Документ взят из кэша поисковой машины. Адрес оригинального документа : http://qfthep.sinp.msu.ru/talks2015/1435230365_gorokhov_2015-6-25.pdf Дата изменения: Thu Jun 25 14:06:05 2015 Дата индексирования: Sat Apr 9 23:32:27 2016 Кодировка:

V.A. Fock's discovery of "hidden" O(4) symmetry of the H-atom and dynamical group theory
Alexander V. Gorokhov,
Samara State University, Russia gorokhov@samsu.ru


Contents
Introduction V.A. Fock and V. Bargmann approaches to the description of the H- atom symmetry Dynamical algebras and groups in Quantum Physics Coherent States (CS), Path Integrals and "Classical" Equations Open Systems. Fokker ­ Planck equations in CS representation Summary
2


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.
3


V.A. Fock and V. Bargmann approaches to the description of the H- atom symmetry r r
^ H (r ) = E (r );
2 4 E0 Z 2 e0 1 r 2 Ze0 ^ ^ H= p- = 27 Z 2 . , En = - 2 , n = 1, 2, ... E0 = r 2 2n h2

R SO(3), T l ( R), dimT l = 2l + 1;

(
l =0

n -1

2l + 1) = n2 . "Accidental" degeneracy of H-atom levels
1935, V.A. Fock. Analytical approach. SO(4)

r' r p Ze ( p ) dp r - E ( p) = p - p' 2 rr 2 h 2
2 2 0 2

'

4


5

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

r2 ( ) = ( p + r = (0 , ) = ( r2 2 p -p 0 = 0 r 2 , 2 p0 + p

r p ) ( p),
2 0

cos , s r 2 = 2 p0

in sin cos , sin sin sin , sin cos ) , + 2 = 1; r 2 p0 p p0 . r 2 , p0 = 2 | E | , E = - +p 2
2 0

r

( ) = 2 =
p
0

2



(

( ' ) d ' ( , , ) = ' 2 ' 2 - )

2



( ' , ' , ' )d ' ; 2 4 s i n ( / 2 )

2 Ze0 , = ; d = s in 2 d s in d d , h

co s = c o s co s ' + s i n si n ' c o s , c o s = c os c o s ' + s i n s i n ' c os ( - ' ) .
nl m

( , , ) = l ( n , ) Ylm ( , ) ,

2



nl m

(

r p) = N

n lm

r ( p2 + p

2 0

)







0

l2 ( n , ) si n 2 d = 1 .
6

-2

Y

n lm

( )


V. Bargmann, 1936

7


V. Bargmann, 1936. Algebraic approach

r rrrr ^ ^ ^^ ^ , L = r в p, p = -ih, A = 1 H 2 r r ^ H , L = H , A = 0. ^ ^^ ^ ^^ ^ ^^ Li , L j = ih ijk Lk , Li , A j = ih ijk Ak ,

(

r rrrr r ^^^^ Lв p - pв L + r

)

^^ ^ ^ Ai , A j = -2 H i h ijk Lk .

E < 0, h = 1. ^= 1 Ni ^ -2 H ^^ Li , L j = i

^ Ai ,
ijk

^ ^ ^ ^ ^ ^ ^ Lk , Li , N j = i ijk N k , N i , N j = i ijk Lk .

8


rr rr r2 r 2 E0 ^^ ^^ ^ ^ L N = N L = 0, L + N = - . ^ 2H r (1) 1 r r r ( 2) 1 r r ^ ^^ ^ ^^ J = L+N , J = L-N ; 2 2 ^ ^ ^ J i( a ) , J (j b ) = i ab ijk J k( b ) , (a, b = 1, 2

(

)

(

)( ) ( )
r ^ J
(1) 2

r ^ =J

( 2)

2

.)
j1 j2

n -1 . = T T ; j1 = j2 = SO(4) = SO(3) в SO(3) T 2 dim T ( j1 , j2 ) = (2 j1 + 1) (2 j2 + 1) n 2 .
( j1 , j2 )

E > 0, SO (4) SO (3,1); E = 0, SO (4) G0 ,
9


Symmetry group applications
Group symmetry of N- dimensional oscillator ­ SU(N), Jauch, Hill, 1940. Classification of states, selection rules for atoms, molecules solids. Lorentz and PoincarИ groups unitary representation and classification of elementary particles. ticles

