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Modern physics and Quantum Mechanics
Physical Systems, 8 Mar.2007 EJZ
• More angular momentum and H atom
• Compare to Bohr atom
• Applications: Bohr magneton, Zeeman
effect
• Brief review of modern physics and QM
• Conferences next week
• Next quarter
Quantization of angular momentum
Show that for ANY radial potential V(r) in the spherical
Schrödinger equation, both the total angular momentum and the
z-component are quantized.
Last week we discussed the momentum operators…
Spherical harmonics solve spherical Schrödinger
equation for any V(r)
Possible orientations of L and Lz (for l=2)
Example 7.1 (p.300), #7.12, 7.14 (p.332)
H-atom: quantization of energy for V= - kZe2/r
Solve the radial part of the spherical Schrödinger equation (next
quarter):
Do these energy values look familiar?
QM H-atom energy levels: degeneracy for states
with different qn and same energy
Selections rules for
allowed transitions:
l must change by one,
since energy hops are
mediated by a photon of
spin-one.
Dn = anything
Dm can = ±1 or 0
H-atom: wavefunctions Y(r,q,f) for V= - kZe2/r
We already have the angular part of the wavefunctions for any radial potential in
the spherical Schrödinger equation:
Y (r ,q ,  )  R(r )Ylm (q ,  ) where
Ylm (q ,  )  spherical harmonics
We can solve (next quarter) for R(r) ~ Laguerre Polynomials
H-atom wavefunctions ↔
electron probability distributions
Discussion: compare Bohr model to Schrödinger model for H atom.
A fourth quantum number: intrinsic spin
Since L 
l (l  1), let S 
s(s  1)
If there are 2s+1 possible values of ms,
and only 2 orientations of ms = z-component of s (Pauli),
What values can s and ms have?
Stern-Gerlach showed splitting due to spin, even
when l=0
l = 1, m = 0, ±1
l = 0, m = ±1/2
Spinning particles shift energies in B fields
Cyclotron frequency: An electron moving with speed v
perpendicular to an external magnetic field feels a Lorentz
force:
F=ma
(solve for w=v/r)
Solve for Bohr magneton…
Magnetic moments
shift energies in B fields
Spin S and orbit L couple to total angular momentum
J = L+ S
Spin-orbit coupling: spin of e- in magnetic field of p
Fine-structure splitting (e.g. 21-cm line)
(Interaction of nuclear spin with electron spin (in an atom) →
Hyper-fine splitting)
Total J + external magnetic field → Zeeman effect
Total J + external magnetic field → Zeeman effect
Total J + external magnetic field → Zeeman effect
History of Light quantization
• Stefan-Boltzmann blackbody had UV catastrophe
• Planck quantized light, and solved blackbody problem
• Einstein used Planck’s quanta to explain photoelectric effect
• Compton effect demonstrated quantization of light
• Corrollary: deBroglie’s matter waves, discovered by Davisson &
Germer
hc/l = Kmax + F
History of atomic models:
• Thomson discovered electron, invented plum-pudding model
• Rutherford observed nuclear scattering, invented orbital atom
• Bohr quantized angular momentum, for better H atom model.
• Bohr model explained observed H spectra, derived En = E/n2
and phenomenological Rydberg constant
• Quantum numbers n, l, ml (Zeeman effect)
• Solution to Schrodinger equation showed that En = E/l(l+1)
• Pauli proposed spin (ms=1/2), and Dirac derived it
deBroglie’s matter waves 
Bohr’s angular momentum
quantization
Compton Effect
h
Dl 
1  cosq 
me c
Quantum wells