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 L z (for l=2) Example 7.1 (p.300), #7.12, 7.14 (p.332)

H-atom: quantization of energy for V= - kZe 2 /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.  n = anything  m can = ±1 or 0

H-atom: wavefunctions  (r,  ) for V= - kZe 2 /r We already have the angular part of the wavefunctions for any radial potential in the spherical Schrödinger equation: 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 If there are 2s+1 possible values of m s, and only 2 orientations of m s = z-component of s (Pauli), What values can s and m s have?

Stern-Gerlach showed splitting due to spin, even when l=0 l = 1, m = 0, ±1l = 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  =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

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/ = K max + 

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 E n = E/n 2 and phenomenological Rydberg constant Quantum numbers n, l, m l (Zeeman effect) Solution to Schrodinger equation showed that E n = E/  l(l+1) Pauli proposed spin (m s =  1/2), and Dirac derived it

Compton Effect deBroglie’s matter waves  Bohr’s angular momentum quantization

Quantum wells