Page 1
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
[email protected] Unit 5(i) Lecture 24
Atomic Structure…..
Perturbative treatment of relativistic effects…
…..Schrodinger’s & Dirac’s QM
Select/Special Topics in Atomic Physics
1Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 2
nmeV
THz
V I B G Y O R
2Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 3
H atom
4
2 22= −
eEn
μvisible
4101.2 4340.1 Å
4860.74 and 6562.10 Å
→λ
3
← hν
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 4
H
Na ( ) =E E n
( , ) =E E n l
4Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 5
Non-relativistic QM of H atom : SO(3); SO(4)
5
NaE E(n, )=Non RelHE E(n)− =
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
2 2 1 for 2 1 for
RelativisticHE E(n, j) Degeneracy :( j ) fold n ( j ) fold n
=
+ − ≠ κ + − = κ
Jing-Ling Chen, Dong-Ling Deng, and Ming-Guang Hu, Phys.Rev. A 77, 034102 2008A. StahlhofenPhys. Rev. A 78, 036101 2008Relativistic QM of H atom : SO(4)
Relativistic QM of H atom : SO(4)?
Page 6
6Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Foldy-Wouthysen transformations of the Dirac Hamiltonian enabled the recognition of the relativistic effects in terms of
Relativistic K.E., spin-orbit interaction,
Darwin……
Hyperfine structure
electron-nucleus interaction
Page 7
7
n
n,j
E E (Schrodinger)E E (Dirac)=
=
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 8
8
n
n,j
E E (Schrodinger)E E (Dirac)=
=
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 9
9
n,l
n,j
E E (Schrodinger)E E (Dirac)=
= Albert Abraham MichelsonDec. 19, 1852 – May 9, 1931
"for his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid". – Nobel Prize 1907
Observed HF structure
Michelson-Morley expt: 1887Hyperfine structure: 1881
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 10
Above interactions: INTERNAL to the (e,p) system.
They can be viewed as ‘corrections’,‘modifications’
due to ‘perturbations’ over the previous-level-approximation.
10
Relativistic effects / Fine structure
Relativistic K.E., spin-orbit interaction, Darwin…..
Hyperfine structure electron-nucleus interaction
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 11
11
( )2
0 2
iH V( r )
m
− ∇= +
One-electron central field non-relativistic Hamiltonian
H' →
0H H H'= +Relativistic, FINE STRUCTUREHYPERFINE STRUCTUREExternal fields:
( )E,B
internal / external perturbations
Lamb shift.
E : Stark effect
B : Zeeman effect
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Perturbative Methods
Page 12
0H H H'= +
1 2' ' ' '
a bH' H H ... H H ....= + + + + +
….. Larger corrections / perturbationsmust not be ignored!
….. ALL corrections of comparable strength must be included!
12Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 13
13
What can a perturbation do to eigen-states ?
– does it change the eigenvalues / eigen spectrum?
- does it change the eigenfunctions?
- or, both eigenvalues and eigenfunctions?
Energy levels can changeas a result of the perturbation
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 14
14
What can a perturbation do to eigen-states ?– eigenvalues/eigen spectrum
- eigenfunctions- both eigenvalues and eigenfunctions
The change in energy can be in any direction, depending on the details of the perturbative interaction
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 15
15
What can a perturbation do to eigen-states ?– eigenvalues/eigen spectrum
- eigenfunctions- both eigenvalues and eigenfunctions
Degeneracy:
Removed – partially or wholly
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 16
16
What can a perturbation do to eigen-states ?
Transitions
- eigenvalues, eigenfunctions, and also occupation probabilities may change under perturbations
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 17
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 17
( )
2 23
0 0 0 02 2
2 22 2
0 02 2
2
0 2
'''Darwin n, ,m n, ,m
nn, ,m
Zeh (r )m c
EZe (r ) Zm c n
= = = =
= =
π= ψ δ ψ
π= ψ = = − α
( )2
2
1 3142
Rel. nK.E.
nE E Zn
⎡ ⎤⎢ ⎥⎢ ⎥Δ = − α −
⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( )( )
2
31 14
12 12
spin orbit n
j( j ) ( )H E Z
n−
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= − α
⎛ ⎞+ +⎜ ⎟⎝ ⎠
0=
0≠
all
In Unit 3, we only ‘mentioned’ these results ……
Page 18
each termdepends on
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 18
( )2
2
1 3142
Rel. nK.E.
