V.A. Fock's discovery of "hidden" O(4) symmetry of the H-atom and

dynamical group theory

Alexander V. Gorokhov,Samara State University, Russiagorokhov@samsu.ru

2

Contents

Introduction V.A. Fock and V. Bargmann approaches to the description of the H- atom symmetryDynamical algebras and groups in Quantum PhysicsCoherent States (CS), Path Integrals and “Classical” EquationsOpen Systems. Fokker – Planck equations in CS representation Summary

3

Introduction“In the thirties, under the demoralizing influence of quantum – theoretic perturbation theory, the mathematics required of a theoretical physicist was reduced to a rudimentary knowledge of the Latin and Greek alphabets.” R. Iost

Lie groups and Lie algebras have been successfully applied in quantum mechanics since its inception.

4

V.A. Fock and V. Bargmann approaches to the description of the H- atom symmetry

( )

2 2 42 20 0 0

02 2

12

0

ˆ ( ) ( );1 ˆˆ , , 1,2,... 27 .

2 2(3), ( ), dim 2 1;

2 1 . "Accidental"degeneracyof H-atomlevels

n

l l

n

l

H r rZ e Z eH p n Z эвr n

R SO T R T l

l n

μμ

−

=

Ψ = Ψ

= − = − = = = ⋅

∈ = +

+ =∑

r r

r

h

E

EE E

1935, V.A. Fock. Analytical approach. SO(4)

22 ' '0

22 '

( ) d( )2 2

Zep p ppp pμ π

⎛ ⎞ Ψ− Ψ =⎜ ⎟

⎝ ⎠ −∫

r rr

r rhE

5

6

( ) ( )

( )

' ' ' ' ' '

22 ' 2 2'

220

0

' '

' ' '

2l0

( ) d ψ ( , , )dψ ( ) ψ ( , , ) ;2 2 4 sin / 2

, ; d sin d sin d d ,

cos cos cos sin sin cos ,

cos cos cos sin sin cos .

2ψ ( , , ) П ( , ) Y ( , ), П ( , )nlm l lm

Zep

n nπ

η ψ ξ ξ η χ θ φξ χ θ φπ ξ π ωξ ξ

α μη α χ χ θ θ φ

ω χ χ χ χ γ

γ θ θ θ θ φ φ

χ θ φ χ θ φ χπ

Ω= ⇒ =

−

⋅= = Ω =

= +

= + −

= ⋅

∫ ∫

h

( ) ( ) ( )

2

22 20

sin d 1.

N Ynlm nlm nlmp p p

χ χ

ξ−

=

Ψ = ⋅ + ⋅

∫r r

Group SO(4) as symmetry group of H – atom:

( )( )

2 20

2 20 0

2 2 20 0 0

0 02 2 2 20 0

( ) ( ),

( , ) cos ,sin sin cos ,sin sin sin ,sin cos , 1;

2, , 2 | |, = .2

p p p

p p p p ppp p p p

ψ ξ

ξ ξ ξ χ χ θ φ χ θ φ χ θ ξ ξ

ξ ξ μμ

= + Ψ

= = + =

−= = = −

+ +

r r

r r

r rrr r E E

7

V. Bargmann, 1936

8

V. Bargmann, 1936. Algebraic approach

( )1ˆˆ ˆ ˆˆ ˆ ˆ ˆˆ , , ,2

ˆˆˆ ˆ, , 0.

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ, , , , , 2 .i j ijk k i j ijk k i j ijk k

rH L r p p i A L p p Lr

H L H A

L L i L L A i A A A H i L

μ αμ

ε ε ε

⎛ ⎞= × = − ∇ = × − × +⎜ ⎟

⎝ ⎠⎡ ⎤⎡ ⎤ = =⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤⎡ ⎤ = = = − ⋅⎣ ⎦ ⎣ ⎦ ⎣ ⎦

rrr r rr r r r rh

rr

h h h

0, 1.1 ˆˆ ,

ˆ2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , , , , .

i i

i j ijk k i j ijk k i j ijk k

N AH

L L i L L N i N N N i Lε ε ε

< =

=−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎣ ⎦ ⎣ ⎦ ⎣ ⎦

hE

9

( ) ( ) ( ) ( )

1 2 1 2

1 2

2 2 0

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

( ) ( ) ( )

( , )1 2

( , )1 2

ˆ ˆ ˆ ˆ ˆ ˆ0, .ˆ21 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, ;2 2

ˆ ˆ ˆ, , ( , 1,2.)

