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Transformations of angular momentum coherent states under coordinate rotations
Volume 34A, number 6
PHYSICS LETTERS
5April1971
TRANSFORMATIONS OF ANGULAR MOMENTUM COHERENT STATES UNDER COORDINATE ROTATIONS V. V. MIKHAILOV The Kazan Physico-Technical Institute of the Academy of Sciences of the USSR Received 12 March 1971
Using the tnvartance of coherent states of the two harmonic oscillators under the splnor transformations of their boson operators a, a +, b, b+ the transformations of angular momentum coherent states under coordinate system rotations are introduced.
It i s known that t h e r e is a r e p r e s e n t a t i o n of t h r e e - d i m e n s i o n a l r e a l E u c l i d e a n space R(3) by a twod i m e n s i o n a l complex Euclidean space C(2). Also t h e r e is h o m o m o r p h i s m between a group of u n i t a r y u n i m o d u l a r m a t r i c e s operating in C(2) and a group of r o t a t i o n s in R(3). This fact allows the a n g u l a r m o m e n t u m f o r m a l i s m in quantum m e c h a n i c s to be d e s c r i b e d in t e r m s of two independent h a r m o n i c osc i l l a t o r s [11. In ref. [11 a n g u l a r m o m e n t u m o p e r a t o r s J3., J+, J a r e e x p r e s s e d with the help of annihilation and c r e a t i o n o p e r a t o r s a , a + of the f i r s t and b, b* of the s e c o n d o s c i l l a t o r s as follows: ,I3 = ½(a+a - b+b) ,
J+ = a+b ,
J_ = b+a.
(1)
Let us c o n s i d e r coordinate s y s t e m r o t a t i o n s about axes z, y, z d e t e r m i n e d by E u l e r angles g/l' X, 62 r e s p e c t i v e l y . A p r o p e r u n i t a r y o p e r a t o r is U(g~l, X, ~2 ) [1]. O p e r a t o r s a and b a r e t r a n s f o r m e d u n d e r t h e s e r o t a t i o n s as s p i n o r components
-b'
= b
(2)
-=
U-1
-v*
where u, v a r e the known functions of E u l e r angles [1]. In ref. [2] coherent s t a t e s of a n g u l a r m o m e n t u m w e r e by analogy with coherent states of h a r m o n i c o s c i l l a t o r [3] introduced: J
o~J+m
"
j=0 m=-j [(J + m)l (j - m ) t ] l / 2 ] j m )
(3)
w h e r e n = [ a ]2 + IBI 2, [jrn) i s the a n g u l a r m o m e n t u m e i g e n s t a t e . In c a s e the s t a t e s [jrrb a r e c o n s t r a c t e d with the help of o p e r a t o r s a +, b + [1], s t a t e s I orB) can be shown to have the f o r m laB) = exp (c~a+ -c~*a) exp (Bb+ - B'b)[00) = D(c~a)D(Bb) I 00)
(4)
U n i t a r y o p e r a t o r D coincides with the d i s p l a c e m e n t o p e r a t o r for one h a r m o n i c o s c i l l a t o r [3]. Let us examine some p r o p e r t i e s of coherent s t a t e s (4) from the viewpoint of coordillate s y s t e m r o tations. Relation a[ ~B) = ~1 orB) and b[ ~B) =BI ~13) a r e true in fixed c o o r d i n a t e s y s t e m and m u s t be true in a rotated s y s t e m . It follows that complex n u m b e r s ot and B m u s t be t r a n s f o r m e d under r o t a t i o n s in the s a m e way as the o p e r a t o r s a and b. Hence we can see that the value a a + + Bb + and consequently the o p e r a t o r D ( ~ a ) D ~ b ) a r e i n v a r i a n t s of spinor t r a n s f o r m a t i o n s (2). So the f o r m of coherent s t a t e s does not depend on coordinate s y s t e m r o t a t i o n s I ~ ' ~ ' ) ' = laB). Taking the r e l a t i o n for ground s t a t e U I 0 0 ) = [00)into account, the action of o p e r a t o r U on the state loeB) can be d e r i v e d in the f o r m U[aB) = [ orB)' m D(~a')D(flb')100) = ]ot"B") =-D ( ~ " a ) D ( f l " b ) ] 0 0 )