nH E Zn
⎡ ⎤⎢ ⎥⎢ ⎥= − α −
⎛ ⎞⎢ ⎥+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( )( )
2
31 14
12 12
spin orbit n
j( j ) ( )H E Z
n−
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= − α
⎛ ⎞+ +⎜ ⎟⎝ ⎠
( )2
2
31 42
Relativistic Re lativisticn
all Z nE E En jtotalii
⎛ ⎞⎜ ⎟α
Δ = Δ = −⎜ ⎟⎜ ⎟+⎝ ⎠
∑
( )2 ''' nDarwin
Eh Zn
= − α
( )2
2
31 1 42
Relativisticn
Z nE En jnj
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟α
= + −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦
depends on j
Page 19
19
What does the perturbation
do to
“good quantum numbers” ?
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 20
| = | a aA ⟩ ⟩| = |
eigenvalue equa a
aa
tionA ⟩ ⟩label
Measurement: system ‘collapses’ into an eigenstate
What else can be measured?C.S.C.O.
Complete Set of Compatible Observables
Complete Set of Commuting Operators
| = | a,b b a b,B ⟩ ⟩ [ ],
: { , , ,......}−= −A B AB BA
CSCO A B C
| label(s)? ⟩
20Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 21
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 21
Non-relativistic (Schrodinger) formalism does not include spin.
Perturbative approach: ad-hoc introduction of the spin
0
non-relativitic (unperturbed)
s sn, ,m ,m n n, ,m ,mH (r, ) E (r, )ψ ζ = ψ ζ
Quantum numbers:
( )2 2 z zeigenvalues of H,L ,L , s ,ssn, ,m ,(s),m
12
1 1 22
1 00 1
ss
s
n, ,m ,m n, ,m m
m
(r, ) (r ) ( )
( ) c c
ψ ζ = ψ χ ζ
⎛ ⎞ ⎛ ⎞χ ζ = +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Page 22
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 22
Quantum numbers:2 z zeigenvalues of H,L ,L ,s
sn, ,m ,m
Alternative set of Quantum numbers:
2 2 zeigenvalues of H,L ,J , jjn, , j,m
Page 23
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 23
0
0 0 0
0 0
H H H'H E
E H'
= +
ψ = ψ
Δ = ψ ψ
First Order Perturbation Theory
0 sn, ,m ,m?
ψ =
Relativistic K.E. correction Darwin correction
spin-orbit correctionH'
0 jn, , j,m?
ψ =
Page 24
2p 'H
1p 'H
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 24
( )
2
42
3 2
2 2
2 2 2 2 2 2
12 8 2
1c l8 4
r 8
u
H''' mc e Bm m c mc
e ie ee div E Em c m
ep Apc
Vrc rc m
⎛ ⎞⎜ ⎟⎜ ⎟= β + − −β σ ⋅ +⎜ ⎟⎜ ⎟⎝ ⎠
+
⎛ ⎞−⎜ ⎟⎝ ⎠
∂∂
φ − − σ ⋅ + σ ⋅
Magnetic dipole term
Relativistic K.E. correction
Spin-orbit interaction (Thomas)
See: Slide No. 139 of STiAP Unit 3 on Relativistic H atom
3p 'H
‘internal’ perturbations
Darwin correctionzitterbewegung
Page 25
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 25
( )2
2 2
2 2 2 23
2 2
3
2 2 2
......Darwin8
8 2
'''Darwin
p '
r
eh div Em c
e Ze Zee (r )m c m c
H
r
= −
⎛ ⎞− π= − ∇ ⋅ = δ⎜
⎠
=
⎟⎝
2 221 .....spin
4-orbit interactionp ' VH
r re
m c∂
σ=∂
⋅
3 21
4
.....Relativistic 8
K.E.p 'Hm cp
−=
0
0 0
0 0
0
H H H'H E
E H'
= +
ψ = ψ
Δ = ψ ψ0
0
s
j
n, ,m ,m
or, n, , j,m ?
ψ =
ψ =
which states do we choose?
