Quantum Review

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GRE ATOMIC AND QUANTUM PHYSICS REVIEW

SOCIETY OF PHYSICS STUDENTS

1. Experiments and “Effects”

1.1. Stern-Gerlach. Beam of electrons goes through an inhomogenous magnetic

field. (Originally done with silver atoms, which have one valence electron.) The

beam is seen to split into two discrete beams. Conclusion: Electrons have spin ±1/2.

1.2. Zeeman Effect.

H =e

2m( L + 2 S ) · Bexternal

Splitting in energy levels is µBBexternal, where µB = e /2m.

1.3. Stark Effect. H = eE externalr cos θ. Energy splitting goes like |e|Ea0.

1.4. Rutherford Scattering. Differential cross-section goes like 1/ sin4 θ.

1.5. Franck-Hertz. Electrons accelerated through mercury vapor. At certain volt-

ages, the electrons would have the proper energy to excite the Hg atoms. Conclusion:

another clue that there are discrete energy levels to which an atom can be excited.

1.6. Photoelectric Effect. Light hitting some metallic surface. Energy of electrons

scattered depends on wavelength of light, not intensity. E photon = hc/λ = 12400/λ.

(λ in Angstroms.)

1.7. Young’s Double-split Experiments. p. 90 in French & Taylor. Two slits a

distance d apart, a screen a distance D away. y is the distance from the center of the

screen.

Phase difference: δ =2π

λ· yd

D

Intensity: I (y) = 4I 0 cos2 πd

Dλ y

1.8. Compton Scattering. Photon hits an electron at rest. Scattered photon has

a changed energy. ∆λ = hmec

(1 − cos θ). Signals the “particle” behavior of photons.

Date : November 1, 2004.



2 GRE Quantum Review

1.9. Mossbauer. Crystal lattice absorbs energy of recoiling nucleus emitting photon:

E recoil = 1

2mv2/N = p2c2

2mc2N . Introduce a Doppler shift: ν observed = ν source

1+v/c1−v/c

≈ν sβ and reabsorption/resonant absorption can be detected: E recoil = h(ν o − ν s) =

hν sβ

1.10. Ramsauer-Townsend Effect. Scattering of electrons by noble-gas atoms.

See also §§3.4

1.11. “Wave-particle Duality”. deBroglie wavelength: λ = h/p = h/mv = h/√

2mE

2. Hydrogen

2.1. Bohr Model.

Energy levels: E n = − m

2 2

e2

4π0

21

n2

Quantization of e− wavelength: nλ = 2πrn

Radius of nth shell: rn = n2a0 = n2

4π0 2

me2

Bohr radius: r1 = a0 = 0.529 × 10−10m

2.2. Fine Structure. On the order of α4mc2. α is the fine structure constant. Due

to both the relativistic correction and spin-orbit coupling:

Relativistic correction: H = − p4

8m3c2

Spin-orbit correction: H =

e2

8π0

1

m2c2r3 S · L ∼ µ · B

2.3. Hyperfine Splitting. On the order of (me/m p)α4mc2. Due to magnetic inter-

action between the dipole moments of the electron and the proton.

3. Some standard potentials

3.1. Infinite Square Well. Width L.

Energy levels: E n =n2π2 2

2mL2

Wavefunction: ψn =

2

Lsinnπx

L



GRE Quantum Review 3

3.2. Harmonic Oscillator.

Potential: V =1

2kx2 =

1

2mω2x2, ω ≡

k

m

Ground state wavefunction: ψ(x) =mω

π 1/4

exp−

mω

2 x

2Energy levels: E n = (n +

1

2) ω

3.3. NMR. γ is the gyromagnetic ratio, i.e., µ = γ S , where µ is the magnetic dipole

moment and S is the spin. Larmor Frequency: ω = γB0

3.4. Transmission and Reflection. ψin = Aeikx, ψr = Be−ikx, and ψt = F eikx.

The transmission coeffecient T is defined to be T =|ψt|2|ψin|2

= F F ∗AA∗ . The reflection

coefficient R is such that R + T = 1.

4. Atomic Stuff

4.1. Work Function. Energy needed to remove an electron from the Fermi surface

of a metal. Basically just the photoelectric effect, with the “work function” being the

minimum energy needed to eject an electron in the first place. See also §§1.6.

4.2. Atomic Spectra.

absorption wavelengths:1

λ= R

1

k2− 1

n2

, n = k + 1, k + 2, . . .

This holds for one electron atoms. R is the Rydberg constant. k = 1 is the Lymanseries, k = 2 is the Balmer series. Remember that Moseley’s Law is basically the

same thing, but with the atomic number Z decreased due to screening.

4.3. Shell Models.

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