σn | n > < n | = 1 = unit operator

Expansion of a wave function ψ : | ψ > = 1 ⋅ | ψ > = Σn | n > < n | ψ > = Σn an | n >

Expansion into Plane Waves | k > : | k > = (2π)-½ exp[ikx] (k continuous Σn ⇒ ∫ dk )

Arbitrary function f(x) = | f > : | f > = ∫dk | k > < k | f >

f(x) = ∫ (2π)-½ exp[ikx] ⋅ ∫ (2π)-½

exp[−ikx] ⋅ f(x) dx dk

= Fourier inversion theorem (p. 7)

Schrödinger versus Heisenberg Representation:

Expand wave functions ψ and operators A : A = 1 ⋅ A ⋅ 1 = Σm,n | m > < m | A | n > < n | =

= Σm,n Amn | m > < n |

ψ = = Vector A = = Tensor

Hilbert Space:

Vector space of wavefunctions ψa(x), ψb(x′), ψc(x′′), … = |a>, |b>, |c>, …

with the scalar product ∫ ψa*(x) ⋅ ψb(x′) dx = < a | b >

Heisenberg Schrödinger

Scalar = Number = Observable

Vector = Wave Function ψ = Physical State

Tensor = Operator A = “Measurement”

The measurement process (= applying an operator A to ψ) generally changes ψ. Subsequentmeasurement of B is inaccurate because ψ changed, therefore the uncertainty relation.

an

F(k)

a1a2

A11 A12A21

Omit unit vectors | m > , < n | .

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General Properties of the Schrödinger Equation:

Discrete versus Continuous Eigenvalues:

V(x)Scattering States: Continuous E, p

Bound States: Discrete En

(ψ→ 0 for x→ ±∞)

Finding Eigenfunctions: Harmonic Oscillator, n=0

Oscillatory versus Damped Solutions:

−ћ2/2me ψ'' + V ψ = E ψ ⇒ ψ'' ~ (V-E) ψ ⇒ (V-E) determines the curvature of ψ

E < V E > V E = V

ψ'' ~ +ψ ψ'' ~ −ψ ψ'' = 0

Concave Convex Inflection Point

ψ ~ exp(±kx) ψ ~ exp(±ikx)

x

ψ(x)E = 0.98 E0

E = 0.99 E0

E = 0.995 E0

E = E0 (normalizable)

E = 1.005 E0

E = 1.01 E0E = 1.02 E0

x

k = √ 2me|V-E| / ћ

= 5.1 nm-1 √ |V-E| / eV

x

V(∞)

13

One-Dimensional Potentials

Particle in a Box:

Counting modes in a cavity,quantum wells in solids

V(x) = 0 for 0 ≤ x ≤ L ∞ elsewhere

En = n2 ⋅ (πћ)2 / 2meL2

ψn(x) = (2/L)1/2 ⋅ sin[n ⋅ (πx/L)]

Harmonic Oscillator:

Vibrations and oscillations,local energy minima

V(x) = ½ κ ⋅ x2

En = ( n + ½ ) ⋅ ћω ω = √ κ/me

ψn(x) = a ⋅ exp[−½ (x/σ)2 ] ⋅ Hn[x/σ] σ = (ћ/meω)1/2

a = (σ 2n n!√π)-1/2

Hn[z] = 1, 2z, 4z2-2,... (Hermite polynomials)

Coulomb Potential:

Radial equation for the H atom (l=0); Image potential states at a metal surface ( −¼ e2/z )

V(r) = − e2 / r

En = − R / n2 R = Rydberg = mee4/2ћ2 = ½ α2 mec2 = 13.6 eV

ψn(r) = a ⋅ r ⋅ exp[-r/na0] ⋅ Ln[r/na0] a0 = Bohr radius = ћ2/mee2 = α-1 λC = 0.05 nm

ψH(r) = ψn(r) / r a = (a03nπ)-1/2 for ψH in 3 dimensions

Ln[z] = 1, (-z+1)/2, (2z2-6z+3)/9, …

Classically forbidden

Zeropointenergy

Energyquantum ћω

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Tunneling

E < V: Classically-forbidden region (Ekin < 0), exponential decay in quantum mechanics.

Scanning Tunneling Microscope (STM):

tip (−) vacuum sample (+)

EVacuum V

Φ = Work Function ψ

EFermi

0 d x

Attenuation of the charge density (~ tunneling current):

|ψ(d)|2 / |ψ(0)|2 = exp(−2kd)

ψ(x) ~ exp(−kx) k ≈ 0.51 Å−1 √ Φ/eV

For a typical energy barrier Φ ≈ 4eV the attenuation is about a factor of 100 per atom diameter.

That makes it possible to detect height changes

much smaller than an atom. The current is kept

constant during a scan by adjusting the tip height

d via a piezoelectric material.