Page 26
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 26
“ Remember the cardinal rule: Choose
unperturbed kets that diagonalize the
perturbation
-J.J.Sakurai in ‘Modern Quantum Mechanics’, (1985)
- - - - - -”
Page 27
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 27
3 21
4
.....Relativistic 8
K.E.p 'Hm cp
−=
( )22 2 2 0L,p L, i L,− −−
⎡ ⎤⎡ ⎤ ⎡ ⎤= − ∇ = − ∇ =⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
1
4
3 2 is diagonal in ,m8
p 's
pHenm c
ce H ,m= −
4 4
1 3 2 3 28 8s sp pE n m m n m m
m c m cΔ = − = −
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
Page 28
( )
2 24 2 2
1 3 2 2 2
2 2 22 2
1 18 2 2 2 2
1 1 22 2n n n
p p pEm c m m c m c m
E V E E V Vm c m c
⎛ ⎞ ⎛ ⎞−Δ = − = − =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
− − ⎡ ⎤= − = − +⎣ ⎦
2ZeVr
= −
2 23 2
1 1 1 112
andr n a r n a
= =⎛ ⎞+⎜ ⎟⎝ ⎠
FIND:
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 28
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
Page 29
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 29
Page 30
( )
2 24 2 2
1 3 2 2 2
2 2 22 2
1 18 2 2 2 2
1 1 22 2n n n
p p pEm c m m c m c m
E V E E V Vm c m c
⎛ ⎞ ⎛ ⎞−Δ = − = − =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
− − ⎡ ⎤= − = − +⎣ ⎦
2ZeVr
= − 2 23 2
1 1 1 112
r n a r n a= =
⎛ ⎞+⎜ ⎟⎝ ⎠
&
( )
2 4 2 2 2
2 2 2 2
2 22 22 2
2 2
12 2
2 2
nmZ e me Z eE
n nZe Zmc mc
c n n
= − = −
α⎛ ⎞= − = −⎜ ⎟
⎝ ⎠
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 30
2ec
⎛ ⎞α = ⎜ ⎟
⎝ ⎠2 3
142
Rel. nK.E.
Z nE En
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
Page 31
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 31
2 2p, s
Non--relativisticSO(4)
4
1
2
3 2
13
14
8
2
p '
n
m cpH
Z nE En
=
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝
−
⎠⎣ ⎦
relativistic K.E.
Page 32
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 32
2s
Non-relativistic
Relativistic K.E.correction
4
1
2
3 2
13
14
8
2
p '
n
m cpH
Z nE En
=
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝
−
⎠⎣ ⎦
relativistic K.E.
( )
221 2
2
2
3 212 4 02
3 252
s ZE E
ZE .
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
α⎛ ⎞= − −⎜ ⎟⎝ ⎠
E=0
The intrinsically negative energy becomes more negative
Page 33
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 33
2p
Non-relativistic
Relativistic K.E.correction
4
1
2
3 2
13
14
8
2
p '
n
m cpH
Z nE En
=
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝
−
⎠⎣ ⎦
relativistic K.E.
221 2
2
2
3 212 4 12
72 12
p ZE E
ZE
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
α⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
E=0
The intrinsically negative energy becomes more negative
Page 34
2 2 degeneracy removed? s, p
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 34
2 2s, p
Non--relativisticSO(4)
Relativistic K.E.correction
4
1
2
3 2
13
14
8
2
p '
n
m cpH
Z nE En
=
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝
−
⎠⎣ ⎦
relativistic K.E.
221 2
72 12
p ZE E α⎛ ⎞ ⎛ ⎞Δ = ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
E=0
( )2
21 2 3 25
2s ZE E .α⎛ ⎞Δ = ⎜ ⎟
⎝ ⎠
Page 35
2 221 ...spin
4-orbit interactionp ' VH e
rm c r∂
⋅=∂
σ
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 35
( )2
2 2
2 2 2 23
2 2 2 2 2
3 ......Darwin8
8 2
'''Darw
'n
r
pi
eh div Em c
e Ze Zee (r )m c r
H
m c
= −
⎛ ⎞− π= − ∇ ⋅ = δ⎜ ⎟
⎠
=
⎝
3 21
4
.....Relativistic 8
K.E.p 'Hm cp
−=
0
0 0 0
0 0
0
H H H'H E
E H'
how to choose ?