1(4) (3) (3) ; .2

dim (2 1) (2 1

a b bi j ab ijk k

j j j j

j j

L N N L L NH

J L N J L N J J

J J i J a b

nSO SO SO T T T j j

T j j

δ ε

⋅ = ⋅ = + = −

= + = − =

⎡ ⎤ = ⋅ =⎣ ⎦−

= × → = ⊗ = =

= + ⋅ +

r r r r r r

r r r r r r r r

E

2) .n≡

0

0, (4) (3,1);0, (4) ,SO SOSO

> →= →EE G

10

Symmetry group applications

Group symmetry of NGroup symmetry of N-- dimensional dimensional oscillator oscillator –– SU(N), Jauch, Hill, 1940.SU(N), Jauch, Hill, 1940.Classification of states, selection rules for Classification of states, selection rules for atoms, molecules solids.atoms, molecules solids.Lorentz and PoincarLorentz and Poincaréé groups unitary groups unitary representation and classification of representation and classification of elementary particleselementary particles.

J.L. Birman, CUNY, USA

Ya.A. Smorodinsky, JIRN, Dubna

Yu.N. Demkov, Leningrad University

11

Dynamical Symmetries

• Classification of hadrons: SU (3), SU (6) symmetry of the flavors, the quarks, the mass formulas

• Dynamical symmetries of quantum systems

• Spectrum generating algebra

• Coherent states

M. Gell-Mann

Y. Neeman

Asim Orhan Barut

12

Eightfold way and dynamical (spectrum generating) groups

The baryon octet

Eightfold Way, classification of hadrons into groups on the basis of their symmetrical properties, the number of members of each group being 1, 8 (most frequently), 10, or 27. The system was proposed in 1961 by M. Gell-Mann and Y. Neʾeman. It is based on the mathematical symmetry group SU(3); however, the name of the system was suggested by analogy with the Eightfold Path of Buddhism because of the centrality of the number eight..

SU(3) … SUI(2)8 = 1 ∆ 2 ∆ 2 ∆ 3

13

A Simple Example - Harmonic Oscillator

( )

( ) ( ) ( )

2 2

2 2 2 20 1 2

1 2

0 0

k+

1ˆ ˆ ˆ , ( 1)21 1 1ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆˆ, , ;4 4 4

1 1ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ, , ,2 2

ˆ ˆ ˆ ˆ ˆ ˆ, , , 2 .

(1,1) (2, ) (2, ) (2,1).1 3T , , .4 4

H p x m

K p x K p x K px xp

K K i K K a a K aa

K K K K K K

SU SL R Sp R SO

k

ω

+ +± + −

± ± + −

= + = = =

= + = − = +

⎧ ⎫⎪ ⎪⎪ ⎪= ± = =⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭⎡ ⎤ ⎡ ⎤=± =−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= = ≈

=

Ndim.oscillator: W ^ (2 , )N Sp N R−

14

Dynamical symmetries:Dynamical symmetries:condensed mattercondensed matter, , quantum opticsquantum optics

Murray Gell-Mann,

lecturing in 2007,Ennackal Chandy

George Sudarshan

SU(2), SU(3), SU(n), SO(4,2), SU(m,n), Sp(2N,R), W(N)^Sp(2N,R),…

15

16

Dynamical symmetry

In a linear case it is possible to find exact solution:• Energy levels and corresponding wave functions (time independentHamiltonian);

• Transition probabilities (time dependent Hamiltonian);

• Quasi energy and quasi energy states (periodic Hamiltonian).Aharonov - Anandan geometric phase

R. Feynman’s method of operator exponent disentanglement

17

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

ˆ ˆˆ ( ),' "

' "ˆ ...

A,A A A ...,

A A A

k kH f A A

H f C f C f Cα α α

= ∈

⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⊃ ⊃ ⊃

Lie groups and an energy levels calculation

Molecular rotational- vibrational levels

F. Iachello, R.D. Levine

(4) (4) (3) (2)u so so so⊃ ⊃ ⊃

H-atom, SO(4), SO(4,2)

V.A. Fock, (1935) V. Bargmann, (1936)

A.O. Barut, H. Kleinert, Yu.B. Rumer, A.I. Fet (1971)

SU(1,1), SU(N,1), Sp(2n,R), quantum optics, superfluidity

18

Molecular spectra & vibronic transitions(2 , )nW Sp N∧ R

( )

( ) ( )

[ ]( ) [ ] [ ] [ ]

'1 2

1 1 '0 0

' '

;ˆ , , ,..., ;

ˆ ˆ ˆ ˆ' , ' ,

, |

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆV =T( )D( ) U(R)ˆI V

Nq Rq q q q q q

R L L q

H

L r r r r

m n

VH

m n m n

V

δ ρ

− −

+

= +Δ =

= Δ = Δ Δ = −

⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦

′ =

< >=

r r r r

r r r r r

r r

( ) *[ ] [ ]2 2

[ ],[ ]

1exp [ ] [ ]2 [ !] [ !]