(5)
343
Volume 34A. n u m b e r 6
PHYSICS
LETTERS
5 A p r i l 1971
where one p r i m e denotes direct t r a n s f o r m a t i o n by means of matrix V as in eq. (2) and two p r i m e s inv e r s e t r a n s f o r m a t i o n V -1. Relation (5) can be interpreted in such a way. The change of angular momentum coherent state Is/J) under t h r e e - d i m e n s i o n a l coordinate system rotation is equivalent to the inv e r s e spinor t r a n s f o r m a t i o n of complex arguments ~ and/3 which c o r r e s p o n d s to the inverse rotation in R(3). It can be easily shown that the angular momentum expectation values in coherent state with t r a n s f o r m e d arguments (~'/3' I J I ~ ' / 3 ' ) a r e expectation values in the rotated s y s t e m and are c o r r e c t l y exp r e s s e d in t e r m s of expectation values in a fixed system. Using relation (5) we can find the following expression for matrix elements of operator U (~1/31 I UI ot2/32) = (or{/3{ I ~2/32) = (Crl/31 I ~ / 3 ~) = exp (-½(n 1 + n2) + ~ * ot2 + /3{*/32) =
(6)
: exp (-½(n 1 +n2) + ~[ot~ + /3[/3~). Here the definition of s c a l a r product (al t a2 ) f r o m ref. [3] was used. F r o m eq. (6) it may be seen that the matrix elements of rotation operatorU are just s c a l a r product of coherent states with the t r a n s f o r m e d and fixed a r g u m e n t s a and /3. The coherent states la/3) form a complete set [3] and in consequence it is easy to verify unitarity of operator U matrix elements. Using the completeness of states 1~/3) and ]jrn), a number of relations between the states in rotated and fixed coordinate s y s t e m may be derived. Most of these relations are already well known [e.g. 1,4], the possibility of broad application of Dirac b r a - k e t formalism, making its derivation much c l e a r e r and simpler. In p a r t i c u l a r for generalized spherical h a r m o n i c s U~lrn(~kl, X, ~k2) we have a relation identical to (8.18) f r o m ref. [4]
~rn
(7) = (- 1)~- m exp (- i ~ l
~,~, ~y~ [(j + ~)!(j - ~7)!(j + m ) I ( j -re)I] 1/2 (cos ½X)2j-2k+m-~ (sin ½X)2k+~-m - ,,,,w2j ~ k!(j -7; -k)! (j +m -k)! (7; - m +k)!
Now consider the dependence of angular momentum wave function (0 q91~/3) on spherical coordinates 0 and 99. The function t r a n s f o r m a t i o n under s y s t e m coordinate rotation is generated by operator U(~k1, X, ~k2). Since the function value does not depend upon the choice of coordinate s y s t e m we have: U(~V1, X, ~2)<0'~']~/3)=
<0'~' q~/3>-- <0¢ I ~'/3'> • The dual way of wave function t r a n s f o r m a t i o n may be helpful in angular momenta evolution problems. The p r o p e r t i e s of angular momentum coherent states shown in this letter say that their usage in quantum mechanics is c o r r e c t . These r a t e s may be especially appropriate in solving the p r o b l e m s in which both quantum and classical p r o p e r t i e s of objects a r e important.
.l:defe~'e'nces [1] [2] [3] [4]
344
J. Schwinger, On Angular Momentum, U.S. Atomic Energy Commission, NYO'-3071 (1952). R. Bonifaeio, D. M. Kim and M. O. Scully, Phys. Rev. 187 (1969) 441. R . J . Glauber, Phys. Rev. 131 (1963} 2766. A. Jucys and A. Bandzaitis, T e o r i j a m o m e n t a koliehestva dvitschenija v kwantovoi mekhanike (Vilnius, 1965).
