Download QM lecture - The Evergreen State College

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
Schrödinger eqn. in spherical coords
with Coulomb potential
The time-independent SE
has solutions
 2 2
  n (r)  V n (r)  En n (r)
2m
 nlm (r )   (r ,q   )  Rnl (r ) Yl m (q   )
where
and Rnl(r)=
Plm = associated Legendre functions of argument (cosq) and
L=Laguerre polynomials
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
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
L  l (l  1) where l  0,1, 2,..., n  1
Lz  ml  L cosq
E
E1
n  l 
2
where E1  Bohr ground state
Angular momentum L:
from Classical physics to QM
L=rxp
Calculate Lx, Ly, Lz and their commutators:  Lx , Ly   i Lz
Uncertainty relations:  L  L  Lz
x
y
2
Each component does commute with
L2:
 L2 , L   0
Eigenvalues:
H nlm  En nlm , L2 nlm 
2
l (l  1) nlm , Lz nlm  m nlm
Spin - review
• Hydrogen atom so far: 3D spherical solution to Schrödinger
equation yields 3 new quantum numbers:
l = orbital quantum number
L  l (l  1)
ml = magnetic quantum number = 0, ±1, ±2, …, ±l
ms = spin = ±1/2
• Next step toward refining the H-atom model:
Spin
with
1
1 1
s

m


s  2 ( 2  1)
z
s
2
Total angular momentum J=L+s
J 
j ( j  1)
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
  s m 
1 1
2 2
 
  s m 
1 1
2 2
 
An electron (for example) can have spin up or spin down
  a    b   a   b 
NEW: 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 m  e  9.27 x1024 Joule  5.79 x109 eV
B
2me
Tesla
Gauss
Stern Gerlach measures me = 2 m 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 melectron with mproton.
Very strong external B, or “normal” Zeeman effect, decouples L
and S, so geff=mL+2mS.