ˆ ˆV Vv n

v n

v nv nβ αβ α β α⎡ ⎤= − + ×⎢ ⎥ ⋅⎣ ⎦

∑r rr rr r

-group and Franck – Condon overlap Integrals,

(harmonic approximation for vibrations)

( ) ( )200' 00

'ˆ , ' , ( ') (2 ') exp ,2( ')! !

' 2 2 ', ' ; ( 1).' '

mn mnII m V n H I xm n

x x M

ωωω ω ωωω ω

ω ω ω ωω ω ω ω

⎡ ⎤= = Δ Δ = + − Δ⎢ ⎥+⎣ ⎦

Δ = Δ Δ = − Δ = =+ +

h

19

20

Model hamiltonians, dynamical groups and coherent states

Klauder, Glauber, Sudarshan, Perelomov, Berezin, Gilmore, …

1972

21

J.R. Klauder

R. Glauber, 2005, October 5

Heisenberg – Weyl group W1 and coherent states

Coherence properties of quantum electromagnetic fields, lasers

1963

22

Holomorphic functions representation

G-invariant Kähler’s 2-form

23

Path Integrals in CS - representation

0 0 1 1 2 1 0ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) ( , )N N N N NU t t U t t U t t U t t U t t− − −≡ = ⋅⋅⋅

24

“Classical” Equations

2

1

2

1

( , ) ln ( , ) ,

( , ) ln ( , ) .

n

n

i z z K z z zz z z

i z z K z z zz z z

βα α β

β

βα α β

β

=

=

∂ ∂=

∂ ∂ ∂

∂ ∂− =

∂ ∂ ∂

∑

∑

&h

&h

2 ln ( , ) , , .K z zg g g g gz z

α β α α β βκ αγ γαβ κβα β δ δ∂

= ⋅ = ⋅ =∂ ∂

25

Quantum oscillator in a field of external force

Oscillator coherent states, R. Glauber, 1963

26P(x,t) = |<x|z(t)>|2

27

28

<n(t)>

<W0(t)>

29

0-1 1x

00

t

0-1 1

00

2( , ) a quantum carpetsx tΨ ←

30

f(t) = A sin(ω t)

31

N-level atoms in classical fields. G = SU(N), ( |k><l| œ u(N) )

32

G= SU(2), (2j+1)-level atom

Coherent state dynamics. (a) – trajectory, (b) – Upper level probability P(t).

(z(0) = 1+i, ω0 = 1. ω = 2/3, A = 2)

j=1/2

33

Complex plane & Bloch’s sphere. Qubits

tan2

iz e φ ϑ⎛ ⎞= ⎜ ⎟⎝ ⎠

34

SU(2) CS generation: (a) – trajectory, (b) Upper level probability P(t).

(z(0) = 0, ω0 = 1. ω = 2, A = 1.5, t0 = 5, t = (3/5)1/2)

35

Three-level atom, G = SU(3). Qutrits

36Case of V – atom transitions and dynamics of the level populations.

We have not here a funny pictures for CS trajectories as in SU(2) case…

coherent population trapping

37

20 0( ) ( ) ( , ; , | ) , .t d d f tα μ ς α ζ α ζ α ζΕΨ = ∫∫

( )2

22

Re Im 2 1 Re Im, ( ) .1 | |

d d j d dd dα α ς ςα μ ςπ π ς

+= =

+

0 0 2 0 2 00lim ( , | , ; ) ( ) ( )t

f tα ζ α ζ δ α α δ ζ ζΕ→= − −

2( ; ) ( ); ( ; ) exp ( ) ( ) ; ( ) ,j

j jE m m m m cl m

m jf f t f t t t j mα ψ ζ α κ α α ψ ζ ζ

=−

⎡ ⎤= ∼ − ⋅ − =⎣ ⎦∑

( )( )

ˆ ˆ| | ; | 2 | ;

ˆ ˆ | 2 | .

a a

a a

α α α α α α α

α α α αα α

+

+

>= > >= ∂ ∂ + >

>= ⋅∂ ∂ + >

( ) ( )( )

