Physics 452

Quantum mechanics II

Winter 2011

Karine Chesnel

Homework

Phys 452

Thursday Jan 27: Homework # 6Pb 6.16, 6.17, 6.18, 6.21

Phys 452

Quiz 8a

Which statement is true regarding the spin-orbit coupling?

A. The SO coupling is the effect of an external magnetic field on the spin of the electron

B. The SO coupling does not occur in the presence of an external magnetic field

C. The SO coupling includes the relativistic correction

D. The SO coupling describes the interaction between different angular momenta

E. The SO coupling is magnetic interaction that lift completely the degeneracy of atomic levels

Phys 452

The spin-orbit coupling

Magnetic interaction

.magE M B

S L

Interaction betweenthe spin S and the orbital angular momentum L

of the electron in the hydrogen atom

S

“Classical view”

B

e+

e-

2'

2 2 30

1.

8SO

eH L S

m c r

Pb 6.16

Phys 452

The spin-orbit coupling

• First-order correction to the energies:

20

12

( 1) ( 1) ( 1)

( 1/ 2)( 1)

n

SO

E n j j l l s sE

mc l l l

2

' 2 2 22 2 3

0

1 1

8 2SO

eH J L S

m c r

• Perturbation:

Phys 452

The fine structure of hydrogen

Bohr’senergy

E =

2 20

2

1

2n

mcE

n

Relativisticcorrection

+

20

12

43

2 1/ 2n

n

E nE

mc l

Fine structure

Spin-orbit coupling

+

20

12

( 1) ( 1) ( 1)

( 1/ 2)( 1)

n

n

E n j j l l s sE

mc l l l

Phys 452

The fine structure of hydrogen

00

2

41 32 1/ 2

nnj n

E nE E

mc j

New relevant quantum numbers: n, l, s, j and mj

Pb 6.18

E

1n

0l

1/ 2j

2n

1l

1/ 2j 3 / 2,1/ 2j

3n

2l

1/ 2j 3 / 2,1/ 2j 5 / 2,3 / 2j

Total energy:

Pb 6.17

Phys 452

Quiz 8b

Does the fine structure correction lift the degeneracyof the atomic levels ?

A. Not at all

B. Yes, partially

C.Yes, completely

D. It depends on the speed of the electron

E. It depends on external fields

Phys 452

Quiz 8c

For a given (n, l, j), what isthe degeneracy of the level Enj?

A.

B.

C.

D.

E.

22d n

2 (2 1)d n j

(2 1)d j s

(2 1)(2 1)(2 1)d j l s

2 1d j

Phys 452

Zeeman effect

S

eM S

m

ˆ ˆ

2L

eM L

m

• Effect of an external field

“Classical view”

e-

S

L

Bext .Zeeman extE M B

' ( 2 ).2Z ext

eH L S B

m

Phys 452

Zeeman effect

“Classical view”

e-

S

L

Bext

Weak-field Strong fieldIntermediatefield

Fine structuredominates

Zeeman effectdominates

• Comparing: and extB intB

' '' SO ZH H H

Pb 6.20

Phys 452

Zeeman effect

Weak -field

intextB B

20

21tot n nj B j j extE E c g m Bn

e-

S

L

Bext

, , , jn l j mGood eigenstates:

1Z B j j extE g m B

with Lande factor:( 1) ( 1) 3 / 4

12 ( 1)j

j j l lg

j j

Pb 6.21

Phys 452

Zeeman effect

e-

S

L

Bext

Strong -field

intextB B 1 2Z B l S extE m m B

01 21

3

( 1)3

4 ( 1/ 2)( 1)l s

fs

l l m mEE

n n l l l

, , , ,l sn l m s mGood eigenstates:

Pb 6.23,6.24

Phys 452

Zeeman effect

e-

S

L

Bext

intermediate -field

intextB B

0 21l stot n nj B ext m mE E c B d

' '' SO ZH H H

Clebsch-Gordan

, , , jn l j m , , , ,l sn l m s m

Pb 6.25