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

Schrödinger eqn in spherical coordinatesSeparation 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 3D2

2 ( , )( , ) ( , )

2

tt V t i

m t

r

r r

has solutions of form

where n(r,t) solves

Recall how to solve this using separation of variables…

1

( , ) ( )ni E t

n nn

t c e

r r

22 ( ) ( ) ( )

2 n n n nV Em

r r r

Separation of variables

To solve

Let

Then the 3D diffeq becomes two diffeqs (one 1D, one 2D)

Radial equation

Angular equation

22 ( ) ( ) ( )

2 n n n nV Em

r r r

( ) ( , ) ( ) ( )r R r Y r

Solving the Angular equation

To solve

Let Y() = and separate variables:

The equation has solutions = eim (by inspection)

and the equation has solutions = C Plm(cos) where

Plm = associated Legendre functions of argument (cos).

The angular solution = spherical harmonics:

Y()= C Plm(cos) eim where 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 Pl

m=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

1

12

( 1) 0,1,2,..., 1

cosz l

l l where l n

L m

EE where E Bohr ground state

n l

L

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:

,x y zL L i L

2x yL L zL

2 , 0L L

2 2, ( 1) ,nlm n nlm nlm nlm z nlm nlmH E L l l L m

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|

( 1)l l L

1 12 2( 1)s 1

2z ss m

( 1)j j J

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…

1 12 2s m 1 1

2 2s m

a b a b

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 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 geff=mL+2mS.

24 99.27 10 5.79 102B

e

e Joule eVx x

m Tesla Gauss