The picture shows iron atoms being assembled

into a ring at a copper surface. The STM was

used for imaging and assembly. The ripples are

electron density waves proportional to the wave

function |ψ|2 of surface electrons.

(From Crommie and Eigler)

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Three-Dimensional Potentials

Separate the variables, i.e. find a coordinate system where the wave function is a product of one-

dimensional wavefunctions: ψ(x,y,z) = ψx(x) ⋅ ψy(y) ⋅ ψz(z)

Instead of a partial differential equation (several variables) one solves three ordinary differential

equations (one variable each), which is much easier.

Particle in a Box:

V(x,y,z) = 0 if 0 ≤ x ≤ Lx and 0 ≤ y ≤ Ly and 0 ≤ z ≤ Lz ; V = ∞ elsewhere.

Using a product wave function ψ(x,y,z) = ψx(x) ⋅ ψy(y) ⋅ ψz(z) the Schrödinger equation separates

into three one-dimensional equations with E replaced by Ex, Ey, Ez. The total energy is:

E = Ex + Ey + Ez = (πћ)2/2me ⋅ [ nx2 / Lx

2 + ny 2/ Ly

2 + nz 2 /Lz

2 ] (compare p. 13)

This separation method works for any potential of the form V(x,y,z) = Vx(x) + Vy(y) + Vz(z).

Coulomb Potential:

V(x,y,z) = − e2 / r ; r = √x2+y2+z2

Polar coordinates r,θ,ϕ separate the variables for any spherically-symmetric potential V(r) and

provide one-dimensional eigenvalue equations for ψr , ψθ , ψϕ with ψ(r,θ,φ) = ψr(r) ⋅ ψθ(θ) ⋅ ψϕ(ϕ).

Separate the kinetic energy into radial and angular parts using the angular momentum L = r×p :

L2 = (r×p)2 = r2 p2 − (r⋅p)2 ⇒ Ekin = p2 / 2me = [ (r⋅p)2 + L2 ] / 2me r2

In quantum mechanics: p = -iћ r×∇ radial angular

(The ordering of r,∇ becomes tricky, but the separation into radial and angular terms remains.)

r: En,l ⋅ ψr = H ψr (H = Hamiltonian = Energy Operator)

θ: ћ2 ⋅ l(l+1) ⋅ ψθ = L2 ψθ (L = −iћ r×∇ = Angular Momentum Operator)

φ: ћ2 ⋅ ml 2 ⋅ ψϕ = Lz

2 ψϕ (Lz = z - Component of L)

r: En,l ⋅ ψr = − (ћ2/2me) ⋅ { (r ψr)'' / r − [ l(l+1) / r2 ] ⋅ ψr } − e2/r ⋅ ψr

θ: l(l+1) ⋅ ψθ = − [ (sinθ ⋅ ψθ')' / sinθ − (ml2 / sin2θ) ⋅ ψθ ]

φ: ml 2 ⋅ ψϕ = − ψϕ''

radial angular

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Solutions:r: ψr ~ exp(−r/na0) ⋅ Ln [2r/na0] Ln = Laguerre Polynomials (p. 13)

θ: ψθ ~ Plm [cosθ, sinθ] Pl

m = Legendre Polynomials

ϕ: ψϕ ~ exp[iml ϕ]

Eigenvalues:

r: En,l = −R / n2 n = 1, 2, ... (R, a0 : p. 13)

θ: L2 = ћ2 ⋅ l(l+1) l = 0, 1, ... , (n-1) = s, p, d, f, ...

φ: Lz = ћ ⋅ ml ml = −l , ... , +l

A complete set of eigenvalues for the H atom includes additional quantum numbers for electron spin,

total angular momentum, and nuclear spin (see p. 18-20).

Radial probability distributions: r2 |ψr|2 Angular wave functions: Re [ψθ ⋅ ψφ]

r2 |ψr|2

z

xr/a00 4

17

Spin

Spin is the intrinsic angular momentum of the electron. Like L, is has two quantum numbers:

S2 = ћ2 ⋅ s(s+1) s = ½

Sz = ћ ⋅ ms ms = ± ½ (spin up or down)

Spin is a relativistic effect that comes out naturally in the Dirac equation, the relativistic

generalization of the Schrödinger equation (see relativity notes p. 11 and below). Spin can be added

to the non-relativistic Schrödinger equation by multiplying the spatial wave function with a two-

component spin wave function χ :

ψtotal = ψ(r,t) ⋅ χ χ+ = 1 for ms= +½ (spin up) χ− = 0 for ms= −½ (spin down) 0 1

The spin operator S is different from the orbital angular momentum operator L. It consists of three

2×2 matrices, the Pauli matrices σx,y,z which act on the spin wave function χ:

S = ½ ћ ⋅ (σx , σy , σz) σx = 0 1 σy = 0 −i σz = 1 0 1 0 i 0 0 −1

[σx,σy] = σx σy − σy σx = 2i σz etc.

