1. Quantum Ch.4 - continued Physical Systems, 27.Feb.2003 EJZ Recall solution to Schrödinger eqn in spherical coordinates with Coulomb potential (H atom) Work on HW help sheet (linked to Help page) – Probs.1 and 10. Angular Momentum - Minilecture by Don Verbeke (Do Prob 4.18, and 4.20 p.150 as you did Prob.1 above) Spin - Minilecture by Andy Syltebo – Do the example on p.157, try problem 4.28 together

2. Schrödinger eqn. in spherical coords with Coulomb potential The time-independent SE has solutions where and R nl (r)= P l m = associated Legendre functions of argument (cos  ) and L=Laguerre polynomials

3. Quantization of l and m In solving the angular equation, we use the Rodrigues formula to generate the Legendre functions: “Notice that l must be a non-negative integer for [this] to make any sense; moreover, if |m|>l, then this says that P l m =0. For any given l, then there are (2l+1) possible values of m:” (Griffiths p.127)

4. Solving the Radial equation…

5. …finish solving the Radial equation

6. Hydrogen atom: a few wave functions Radial wavefunctions depend on n and l, where l = 0, 1, 2, …, n-1 Angular wavefunctions depend on l and m, where m= -l, …, 0, …, +l

7. Angular momentum L: review from Modern physics Quantization of angular momentum direction for l=2 Magnetic field splits l level in (2l+1) values of m l = 0, ±1, ± 2, … ± l

8. Angular momentum L: from Classical physics to QM L = r x p Calculate L x, L y, L z and their commutators: Uncertainty relations: Each component does commute with L 2 : Eigenvalues:

9. Spin - review Hydrogen atom so far: 3D spherical solution to Schrödinger equation yields 3 new quantum numbers: l = orbital quantum number m l = magnetic quantum number = 0, ±1, ±2, …, ±l m s = spin = ±1/2 Next step toward refining the H-atom model: Spin with Total angular momentum J=L+s with j=l+s, l+s-1, …, |l-s|

10. Spin - new Commutation relations are just like those for L: Can measure S and S z simultaneously, but not S x and S y. Spinors = spin eigenvectors An electron (for example) can have spin up or spin down NEW: operate on these with Pauli spin matrices …

11. Total angular momentum: Multi-electron atoms have total J = S+L where S = vector sum of spins, L = vector sum of angular momenta Allowed transitions (emitting or absorbing a photon of spin 1) ΔJ = 0, ±1 (not J=0 to J=0) ΔL = 0, ±1 ΔS = 0 Δm j =0, ±1 (not 0 to 0 if ΔJ=0) Δl = ±1 because transition emits or absorbs a photon of spin=1 Δm l = 0, ±1 derived from wavefunctions and raising/lowering ops

12. Review applications of Spin Bohr magneton Stern Gerlach measures  e = 2  B : Dirac’s QM prediction = 2*Bohr’s semi-classical prediction Zeeman effect is due to an external magnetic field. Fine-structure splitting is due to spin-orbit coupling (and a small relativistic correction). Hyperfine splitting is due to interaction of  electron with  proton. Very strong external B, or “normal” Zeeman effect, decouples L and S, so g eff =m L +2m S.