2

0 2

ˆ ˆ| (1 ) | ; | (1 ) | ;

ˆ | (1 ) (1 ) | .j

J j J j

J

ζ ζ ζ ζζ ζ ζ ζ ζ ζ ζζ ζ

ζ ζ ζ ζζ ζζ ζ

+ −>= ∂ ∂ + + > >= − ⋅∂ ∂ + + >

>= ⋅∂ ∂ + − + >

The initial condition:

(0) (0) (0) .α ζΨ = | > ⊗ | > ˆ ˆ(0) (0) ( ) ( ) ;E H t H tΕ = = Ψ Ψ = Ψ Ψ

ˆ( ) ( )i t H tt∂

Ψ = Ψ∂

h

Q-corrections

38

Open systemsCoherent relaxation of N-level systems,

(the Markovian approximation)

39

Glauber – Sudarshan P-representation for density operator

40

G=SU(2), (2j+1)-level atom.( j=1/2, 1; j •)

The Fokker-Planck (FP) equation:

41

Method on spherical functions expansion:

FP-equation in the case of squeezed bath

42

FP-equation propagator for a qubit in a squeezed bath

43

Contour of the emission line for j=1/2 “atom” in a squeezed bath

No squeezing

squeezing

The ratio of radiation lines is independent on the bath temperature, r is a squeezing

parameter

44

Parametric Amplifier in a thermal bath

2 20

1ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )2

i t i tH t a a g aae a a e+ + + −= + + +ω ωω

1 1ˆ ˆ ˆ ˆ ˆU ( , ) exp2 2I t t t a a aaξ ξ+ +⎛ ⎞

⎜ ⎟⎝ ⎠

+Δ ≈ − 02 exp[ 2 ( ) ]ig t i tξ ω ω= − Δ − −

( ) ( )0 02 2 22 22 2

1 ˆ[e 2 e 2 ]2

i t i tg z z z z P LPt z z z z z zz z

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

∂Ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + + + +Ν ≡∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂

( ) ( ) ( )0 0 ;ˆ ,P t U t t P t= ( ) ( )00

ˆ , ˆ ˆ ˆ ˆ, , .U t t

LU U t t It

∂= =

∂

( ) ( )

0 02 ( ) t 2 ( ) t

0 02 2

2

2 , 2 ,

2 , 2 ,12 ,2

1 .2

d d d d

d d d d

i id d d d

i t i tr r

r ae r

ge ge

a ad be ge a ad be ge

b d d b ae N

re re d dω ω ω ω

ω ω ω ω

γ

γ

− −

− −

− − −− −

− − −

⎛ ⎞ +⎜ ⎟⎝ ⎠

= =

+ + = + + =

+ + + =

= =

&&

&&

&& &&

&& &

&&&&

45

( ) ( )( )

( ) ( ) ( )

12 2 2 2

12 2 2 2

1( ) 16 { 1 4 4 4 },4

1( ) 16 8 1 4 4 2 4 ,2

t t

t t

a t g g gN e ch gt N g e sh gt

b t g N g e ch gt g gN e sh gt

γ γ

γ γ

γ γ γ γ

γ γ γ γ

− − −

− − −

⎛ ⎞⎜ ⎟⎝ ⎠

⎧ ⎫⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

= − − − + −

= − − − + −

If 0 0ω ω− =

( ) ( ) ( ) 20, , | ', ',0 ', ' ',, , K z z t z z P z z zz z tP d= ∫

( ) ( )( )

212 2

0 22 2exp ,, , | , ,0 f fK z z t z z α β

α βα β

−⎡ ⎤

+⎢ ⎥= − −⎢ ⎥−⎢ ⎥⎣ ⎦

22

0 01 1 , , .2 2

rr

ddch r sh rab b f z z e z eα α βα

−−⎛ ⎞ −⎜ ⎟⎝ ⎠

= + − = − = −where

46Time dependence of the package width

The trajectory of the package center

The case of zero detuning

47

Nonzero detuning

The trajectory of the package center

Time dependence of the package width

48

Summary and some problemsWe have presented a mathematical formalism for describing the dynamics and relaxation of quantum systems. Group theoretical method and the CS technique are naturally used in quantum optics, quantum information theory, condensed matter and so on. Search for quantum corrections to the semi-classical dynamics CS in this approach in the general case has not been solved to date.One of the main problems here is the inclusion of non-Markovian effects into consideration.Possible generalizations of the concept of dynamical symmetries (super-algebras, associative algebras, …) to more complicated and realistic systems also a worthy of special consideration.

49

Thank You!

zG