{σi,σk} = σi σk + σk σi = 2 δik

Dirac equation:

This is the relativistic wave equation for spin ½ particles. The wave function ψ = (ψ1,ψ2,ψ3,ψ4) has

4 components (two for spin up/down and two for particle/antiparticle). The 4×4 matrices Dirac

matrices γµ act on ψ. They consist of 2×2 blocks containing σx,y,z and the 2×2 unit matrix 1:

0 σx,y,z 1 0

−σx,y,z 0 0 –1{γµ,γν} = 2 gµν (gµν = metric)

γµ pµ ψ = mec ⋅ ψ pµ = iћ ∂µ − q/c Aµ

γµ = (γ0, γ1, γ2 , γ3) γ1,2,3 = γ0 =

Aµ = electromagnetic four-potential, q = −e for e−

(Relativity notes p. 10,11)

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Total Angular Momentum

General Properties of Angular Momentum Operators J, L, S:

J = (Jx,Jy,Jz) [Jx,Jy] = iћ ⋅ Jz (cyclical)

[Jy,Jz] = iћ ⋅ Jx

[Jz,Jx] = iћ ⋅ Jy

From these properties one can derive:

J2 = ћ2 ⋅ j(j+1) j integer or half-integer

Jz = ћ ⋅ mj mj = −j , ... , +j in integer increments, (2j+1) states

J = J1 + J2 ⇒ j = |j1−j2| , ... , j1+j2

The sum of two angular momenta is again an angular momentum vector, e.g., for an electron the sum

of orbital angular momentum L and spin S:

J = L + S s = ½

J2 = ћ2 ⋅ j(j+1) j = |l−½|, l+½

Jz = ћ ⋅ mj mj = −j , ... , +j

Spin-Orbit Interaction:

This is an extra term ~ L ⋅ S in the Hamiltonian, again a relativistic effect originating from the

Dirac equation. It doubles all levels with l>0 into two states with j = l − ½ , l + ½ . The level with

smaller j lies lower. Since Lz and Sz do not commute with L⋅S, ml and ms cease to be good

quantum numbers. They are replaced by mj. Spin-orbit interaction is strong for a large Coulomb

potential (heavy elements, deep levels).

Quantum Numbers of the H Atom:

n, l, s, j, mj Terminology: 2p3/2 stands for n=2, l=1, j=3/2

s, p, d, f, g, ... stands for l = 0, 1, 2, 3, 4, ...

They form a complete set for a single electron in a spherically-symmetric potential. This single

electron label remains useful in many-electron systems (atoms, molecules, solids). Including the

proton requires additional quantum numbers for the nuclear spin (see next).

19

Nuclear Spin, Hyperfine Structure:

Like L and S combining into J=L+S for the electron, J and I, the proton spin, combine to a total

angular momentum F=J+I. Analogous to the spin-orbit interaction ~ L ⋅ S of the electron there is

the hyperfine interaction ~ J ⋅ I which may be viewed as the energy of the electron in the magnetic

field created by the proton spin (see below). The effect

is reduced by the mass ratio mp/me ≈ 2000 . The proton

spin i= ½ doubles all levels once more into two states

with f = j − ½ , j + ½ . The 1s1/2 level in H splits into a

singlet (f=0) and a triplet (f=1) with a transition at the

wavelength of 21 cm (used in radio astronomy).

Electrons in a Magnetic Field (Zeeman Effect):

Energy of a magnetic dipole µ in a magnetic field B (defines z):

E = − µ ⋅ B = + mj ⋅ g ⋅ µB ⋅ B µB = eћ/2me = 5.6 ⋅ 10-5 eV/T (Bohr Magneton)

The sign reversal comes from the negative charge of the electron.

g = = gyromagnetic ratio

g = 1 for pure L g = 2 for pure S

Determine j, mj from the splitting

in a magnetic field: (2j+1) levels.

Perturbation Theory:

Calculate small energy shifts due to an interaction Hamiltonian Hint, such as the spin-orbit,

hyperfine, and magnetic splittings. Take as unperturbed Hamiltonian H0 = −ћ2 ∇2 / 2m − e2/r .