PHYSICS LETTERS
5April1971
TRANSFORMATIONS OF ANGULAR MOMENTUM COHERENT STATES UNDER COORDINATE ROTATIONS V. V. MIKHAILOV The Kazan Physico-Technical Institute of the Academy of Sciences of the USSR Received 12 March 1971
Using the tnvartance of coherent states of the two harmonic oscillators under the splnor transformations of their boson operators a, a +, b, b+ the transformations of angular momentum coherent states under coordinate system rotations are introduced.
It i s known that t h e r e is a r e p r e s e n t a t i o n of t h r e e - d i m e n s i o n a l r e a l E u c l i d e a n space R(3) by a twod i m e n s i o n a l complex Euclidean space C(2). Also t h e r e is h o m o m o r p h i s m between a group of u n i t a r y u n i m o d u l a r m a t r i c e s operating in C(2) and a group of r o t a t i o n s in R(3). This fact allows the a n g u l a r m o m e n t u m f o r m a l i s m in quantum m e c h a n i c s to be d e s c r i b e d in t e r m s of two independent h a r m o n i c osc i l l a t o r s [11. In ref. [11 a n g u l a r m o m e n t u m o p e r a t o r s J3., J+, J a r e e x p r e s s e d with the help of annihilation and c r e a t i o n o p e r a t o r s a , a + of the f i r s t and b, b* of the s e c o n d o s c i l l a t o r s as follows: ,I3 = ½(a+a - b+b) ,
J+ = a+b ,
J_ = b+a.
(1)
Let us c o n s i d e r coordinate s y s t e m r o t a t i o n s about axes z, y, z d e t e r m i n e d by E u l e r angles g/l' X, 62 r e s p e c t i v e l y . A p r o p e r u n i t a r y o p e r a t o r is U(g~l, X, ~2 ) [1]. O p e r a t o r s a and b a r e t r a n s f o r m e d u n d e r t h e s e r o t a t i o n s as s p i n o r components
-b'
= b
(2)
-=
U-1
-v*
where u, v a r e the known functions of E u l e r angles [1]. In ref. [2] coherent s t a t e s of a n g u l a r m o m e n t u m w e r e by analogy with coherent states of h a r m o n i c o s c i l l a t o r [3] introduced: J
o~J+m
"
j=0 m=-j [(J + m)l (j - m ) t ] l / 2 ] j m )
(3)
w h e r e n = [ a ]2 + IBI 2, [jrn) i s the a n g u l a r m o m e n t u m e i g e n s t a t e . In c a s e the s t a t e s [jrrb a r e c o n s t r a c t e d with the help of o p e r a t o r s a +, b + [1], s t a t e s I orB) can be shown to have the f o r m laB) = exp (c~a+ -c~*a) exp (Bb+ - B'b)[00) = D(c~a)D(Bb) I 00)
(4)
U n i t a r y o p e r a t o r D coincides with the d i s p l a c e m e n t o p e r a t o r for one h a r m o n i c o s c i l l a t o r [3]. Let us examine some p r o p e r t i e s of coherent s t a t e s (4) from the viewpoint of coordillate s y s t e m r o tations. Relation a[ ~B) = ~1 orB) and b[ ~B) =BI ~13) a r e true in fixed c o o r d i n a t e s y s t e m and m u s t be true in a rotated s y s t e m . It follows that complex n u m b e r s ot and B m u s t be t r a n s f o r m e d under r o t a t i o n s in the s a m e way as the o p e r a t o r s a and b. Hence we can see that the value a a + + Bb + and consequently the o p e r a t o r D ( ~ a ) D ~ b ) a r e i n v a r i a n t s of spinor t r a n s f o r m a t i o n s (2). So the f o r m of coherent s t a t e s does not depend on coordinate s y s t e m r o t a t i o n s I ~ ' ~ ' ) ' = laB). Taking the r e l a t i o n for ground s t a t e U I 0 0 ) = [00)into account, the action of o p e r a t o r U on the state loeB) can be d e r i v e d in the f o r m U[aB) = [ orB)' m D(~a')D(flb')100) = ]ot"B") =-D ( ~ " a ) D ( f l " b ) ] 0 0 )
