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

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

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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)

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)

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.