v.a. fock's discovery of hidden o(4) symmetry of the h...
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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
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