Select/Special Topics in Atomic Physics · Foldy-Wouthysen transformations of the Dirac Hamiltonian...

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

nmeV

THz

V I B G Y O R


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

H

Na ( ) =E E n

( , ) =E E n l


Non-relativistic QM of H atom : SO(3); SO(4)

5

NaE E(n, )=Non RelHE E(n)− =


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)?


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

7

n

n,j

E E (Schrodinger)E E (Dirac)=

=


8

n

n,j


=


9

n,l

n,j


= 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


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


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


Perturbative Methods

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!


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


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


15

What can a perturbation do to eigen-states ?– eigenvalues/eigen spectrum

- eigenfunctions- both eigenvalues and eigenfunctions

Degeneracy:

Removed – partially or wholly


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 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 ……

each termdepends on


( )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

19

What does the perturbation

do to

“good quantum numbers” ?


| = | 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)? ⟩



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 = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠


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


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 =

2p 'H

1p 'H


( )

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


( )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?


“ Remember the cardinal rule: Choose

unperturbed kets that diagonalize the

perturbation

-J.J.Sakurai in ‘Modern Quantum Mechanics’, (1985)

- - - - - -”


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

( )

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:




( )

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⎛ ⎞= − = −⎜ ⎟

⎝ ⎠


2ec

⎛ ⎞α = ⎜ ⎟

⎝ ⎠2 3

142

Rel. nK.E.

Z nE En

⎡ ⎤⎢ ⎥α⎛ ⎞ ⎢ ⎥Δ = − −⎜ ⎟ ⎛ ⎞⎢ ⎥⎝ ⎠ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦



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.


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


2p

Non-relativistic


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

2 2 degeneracy removed? s, p


2 2s, p



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 = ⎜ ⎟

⎝ ⎠

2 221 ...spin

4-orbit interactionp ' VH e

rm c r∂

⋅=∂

σ


( )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 = ?

?

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



( )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.


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 =

…. You have to be either a fool

or a masochist to use Lz, sz eigenkets as

the base kets for this problem….”


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

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



( ) ) ( ) ( ) )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

=−=−= ∑ ∑

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

43

2

2

0

0

• s, L

• s, s−

−

⎡ ⎤ =⎣ ⎦

⎡ ⎤ =⎣ ⎦


0 & 0z z• s, • s, s− −

⎡ ⎤ ⎡ ⎤≠ ≠⎣ ⎦ ⎣ ⎦

sn, ,m ,m suitable→hence not

jhence n, j,m suitable→

2 0

0z

• s, j

• s, j−

−

⎡ ⎤ =⎣ ⎦

⎡ ⎤ =⎣ ⎦

44

2 2

14spin orbit angular

radial partpart

e VHm c r r−

∂= σ ⋅

∂


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−

−= − ⋅ = ⋅

( )

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 ⋅

∂ ∂

−= − ⋅


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

( )

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

− = ⋅

⎧ ⎫⎪ ⎪⎡ ⎤⎪ ⎪= + − + − +⎨ ⎬⎢ ⎥⎛ ⎞ ⎣ ⎦⎪ ⎪+ +⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭


2s = σ

0 j

Use

n, , j,mψ =

2 2 21s j s2⎡ ⎤= − −⎣ ⎦i

Eigen-value

( )

2 32

2 23 3

31 14

14 12

spin orbit

Z j( j ) ( )ZeHm c n a

−

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭=

⎛ ⎞+ +⎜ ⎟⎝ ⎠


2

2ame

=


( )

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

−

⎧ ⎫+ − + −⎨ ⎬⎛ ⎞ ⎩ ⎭= ⎜ ⎟ ⎛ ⎞⎝ ⎠ + +⎜ ⎟⎝ ⎠


2ec

⎛ ⎞α = ⎜ ⎟

⎝ ⎠

( )

2 4

2 2

22

2

12

2

nmZ eE

nZ

mcn

= −

α= −


( )

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

⎛ ⎞+ +⎜ ⎟⎝ ⎠

Spin-orbit correction- same order as ‘relativistic mass’ correction


( )22

22n

ZE mc

nα

= −

( )( )

2

31 14

12 12

spin orbit n

j( j ) ( )H E Z

n−

⎧ ⎫+ − + −⎨ ⎬⎩ ⎭⇒ = − α

⎛ ⎞+ +⎜ ⎟⎝ ⎠


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 ) ( ) ( )

⎧ ⎫⎧ ⎫ ⎛ ⎞⎛ ⎞= + + − + − = + + − + − =⎨ ⎬ ⎨ ⎬⎜ ⎟⎜ ⎟⎩ ⎭ ⎝ ⎠⎝ ⎠⎩ ⎭

⎧ ⎫⎧ ⎫ ⎛ ⎞⎛ ⎞= − + − + − = − + − + − = − −⎨ ⎬ ⎨ ⎬⎜ ⎟⎜ ⎟⎩ ⎭ ⎝ ⎠⎝ ⎠⎩ ⎭


( )2

2 2

2 2 2 23

2 2

3

2 2 2

......Darwin8

8 2

'''Darwin

p '

r

eh div Em 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



( )2

2 2

2 2 2 23

2 2

3

2 2 2

......Darwin8

8 2

'''Darwin

p '

r

eh div Em 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


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”


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= −


( )2

2 2

2 2 2 23

2 2

3

2 2 2

......Darwin8

8 2

'''Darwin

p '

r

eh div Em c


H

r

= −

⎛ ⎞− π= − ∇ ⋅ = δ⎜

⎠

=

⎟⎝

0 so we need consider only 0

nR (r) r r→ →

=

( )2

n

ZE

nα

= −


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 ) ( )

⎧ ⎫= + + − + − =⎨ ⎬⎩ ⎭⎧ ⎫= − + − + − = − −⎨ ⎬⎩ ⎭


2 2p, s



2 2s, p



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 = ⎜ ⎟

⎝ ⎠


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= +


( )

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=

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


( )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

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 =


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


Relativistic H atom

62

n,l

n,j


=


Albert Abraham Michelson

WillisLamb

21 cm lineImportance in Astronomy

63

n,l

n,j


=


Albert Abraham Michelson0.07oK

Edward Purcell

21 cm lineImportance in Astronomy

Cold, but hot enough!

( )2

2

31 42

3

1

n

(n, , j)

Z nEn j

Relatin

ivistdependent

of

ici

iE E=

⎛ ⎞⎜ ⎟α

= −⎜ ⎟⎜ ⎟+⎝ ⎠

=Δ Δ∑


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

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