H = H0 + Hint H0 ψ0n = En ⋅ ψ0

n H ψn = (En + ∆E) ⋅ ψn ∆E ≈ < ψ0n | Hint | ψ0

n >

magnetic moment in µBangular momentum in ћ

g(j,l,s) between 1 and 2

p1/2

p3/2

p

mj :+3/2

+1/2

−1/2

−3/2

+1/2

−1/2

B=0 B≠0

f = j+½

f = j−½ l

j = l+½

j = l−½

20

Many-Electron Atoms

Adding angular momenta Li, Si, Ji of several electrons: Many-electron levels can be sorted by

their angular momenta. First, the largest interactions (splittings) are considered, then the smaller

ones as perturbations. The most common sequence is (for low Z):

1) Large ∆E for different S = Σ Si → S (e−- e− Coulomb Repulsion)

2) Intermediate ∆E for different L = Σ Li → L (e−- e− Coulomb Repulsion)

3) Small ∆E for different J = L + S → J (Spin-Orbit Interaction)

Hund's three rules determine the signs of the splittings in the ground state (lowest energy):

1) Maximize S Smax = | Σ ms | + Pauli principle

2) Maximize L Lmax = | Σ ml | + Pauli principle

3) Minimize J for < half-filled shell Jmin = |Lmax − Smax| Maximize J for > half-filled shell Jmax = Lmax + Smax

Hund’s first rule is due to the fact that electrons with parallel spin are farther apart to satisfy the

Pauli principle, which reduces their mutual Coulomb repulsion. This rule is responsible for

magnetism. Typical multiplet splittings are about 2eV and produce colors in gems and laser media:

Ruby = Cr in sapphire (Al2O3), titanium in sapphire (fastest lasers), erbium in SiO2 (amplifier for

optical fibers).

Filled shell: All the (2li + 1)⋅2 = 2, 6, 10, 14 … states with different ml and ms are occupied.

The total angular momenta L, S, J are zero. A filled shell is particularly stable since an extra

electron has to move up into a higher-energy shell with different ni, li due to Pauli’s exclusion

principle.

Hole: Taking an electron from a filled shell is equivalent to adding a positive charge (= hole).

The remaining having (n-1) electrons can be substituted by the single hole. Analogs exist in solid

state physics (filled shell ⇒ valence band of a semiconductor) and in particle physics (filled shell

⇒ sea of electrons in negative energy states of the Dirac equation; hole ⇒ positron).

Notation for Many-Electron States: 1s2 2s2 2p5, 2P1/2

1s2 stands for ni = 1, li = 0, occupancy 2 for individual shells i2P1/2 stands for (2S+1)=2 (doublet), L=1, J=1/2 for the combined L,S,J

21

Dipole Selection Rules:

These rules govern transitions between two levels m, n involving an emitted or absorbed photon,

e.g. in optical absorption and emission, photoemission and inverse photoemission, but not in Auger

electron emission (see p. 22). The probability of a transition m → n can be obtained via the classical

electromagnetic radiation of an oscillating dipole moment, which is determined quantum-

mechanically. An energy eigenfunction ψn(x,t) = ψn(x) ⋅ exp(−i t) does not produce an

oscillating dipole moment since the time-dependence of ψn* and ψn cancels out in the probability

density (thereby solving the stability problem of Bohr’s model):

ψ n* ψn ~ ψn*(x) ψn(x) ⋅ exp(+i t) ⋅ exp(−i t) = ψn*(x) ψn(x)

However, the superposition wave function ψ = ψm+ψn creates an oscillating term:

ψ*m ψn ~ ψm*(x) ψn(x) ⋅ exp[i t]

Dipole moment: xmn = e ⋅ < m | x | n > = e ⋅ ∫ ψm*(x) ⋅ x ⋅ ψn(x) dx

Transition probability: ~ | xmn |2

Allowed transitions: xmn ≠ 0

Allowed Transitions: Symmetries generate conservation laws and allowed transitions. For example,

spherical symmetry leads to angular momentum conservation and inversion symmetry to parity

conservation. The photon has angular momentum sphoton = 1 and parity −1 (vector potential A → −A,

dipole e⋅x → −e⋅x ).

Single Electron Quantum Numbers: e−m ↔ e−

n + photon

∆l = ±1 Parity (−1)l : (−1)l±1 = (−1)l ⋅ (−1)

∆j = ±1, 0 Angular momentum: jm = |jn−1| , jn , jn+1

∆mj = ±1, 0

Combined Angular Momenta in Multi-Electron Atoms:

(For LS coupling, where the e−- e− repulsion is larger than the spin-orbit interaction; low Z)

∆S = 0 Total spin (−e⋅x does not affect spin wavefunctions)

∆L = ±1 Total orbital angular momentum

∆J = ±1, 0 (not 0→0) Total overall angular momentum

En ћ

En ћ

En ћ

(Em-En) ћ

Jm = Jn + Sphoton

22

Spectroscopies of Atomic Levels

2 levels eout 4 levels eup

hνin hνin edown

hνout

core level

Photoemission Absorption Emission Auger Decay (UPS, XPS, ESCA) (XAS, NEXAFS)

Used for the identification of elements by the core levels (bottom level), for example in astrophysicsand materials science.

eout

XPS

UPS

(UPS) E vacuum E Fermi + hν