(5)
343
Volume 34A. n u m b e r 6
PHYSICS
LETTERS
5 A p r i l 1971
where one p r i m e denotes direct t r a n s f o r m a t i o n by means of matrix V as in eq. (2) and two p r i m e s inv e r s e t r a n s f o r m a t i o n V -1. Relation (5) can be interpreted in such a way. The change of angular momentum coherent state Is/J) under t h r e e - d i m e n s i o n a l coordinate system rotation is equivalent to the inv e r s e spinor t r a n s f o r m a t i o n of complex arguments ~ and/3 which c o r r e s p o n d s to the inverse rotation in R(3). It can be easily shown that the angular momentum expectation values in coherent state with t r a n s f o r m e d arguments (~'/3' I J I ~ ' / 3 ' ) a r e expectation values in the rotated s y s t e m and are c o r r e c t l y exp r e s s e d in t e r m s of expectation values in a fixed system. Using relation (5) we can find the following expression for matrix elements of operator U (~1/31 I UI ot2/32) = (or{/3{ I ~2/32) = (Crl/31 I ~ / 3 ~) = exp (-½(n 1 + n2) + ~ * ot2 + /3{*/32) =
(6)
: exp (-½(n 1 +n2) + ~[ot~ + /3[/3~). Here the definition of s c a l a r product (al t a2 ) f r o m ref. [3] was used. F r o m eq. (6) it may be seen that the matrix elements of rotation operatorU are just s c a l a r product of coherent states with the t r a n s f o r m e d and fixed a r g u m e n t s a and /3. The coherent states la/3) form a complete set [3] and in consequence it is easy to verify unitarity of operator U matrix elements. Using the completeness of states 1~/3) and ]jrn), a number of relations between the states in rotated and fixed coordinate s y s t e m may be derived. Most of these relations are already well known [e.g. 1,4], the possibility of broad application of Dirac b r a - k e t formalism, making its derivation much c l e a r e r and simpler. In p a r t i c u l a r for generalized spherical h a r m o n i c s U~lrn(~kl, X, ~k2) we have a relation identical to (8.18) f r o m ref. [4]
~rn
(7) = (- 1)~- m exp (- i ~ l
~,~, ~y~ [(j + ~)!(j - ~7)!(j + m ) I ( j -re)I] 1/2 (cos ½X)2j-2k+m-~ (sin ½X)2k+~-m - ,,,,w2j ~ k!(j -7; -k)! (j +m -k)! (7; - m +k)!
Now consider the dependence of angular momentum wave function (0 q91~/3) on spherical coordinates 0 and 99. The function t r a n s f o r m a t i o n under s y s t e m coordinate rotation is generated by operator U(~k1, X, ~k2). Since the function value does not depend upon the choice of coordinate s y s t e m we have: U(~V1, X, ~2)<0'~']~/3)=
<0'~' q~/3>-- <0¢ I ~'/3'> • The dual way of wave function t r a n s f o r m a t i o n may be helpful in angular momenta evolution problems. The p r o p e r t i e s of angular momentum coherent states shown in this letter say that their usage in quantum mechanics is c o r r e c t . These r a t e s may be especially appropriate in solving the p r o b l e m s in which both quantum and classical p r o p e r t i e s of objects a r e important.
.l:defe~'e'nces [1] [2] [3] [4]
344
J. Schwinger, On Angular Momentum, U.S. Atomic Energy Commission, NYO'-3071 (1952). R. Bonifaeio, D. M. Kim and M. O. Scully, Phys. Rev. 187 (1969) 441. R . J . Glauber, Phys. Rev. 131 (1963} 2766. A. Jucys and A. Bandzaitis, T e o r i j a m o m e n t a koliehestva dvitschenija v kwantovoi mekhanike (Vilnius, 1965).