= +
ψ = ψ
Δ = ψ ψ
ψ
QUESTIONS ? Write to: [email protected]
4
1
2
3 2
13
14
8
2
p '
n
m cpH
Z nE En
=
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝
−
⎠⎣ ⎦
relativistic K.E.
0
0
s
j
n, ,m ,m
or, n, , j,m ?
ψ =
ψ = ?
?
Page 36
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
[email protected] Unit 5(ii) Lecture 25
Atomic Structure…..
Perturbative treatment of relativistic effects…
…..Schrodinger’s & Dirac’s QM
Select/Special Topics in Atomic Physics
36Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 37
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 37
( )2
2 2
2 2 2 23
2 2 2 2 2
3 ......Darwin8
8 2
'''Darw
'n
r
pi
eh div Em c
e Ze Zee (r )m c r
H
m c
= −
⎛ ⎞− π= − ∇ ⋅ = δ⎜ ⎟
⎠
=
⎝
2 221 ...spin
4-orbit interactionp ' VH e
rm c r∂
⋅=∂
σ
3 21
4
.....Relativistic 8
K.E.p 'Hm cp
−=
0
0 0 0
0 0
0
H H H'H E
E H'
how to choose ?
= +
ψ = ψ
Δ = ψ ψ
ψ
4
1
2
3 2
13
14
8
2
p '
n
m cpH
Z nE En
=
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝
−
⎠⎣ ⎦
relativistic K.E.
Page 38
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 38
2 221
spin-orbit inte tion4
rac
p ' VHr
em c r
∂=
∂σ ⋅
3 21
4
.....Relativistic 8
K.E.p 'Hm cp
−=
0
0 0 0
0 0
H H H'H E
E H'
= +
ψ = ψ
Δ = ψ ψ
0
1
s
p '
n, ,m ,m
H
ψ =we used
in the case of
0 jor, n, , j,mψ = ... ?0 sn, ,m ,m
?ψ =
Page 39
…. You have to be either a fool
or a masochist to use Lz, sz eigenkets as
the base kets for this problem….”
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 39
2 221 .....spin
4-orbit interactionp ' VH
r re
m c∂
σ=∂
⋅
“Remember the cardinal rule: Choose
unperturbed kets that diagonalize the
perturbation ...
-J.J.Sakurai in ‘Modern Quantum Mechanics’, (1985) Section 5.3, page 305
0 j sUse n, , j,m n, ,m ,mψ = , not
Page 40
40
Good quantum numbers of the unperturbed
one-electron hydrogenic system sn, ,m ,m
sn, ,m ,mState vector C.S.C.O. : 2z zH, , ,s
j s= + In the presence of the spin-orbit interaction
C.S.C.O. 2 2zH, , j , j jn, , j,mState vector
sm ,m : no longer “good quantum numbers”reason: our perception of
“angular momentum” has now altered, having now considered the spin-orbit interaction
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 41
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 41
( ) ) ( ) ( ) )1 2
1 1 2 2
j j
1 2 1 2 1 2 1 2 1 2m j m j
j j jm j j m m m m j j jm =− =−
= ∑ ∑
1 2j j j= +
j s= +
( ) ) ( ) ( ) )s
12
j s s j1m2
ms jm s m m m m s jm
=−=−= ∑ ∑
Page 42
42
R 1 1 2 2 R 1 1 2 2U j m j m U j m j m=1 2
1 2' '1 1 2 2' '
1 1 2 2
j j( j ) ( j )' '
1 1 2 2m m m mm j m j
D j m D j m=− =−
= ∑ ∑
PCD STiAP Unit 2
1 2" "1 1 2 2
( j ) ( j )" "1 1 2 2 R 1 1 2 2 m m m mj m j m U j m j m D (R) D (R)= ×
