Physics 451 Quantum mechanics I Fall 2012 Review 2 Karine Chesnel.
Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel.
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Transcript of Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel.
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