J.L. Birman, CUNY, USA

Yu.N. Demkov, Leningrad University

Ya.A. Smorodinsky, JIRN, 10 Dubna


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

Asim Orhan Barut

Y. Neeman

11


Eightfold way and dynamical (spectrum generating) groups SU(3) ... SUI(2)
8=1223

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. Neeman. 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..
12


A Simple Example - Harmonic Oscillator
^ = 1 ( p 2 + x 2 ) , ( = m = = 1) ^ ^ H 2 ^ = 1 ( p 2 + x 2 ) , K = 1 ( p 2 - x 2 ) , K = 1 ( px + xp ); ^ ^ ^ ^ ^ ^ ^ ^ ^^ K0 1 2 4 4 4 ^ = K ± i K , K = 1 a + a + , K = 1 aa , ^ ^ ^ ^ ^ ^- ^ ^ K± + 1 2 2 2 K , K =± K , K , K = -2 K . ^^ ^ ^^ ^ 0 ±+ - 0 ± SU (1,1) = SL(2, R) = Sp(2, R ) SO(2,1). 13 T , k= , . 44 N - dim . oscillator: WN ^ Sp (2 N , R)
k +

13


Dynamical symmetries: condensed matter, quantum optics

Murray Gell-Mann, lecturing in 2007,

Ennackal Chandy George Sudarshan
14

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


15


Dynamical symmetry

R. Feynman's method of operator exponent disentanglement

In a linear case it is possible to find exact solution:

· Energy levels and corresponding wave functions (time independent
Hamiltonian); · Transition probabilities (time dependent Hamiltonian); · Quasi energy and quasi energy states (periodic Hamiltonian).
Aharonov - Anandan geometric phase
16


Lie groups and an energy levels calculation
H-atom, SO(4), SO(4,2)
SU(1,1), SU(N,1), Sp(2n,R), V.A. Fock, (1935) V. Bargmann, (1936) A.O. Barut, H. Kleinert, Yu.B. Rumer, A.I. Fet (1971) quantum optics, superfluidity

^ H = f1 C

((

A, A A' A" ..., A ) ) + f C ( A' ) + f
^ ^ ^ H = f ( Ak ), Ak
12





22

C

()
A"

+ ...

Molecular rotational- vibrational levels F. Iachello, R.D. Levine

u(4) so(4) so(3) so(2)
17


Molecular spectra & vibronic transitions
Wn Sp (2 N , R) -group and Franck ­ Condon overlap Integrals,
(harmonic approximation for vibrations)

r ^ T( ) D( ) U(R) ^r^^ V=

^ ^^ ^ H = VH V + ; r ^r rr q ' = R q + q , q = ( q1 , q2 , ..., qN ) ; -1 -1 r rrr ^ = L ' L, q = L ' r , r = r - r ' ^ ^r^ R 0 0

()

()

^ I [ m] , n' = < [ m] | n' > = [ m] V [ n
1 r2 r ^r V = exp - + 2 r

(

)

]

(

2

)

в

[ v ],[ n ]





r

*[ v ]



r

[n]

[v !] [n !]

^ [v ] V [ n ]

I

mn '

^ = mV n =

I 00 H m !n ! =

mn

(

, ') , I

00

' 2 = ( + ') ( 2 ' ) exp - ( x ) , 2( + ')
18

' 2 2 ' x, ' = - x; ( h = M = 1). + ' + '


19


Model hamiltonians, dynamical groups and coherent states
Klauder, Glauber, Sudarshan, Perelomov, Berezin, Gilmore, ...

1972

20


Heisenberg ­ Weyl group W1 and coherent states

1963

J.R. Klauder R. Glauber, 2005, October 5

Coherence properties of quantum electromagnetic fields, lasers

21


Holomorphic functions representation

G-invariant KДhler's 2-form

22


Path Integrals in CS - representation
^ ^ ^ U (t , t0 ) U (t N , t0 ) = U (t N , t
N -1

^ )U (t

N -1

,t

N -2

^ ) U (t1 , t0 )

23


"Classical" Equations

i (z, z ) = z h



n

=1

2 ln K ( z , z ) & z, z z
n =1

i (z, z ) - = z h



2 ln K ( z , z ) & z. z z

g

2 ln K ( z, z ) = ,g z z



g = , g



g = .

