QM in 3D Quantum Ch.4, Physical Systems, 24.Feb.2003 EJZ

QM in 3DQuantum Ch.4, Physical Systems, 24.Feb.2003 EJZ Schrödinger eqn in spherical coordinates Separation of variables (Prob.4.2 p.124) Angular equation (Prob.4.3 p.128 or 4.23 p.153) Hydrogen atom (Prob 4.10 p.140) Angular Momentum (Prob 4.20 p.150) Spin

Schrödinger eqn. in spherical coords. The time-dependent SE in 3D has solutions of form where Yn(r,t) solves Recall how to solve this using separation of variables…

Separation of variables To solve Let Then the 3D diffeq becomes two diffeqs (one 1D, one 2D) Radial equation Angular equation

Solving the Angular equation To solve Let Y(q,f) = Q(q)F(f) and separate variables: The f equation has solutions F(f) = eimf (by inspection) and the q equation has solutions Q(q) = C Plm(cosq) where Plm = associated Legendre functions of argument (cosq). The angular solution = spherical harmonics: Y(q,f)= C Plm(cosq) eimfwhere C = normalization constant

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 Plm=0. For any given l, then there are (2l+1) possible values of m:” (Griffiths p.127)

Solving the Radial equation…

…finish solving the Radial equation

Solutions to 3D spherical Schrödinger eqn Radial equation solutions for V= Coulomb potential depend on n and l(L=Laguerre polynomials, a = Bohr radius) Rnl(r)= Angular solutions = Spherical harmonics As we showed earlier, Energy = Bohr energy with n’=n+l.

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

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 ml = 0, ±1, ± 2, … ±l

Angular momentum L: from Classical physics to QM L = r x p Calculate Lx, Ly, Lz and their commutators: Uncertainty relations: Each component does commute with L2: Eigenvalues:

Spin - review • Hydrogen atom so far: 3D spherical solution to Schrödinger equation yields 3 new quantum numbers: l = orbital quantum number ml = magnetic quantum number = 0, ±1, ±2, …, ±l ms = 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|

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

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 Δmj =0, ±1 (not 0 to 0 if ΔJ=0) Δl = ±1 because transition emits or absorbs a photon of spin=1 Δml = 0, ±1 derived from wavefunctions and raising/lowering ops

Review applications of Spin Bohr magneton Stern Gerlach measures me = 2 mB: 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 melectron with mproton. Very strong external B, or “normal” Zeeman effect, decouples L and S, so geff=mL+2mS.

QM in 3D Quantum Ch.4, Physical Systems, 24.Feb.2003 EJZ