Matrix element of the rotation operator in
the (direct) product states is the product
of the matrix elements of the rotation
operator in the ‘factor’ states.Sept.'12 PCD STiAP Unit 5 Perturbative
treatment of relativistic effects
Page 43
43
2
2
0
0
• s, L
• s, s−
−
⎡ ⎤ =⎣ ⎦
⎡ ⎤ =⎣ ⎦
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
0 & 0z z• s, • s, s− −
⎡ ⎤ ⎡ ⎤≠ ≠⎣ ⎦ ⎣ ⎦
sn, ,m ,m suitable→hence not
jhence n, j,m suitable→
2 0
0z
• s, j
• s, j−
−
⎡ ⎤ =⎣ ⎦
⎡ ⎤ =⎣ ⎦
Page 44
44
2 2
14spin orbit angular
radial partpart
e VHm c r r−
∂= σ ⋅
∂
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
2 221
4j jn, ,j,m n, ,j,mV
rE e
m c r∂∂
ψ ⋅Δ = ψσ
2 221
.....spin4
-orbit interaction
p ' VHr r
em c
∂σ=
∂⋅
0 j sUse n, , j,m n, ,m ,mψ = , not
( )2
2 2 3 2 2 3
1 2 1 4 2spin orbit
e s ZeH Ze sm c r m c r−
−= − ⋅ = ⋅
Page 45
( )
2 2 2 2
2 2 3
1 14 4
1 24
spin orbitradial
e V e VHm c r r m c r re sZe
m c r
−
∂ ∂= σ ⋅ = σ ⋅
∂ ∂
−= − ⋅
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 45
2s = σ
0 j
Use
n, , j,mψ =
j s= +
( ) ( )j j s s= + +i i 2 2 2j 2 s s= + +i
2 2 22 s j s= − −i
Page 46
( )
2 2 2 2
2 2 3
1 14
1 24
4spin orbitradial
e V e VHm c r r m c r re sZe
m c r
−
∂ ∂= σ ⋅ = σ ⋅
∂
= ⋅−
∂
−
( ){ }
2
2 2 3
2 3 2
2 23 3
12
1 1 112 212
spin orbitZeH sm c r
Ze Z j( j ) ( ) s(s )m c n a
− = ⋅
⎧ ⎫⎪ ⎪⎡ ⎤⎪ ⎪= + − + − +⎨ ⎬⎢ ⎥⎛ ⎞ ⎣ ⎦⎪ ⎪+ +⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 46
2s = σ
0 j
Use
n, , j,mψ =
2 2 21s j s2⎡ ⎤= − −⎣ ⎦i
Eigen-value
Page 47
( )
2 32
2 23 3
31 14
14 12
spin orbit
Z j( j ) ( )ZeHm c n a
−
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭=
⎛ ⎞+ +⎜ ⎟⎝ ⎠
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 47
2
2ame
=
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
( )
332 2 2
2 2 23
31 14
14 12
spin orbit
Z j( j ) ( )Ze meHm c n
−
⎧ ⎫+ − + −⎨ ⎬⎛ ⎞ ⎩ ⎭= ⎜ ⎟ ⎛ ⎞⎝ ⎠ + +⎜ ⎟⎝ ⎠
2ec
⎛ ⎞α = ⎜ ⎟
⎝ ⎠( )
424 2
3
31 14
14 12
spin orbit
j( j ) ( )eH Z mcc n
−
⎧ ⎫+ − + −⎨ ⎬⎛ ⎞ ⎩ ⎭= ⎜ ⎟ ⎛ ⎞⎝ ⎠ + +⎜ ⎟⎝ ⎠
Page 48
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 48
2ec
⎛ ⎞α = ⎜ ⎟
⎝ ⎠
( )
2 4
2 2
22
2
12
2
nmZ eE
nZ
mcn
= −
α= −
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
( )
424 2
3
31 14
14 12
spin orbit
j( j ) ( )eH Z mcc n
−
⎧ ⎫+ − + −⎨ ⎬⎛ ⎞ ⎩ ⎭= ⎜ ⎟ ⎛ ⎞⎝ ⎠ + +⎜ ⎟⎝ ⎠
( )( )
4 2
3
31 14
14 12
spin orbit
j( j ) ( )H Z mc
n−
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭= α
⎛ ⎞+ +⎜ ⎟⎝ ⎠
( )( )
2
31 14
12 12
spin orbit n
j( j ) ( )H E Z
n−
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭⇒ = − α
⎛ ⎞+ +⎜ ⎟⎝ ⎠
Page 49
Spin-orbit correction- same order as ‘relativistic mass’ correction
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 49
( )22
22n
ZE mc
nα
= −
( )( )
2
31 14
12 12