24


Quantum oscillator in a field of external force

Oscillator coherent states, R. Glauber, 1963

25


P(x,t) = ||2

26


27






28


-1

0

1

t 0 -1 0 x 1

0

( x, t )

2

a quantum carpets

29


f(t) = A sin( t)

30


N-level atoms in classical fields. G = SU(N), ( |k>
31

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

Coherent state dynamics. (a) ­ trajectory, (b) ­ Upper level probability P(t). (z(0) = 1+i, 0 = 1. = 2/3, A = 2)
32


Complex plane & Bloch's sphere. Qubits

z = e tan 2
i

33


SU(2) CS generation: (a) ­ trajectory, (b) Upper level probability P(t). (z(0) = 0, 0 = 1. = 2, A = 1.5, t0 = 5, t = (3/5)1/2)
34

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

35

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

coherent population trapping

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


Q-corrections

(t ) =
d 2 =

ih

^ (t ) = H (t ) t



d 2 d ( ) f ( , ; 0 , 0 | t ) , .

, d ( ) = 2 j + 1 d Re d Im

d Re d Im



(1+

| |

22

)

.

(0) = | (0) > | (0) > .
fE =

^ ^ = E = (0) H (0) = (t ) H (t ) ;

m =- j



j

2 j j f m ( ; t ) m ( ); f m ( ; t ) exp - m (t ) - cl (t ) ; m ( ) = j , m

The initial condition:

lim f ( , | 0 , 0 ; t ) = 2 ( - 0 ) 2 ( - 0 )
t 0

^ ^ a | >= | >; a + | >= ( + 2 ) | >; ^^ a + a | >= ( + 2 ) | > .
^ ^ J + | >= ( + j (1 + ) ) | >; J - | >= ( - 2 + j (1 + ) ) | >; ^ J 0 | >= ( + 2j (1 - ) (1 + ) ) | > .

37


Open systems Coherent relaxation of N-level systems, (the Markovian approximation)

38


Glauber ­ Sudarshan P-representation for density operator

39

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

·)

The Fokker-Planck (FP) equation:

40


FP-equation in the case of squeezed bath

Method on spherical functions expansion:

41


FP-equation propagator for a qubit in a squeezed bath

42


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

43


Parametric Amplifier in a thermal bath
1 ^ ^^ ^^ H (t ) = 0 (a + a + ) + g (aae 2
1 1 ^ ^^ ^^ U I (t +t , t ) exp a + a + - aa 2 2
2it

^^ + a+ a+e

-2it

)

= -2ig t exp[-2i( - 0 )t ]
1 2 z + ] + z + 2 z z zz ^ P LP

-2i(0 - )t 2 = g[e + 2z z z 2 t

2i (0 +e

- )t

2 + 2z z z 2

^ P ( t ) = U ( t , t0 ) P ( t0 ) ;

^ U ( t , t0 ) ^^ ^ ^ = LU , U ( t , t0 ) = I . t
d -d

& & a + 2ad + be

d -d

2i ( & r = ge 0

- )t

& & , a + 2ad + be
& r + 2ae
d -d

& r = ge

-2i (0 - )t

,

& & & b + d + d b + 2ae


d -d

&1 r = N,

& re

d -d

= 2 ge

2 i (0 - ) t

& , re

d -d

= 2 ge

-2 i (0 - ) t

2 &&1 , d = d = . 2

44


P ( z, z , t ) = K ( z, z , t | z ', z ', 0 ) P0 ( z ', z ')d 2 z ',
K z, z, t | z0 , z, 0

(

)(

= -

2

2 -1

)

f + f exp - 2 2 -

2

(

)

2

,

where

1 1 = b+ 2 2

2 a b - , = - , f = z - z0e - d ch r - z0 e

2

-d

r r

sh r .

If

0 - = 0
2

a(t ) = 2 -16 g b(t ) = -16 g
2

(

)

-1

1 {( g - gN ) (1 - e- t ch4 gt ) + 2 N - 4 g 2 e- t sh4 gt}, 4

(

2

)

-1

1 2

2 N - 8 g 2 (1 - e- t ch4 gt ) + ( 4 g - 2 gN ) e- t sh4 gt ,






45


The case of zero detuning

The trajectory of the package center

Time dependence of the package width
46


Nonzero detuning

The trajectory of the package center

Time dependence of the package width
47


Summary and some problems
We 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 nonMarkovian 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.
48


Thank You!

G
z
49