P460 - spin-orbit 1

Energy Levels in Hydrogen • Solve SE and in first order get (independent of L):

• can use perturbation theory to determine:

magnetic effects (spin-orbit and hyperfine e-A)

relativistic corrections

• Also have Lamb shift due to electron “self-interaction”. Need QED (Dirac eq.) and depends on H wavefunction at r=0 (source of electric field). Very small and skip in this course

2

6.13

n

eVEn

P460 - spin-orbit 2

Spin-Orbit Interactions • A non-zero orbital angular momentum L produces

a magnetic field

• electron sees it. Its magnetic moment interacts giving energy shift

• in rest frame of electron, B field is (see book):

• convert back to lab frame (Thomas precession due to non-inertial frame gives a factor of 2). Energy depends on spin-orbit coupling

BSg

BF

Ldr

rdV

remcB

bs

)(112

EvB

rvAB

use

c

21

LSdr

dV

r

g

emcE bs

1

2

12

P460 - spin-orbit 3

Spin-Orbit: Quantum Numbers • The spin-orbit coupling (L*S) causes ml and ms to

no longer be “good” quantum numbers

• spin-orbit interactions changes energy.

• In atomic physics, small perturbation, and can still use H spatial and spin wave function as very good starting point. Large effects in nuclear physics (and will see energy ordering very different due to couplings)

0,

,:

)(2

22

2

HLHLLH

LLionseigenfunctalso

EHandrVHif

zzz

z

m

commutedonotLLLas

LSLSLSLLSL

SLaHH

zyx

zzzyyxxz

,,0

,,

P460 - spin-orbit 4

SL Expectation value • Determine expectation value of the spin-orbit

interaction using perturbation theory. Assume J,L,S are all “good” quantum numbers (which isn’t true)

• assume H wave function is ~eigenfunction of perturbed potential

)(

)(

2

0,

22221

22221

222

SLJSL

SLJSL

orSSLLJ

SLifSLJ

43

21

21

2

)1(,

:20

)1()1()1(2

sswithllj

valueslfor

sslljjSL

P460 - spin-orbit 5

SL Expectation value • To determine the energy shift, also need the

expectation value of the radial terms using Laguerre polynomials

• put all the terms together to get spin-orbit energy shift. =0 if l=0

0)12)(1(

221*1

14

1

320

33

330

2

ldrr

needas

lllnarr

rre

drdV

r

)12)(1(

))1()1((3

432

0

llln

lljjEE

SL

L=1

J=3/2

j=1/2

P460 - spin-orbit 6

Spin Orbit energy shift • For 2P state. N=2, L=1, J= 3/2 or 1/2

• and so energy split between 2 levels is

L=1

J=3/2

j=1/2eVE

EEE

split

split

5105.4

1371

21

23

48

2

48

)2(

485218

)21(

)12)(1(

)1()1((

204

323

212

0

204

325

232

0

3432

0

21

23

EEE

EEE

llln

lljjEE

SL

P460 - spin-orbit 7

Relativistic Effects

• Solved using non-relativistic S.E. can treat relativistic term (Krel) as a perturbation

• <V> can use virial theorom

23

4

23

42

24222

882 cm

pK

cm

p

m

p

mccmpcmET

rel

VEVEEEVEbut

VVEEK

VEK

ESEV

KEvEE

nn

mcrel

mcmp

mcrel

mp

relnn

,0,

)2(

)()(

.).(

22

22

21

2

212

221

2

2

2

2

2

2

nnaZe

rZe EV

o22

0

2

0

2

41

4

P460 - spin-orbit 8

Relativistic+spin-orbit Effects• by integrating over the radial wave function

• combine spin-orbit and relativistic corrections

• energy levels depend on only n+j (!). Dirac equation gives directly (not as perturbation)

)12(22

412

42

30

2

0

2

)()(

lna

Zer

Zeo

V

)( 83

1212

3

44

nlenZ

rel cmK

):()( 2

143

122

43

122

)12)(1()1()1(

3

20

43

3

20

ljuse

EE

njn

E

nlllllljj

n

E

relSL

P460 - spin-orbit 9

Energy Levels in Hydrogen• Degeneracy = 2j+1

• spectroscopic notation: nLj with L=0 S=state, L=1 P-state, L=2 D-state

E

N=1

N=3

N=2

223,31,0

443,32,1

632

21

21

23

23

25

21

23

25

PSjl

DPjl

Djl

222,21,0

421

21

21

23

21

23

PSjl

Pjl

210212

1 Sjl

# states

P460 - spin-orbit 10

Hyperfine Splitting• Many nuclei also have spin

• p,n have S=1/2. Made from 3 S=1/2 quarks (plus additional quarks and antiquarks and gluons). G-factors are 5.58 and -3.8 from this (-2 for electron).

• Nuclear g-factors/magnetic moments complicated. Usually just use experimental number

• for Hydrogen. Let I be the nuclear spin (1/2)

• have added terms to energy. For S-states, L=0 and can ignore that term

Bp

epp

ppp m

mgI

g

58.5

eppnuc bLaE

P460 - spin-orbit 11

Hyperfine Splitting• Electron spin couples to nuclear spin

• so energy difference between spins opposite and aligned. Gives 21 cm line for hydrogen (and is basis of NMR/MRI)

)2

3

2

123(

2,1

2

3

2

12

2,0

))1()1()1((2

2

2

2

ISfalignedspins

ISfoppositespins

iissffIS

SIFletISep

eVffE 6106)]0()1[(