spin orbit n
j( j ) ( )H E Z
n−
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭⇒ = − α
⎛ ⎞+ +⎜ ⎟⎝ ⎠
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
1 3 1 3 3for 1 1 12 4 2 2 4
1 3 1 1 3for 1 1 1 12 4 2 2 4
j , j( j ) ( ) ( )
j , j( j ) ( ) ( )
⎧ ⎫⎧ ⎫ ⎛ ⎞⎛ ⎞= + + − + − = + + − + − =⎨ ⎬ ⎨ ⎬⎜ ⎟⎜ ⎟⎩ ⎭ ⎝ ⎠⎝ ⎠⎩ ⎭
⎧ ⎫⎧ ⎫ ⎛ ⎞⎛ ⎞= − + − + − = − + − + − = − −⎨ ⎬ ⎨ ⎬⎜ ⎟⎜ ⎟⎩ ⎭ ⎝ ⎠⎝ ⎠⎩ ⎭
Page 50
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 50
( )2
2 2
2 2 2 23
2 2
3
2 2 2
......Darwin8
8 2
'''Darwin
p '
r
eh div Em c
e Ze Zee (r )m c m c
H
r
= −
⎛ ⎞− π= − ∇ ⋅ = δ⎜
⎠
=
⎟⎝
0
0 0 0 0 0 H H H'H E E H'= +
ψ = ψ Δ = ψ ψ
0 s
,
n, ,m ,mψ =s
Darwin term is diagonal in , m m
hence we use
Page 51
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 51
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
( )2
2 2
2 2 2 23
2 2
3
2 2 2
......Darwin8
8 2
'''Darwin
p '
r
eh div Em c
e Ze Zee (r )m c m c
H
r
= −
⎛ ⎞− π= − ∇ ⋅ = δ⎜
⎠
=
⎟⎝
0
0 0
0 0
0
H H H'H E
E H'
= +
ψ = ψ
Δ = ψ ψ
2 23
0 02 22Ze (r )
mE
cΔ ψ
πδ= ψ
0 important→
onlyr
Page 52
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 52
smallr
0=l1=l 2=l
r 0 region; very small r
( 0) lR r r→ →
3=l
s
p df
2
2
( 1)' 1_D Schrodinger equation' ( ) ( )2l
l lV r V rm r
+= +
( ~ 0)R r →( ) 0 , .
R r more rapidlygreater the l
“centrifugal”
Page 53
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 53
2 23
3 0 0 0 02 2
2 22
0 02 2
2
02
n, ,m n, ,m
n, ,m
ZeE (r )m c
Ze (r )m c
= = = =
= =
πΔ = ψ δ ψ
π= ψ =0=
( )
2 4
2 2
22
2
12
2
nmZ eE
nZ
mcn
= −
α= −
Ref.: Bransden & Joachain – Physics of Atoms & Molecules / Ch 5
( )2
2 2
2 2 2 23
2 2
3
2 2 2
......Darwin8
8 2
'''Darwin
p '
r
eh div Em c
e Ze Zee (r )m c m c
H
r
= −
⎛ ⎞− π= − ∇ ⋅ = δ⎜
⎠
=
⎟⎝
0 so we need consider only 0
nR (r) r r→ →
=
( )2
n
ZE
nα
= −
Page 54
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 54
4
1
2
3 2
13
14
8
2
p '
n
m cpH
Z nE En
=
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝
−
⎠⎣ ⎦
relativistic K.E.
( )( )
2
2
2 2
2
14
31 14
1
2 12
p '
n
em c
j(
V
j ) ( )E E Z
Hr
n
rσ ⋅
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭Δ = − α
⎛ ⎞+
∂
+⎜ ⎟⎠
∂
⎝
=
1 j=2
s o
correction depends on
−
±
( )
2 23
2 2
2
3
3
2p
n
' Ze (r )m c
ZE
H
En
πδ
Δ = −
=
α
1 3for 1 12 41 3for 1 1 12 4
j , j( j ) ( )
j , j( j ) ( )
⎧ ⎫= + + − + − =⎨ ⎬⎩ ⎭⎧ ⎫= − + − + − = − −⎨ ⎬⎩ ⎭
Page 55
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 55
2 2p, s
Non--relativisticSO(4)
Page 56
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 56
2 2s, p
Non--relativisticSO(4)
Relativistic K.E.correction
4
1
2
3 2
13
14
8
2
p '
n
m cpH
Z nE En
=
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝
−
⎠⎣ ⎦
relativistic K.E.
221 2
72 12
p ZE E α⎛ ⎞ ⎛ ⎞Δ = ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
E=0
( )2
21 2 3 25
2s ZE E .α⎛ ⎞Δ = ⎜ ⎟
⎝ ⎠
Page 57
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 57
np
Non--relativistic
Spin-orbit correction
( )( )
2
2
2 2
2
14
31 14
1
2 12
p '
n
em c
j(
V
j ) ( )E E Z
Hr
n
rσ ⋅
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭Δ = − α
⎛ ⎞+
∂
+⎜ ⎟⎠
∂
⎝
=
( )32
2
2 2 2 12p p p
ZE E E
α= −
E=0
( )12
2
2 2 6p p p
ZE E E
α= +
Page 58
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 58
( )
2 23
2 2
2
3
3
2p
n
' Ze (r )m c
ZE
H
En
πδ
Δ = −
=
α
nE
Non--relativistic
( )2
nn En
EZα
−
E=0Darwincorrection
0only for=
Page 59
Reference:Fig.5.2 in Bransden & Joachain
3 DarwinEΔ2 s oE −Δ
1 Rel.K.E.EΔ
( )1 2 3 + +E E EΔ Δ Δ
59
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
( )2
3 n
ZE E
nα
Δ = −( )( )
22
31 14
12 12
n
j( j ) ( )E E Z
n
⎧ ⎫+ − + −⎨ ⎬⎩ ⎭Δ = − α
⎛ ⎞+ +⎜ ⎟⎝ ⎠
2
13
142
nZ nE En
⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Energyin cm‐1
eV
0.21 0.260 x 10‐41.19 1.475 x 10‐40.12 0.148 x 10‐40.24 0.298 x 10‐4
Energyin cm‐1
eV
0.73 0.905 x 10‐40.09 0.111 x 10‐40.46 0.570 x 10‐4
Page 60
52
3d3 32 2
3 3d , p12
3s
12
1s
1 32 2
2 2p , s
32
2p
1 2 3E E EΔ + Δ + Δ
R
Non-relativistic H atom
Reference:
Fig.5.1 in Bransden & JoachainPhysics of Atoms and Molecules(1985)
3n =
2n =
1n =
60Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Page 61
52
3d3 32 2
3 3d , p12
3s
12
1s
1 12 2
2 2p , s
32
2p
Non-relativistic1 2 3E E EΔ + Δ + Δ
FurtherLamb shift
3n =
2n =
1n =Reference:
Fig.5.1 in Bransden & JoachainPhysics of Atoms and Molecules(1985)
Energy in cm‐1
eV
1.46 1.812x 10‐4
0.365 0.453x 10‐4
0.091 0.113x 10‐4
0.108 0.134x 10‐4
0.036 0.044 x 10‐4
0.018 0.022 x 10‐4
61Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Relativistic H atom
Page 62
62
n,l
n,j
E E (Schrodinger)E E (Dirac)=
=
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Albert Abraham Michelson
WillisLamb
21 cm lineImportance in Astronomy
Page 63
63
n,l
n,j
E E (Schrodinger)E E (Dirac)=
=
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects
Albert Abraham Michelson0.07oK
Edward Purcell
21 cm lineImportance in Astronomy
Cold, but hot enough!
Page 64
( )2
2
31 42
3
1
n
(n, , j)
Z nEn j
Relatin
ivistdependent
of
ici
iE E=
⎛ ⎞⎜ ⎟α
= −⎜ ⎟⎜ ⎟+⎝ ⎠
=Δ Δ∑
Sept.'12 PCD STiAP Unit 5 Perturbative treatment of relativistic effects 64
QUESTIONS ? Write to: [email protected]
( )2
2
31 1 42
Relativisticnj n
Z nE En j
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟α
= + −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦
Perturbation
Methods in
Atomic
Spectroscopy
Next, Unit 6: Probing the atom
Interactions of atoms with EM radiation and with
neutral/charged elementary/composite particles