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Einstein-Podolsky-Rosen correlation - parallelism between the Wigner function and the local hidden variable approaches
PHYSICS LETTERS A
Physics Letters A 170 (1992) 359—362 North-Holland
Einstein—Podolsky—Rosen correlation parallelism between the Wigner function and the local hidden variable approaches —
G.S. Agarwal School of Physics, University of Hyderabad, Hyderabad 500 134, India
D. Home Department of Physics, Bose Institute, Calcutta 700 009, India
and W. Schleich Abteilung Theoretische Physik, III, Universitat Ulm, 79 Ulm, Germany Received 9 June 1992; accepted for publication 9 September 1992 Communicated by J.P. Vigier
We show that by using Wigner functions one can develop a treatment of the Einstein—Podolsky—Rosen correlated state oftwo spin 1/2 systems in a form similar to that of a local hidden variable model. The quantum mechanical results are exactly reproduced at the cost of allowing the probability distribution function to become negative.
Feynman [1] has one commented that the essential difference between a probabilistic classical world and the quantum world “is that somehow or other it appears as if the probabilities would have to go negative”. In the present note we seek to corroborate this contention in the context of the Einstein—Podolsky—Rosen (EPR) type correlated state of two spin 1/2 systems by using Wigner functions. We show that in terms of Wigner functions the quantum mechanical treatment can be cast in a form which is structurally similar to the local hidden variable description with the angular variables in the Wigner function interpreted as hidden variables. This form reproduces all the quantum mechanical correlation functions, but with negative probability distribution functions. This is consistent with the fact that in the proofs of Bell’s inequality one necessarily assumes that the relevant probabilities are positive definite, If this condition is relaxed, Bell’s inequality cannot be deduced anymore and the prohibition to reproduce the quantum mechanical correlation functions
in a form similar to local hidden variable theory does not hold well. Before giving the details of our treatment it should be useful to comment briefly on the previous relevant studies on this issue and place our present work in a proper perspective (for relevant reviews on the EPR paradox and Bell’s theorem, see for example ref. [2]). MUckenheim [31was the first to show by using a specific model that one can reproduce the quantum mechanical results for correlated spin measurements on the EPR-type state by allowing for negative probabilities. Variants of such models were later discussed by Ivanovic [4] and Polubarinov [5]. Generalisations of these models have been discussed by Home, Lepore and Selleri [61. However, such models are, in general, ad hoc. In contrast, our treatment stems naturally from first principles and serves to underscore the point that equivalence between the mathematical structures of quantum mechanics and local hidden variable theory can be achieved at the expense of introducing a nonphysical negative prob-
0375-960 1/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
359
Volume 170, number 5
PHYSICS LETTERS A
ability distribution [7]. This realisation is particularly important in the context of clarifying confusion generated by claims to have reproduced quantum mechanical results for the EPR-type correlated systems by using some form of local hidden variable theory, such as a recent claim by Six [8]. Moreover, it should be interesting to see the instructive role Wigner functions play in this context. Before we present our analysis we summarize the main results of the paper. In phase space formulation each state (density matrix) of a quantum mechanical system is represented by a distribution function which does not necessarily possess the properties of a classical distribution function. Representing by (0, ~) the phase space variables [9] ~ of a spin 1/2 system, we find that the singlet state of a system of two spin 1/2 particles can be represented in phase space by the Wigner function W(0
16 November 1992
the Wigner function is negative in a certain range of (0, ~,) values t2 The mathematical treatment of this paper will be based on the definition of the Wigner function for spins in the form discussed by Agarwal [9,10]. Following Agarwal, we define the Wigner function corresponding to an arbitrary operator G as follows. We first expand the given operator G in terms of the multipole operators TKQ defined by ~
(— I
I —m
K
I
Q
m’) urn> Kim’
TKQ = ><
(
mm
where
(_~, ~
)~“(2k+1)1/2
)
is the Wigner
(3) —
3j symbol, the pa-
rameter Q ranges from IKI to + K~in steps of unity and K ranges from 0 to 2j in steps of one. The operators TKQ form a complete set and hence we write the operator G as —
(1)
1,91,02,92)_—~—(l—3n1~n2),
G=~GKQTKQ,
(4)
KQ
where n1 is the unit vector in the direction (Os, ~“i) corresponding to particle 1 and n2 is that for the partide 2 in the direction (02, 92). It is important to note that W can be negative for example when n1 and n2 are parallel. This is how the quantum correlations enter the distribution function. The spin vector J in the phase space is represented by the vector ,~/i7~n. All quantum 2~> canmechanical be obtainedexpectation in terms of valthe ues like function K J,Y J~ (1) as Wigner
=
~
f dQ1 $ dQ2 ~
~l)
8it ~
where GKQ =
Tr ( GT~Q).
Having obtained the expansion coefficients GKQ, the Wigner function WG(O, ~) is defined as ~ YKQ(O, 9) GKQ, (6) KQ where ~KQ are the spherical harmonics. If G stands for the density matrix p so that WG(O, ø)
Tr(p)=1, n~
W( 0~,~
02, 92),
(5)
(7)
then the corresponding Wigner function is normalized as
(2) where the angular integrations are over the full solid angle. The similarity of (2) to local hidden variable theory is to be noticed. However, all quantum correlations are properly contained in the result (2) as
SI
Professor V. Singh has brought to our attention the work of Varilly and Gracia-Bondia 1101, which also discusses the Wigner functions for spins.
360
52
Phase space description for EPR correlations has been discussed before, Wodkiewicz 111] used the spin analogue of the P-function for a phase space description of the EPR correlations and noted that the underlying P-function becomes negative whichin turn leads to the violation of Bell’s inequalities. Scully and Cohen 171 have used Wigner functions extensively in the context ofthe EPR problem and hidden variable theories. However, the Wigner functions used by Scully and Cohen are defined differently from the one used by us. Our phase space description comes closest to the classical continuous description of spin with a fixed length.
Volume 170, number 5
PHYSICS LETTERS A
j. (0, ~,) dQ= 1 ~J~J~l
(8)
.
16 November 1992
Using eqs. (12)—(l4) and eq. (15), the Wigner function corresponding to J is found to be
w,=~fi7~ (sin0cosØ~+sin0sinØ9+cosO±)
The Wigner functions can be used to evaluate the quantum mechanical expectation values as follows, Tr(pG)= dQ W~(0,9)W0(o, ~) (9)
J
Now let us proceed to find the Wigner function for an EPR correlated singlet state of two spin 1/2 systems, labelled by indices 1 and 2. First, we observe that in this case the Wigner function for each spin is in a four-dimensional space and j= 1/2, K= 0,1. We then note that the density matrix corresponding to the EPR correlated state (1 /,f2) I t >1 ~ >2 I~>II~>2) is given by —
(16)
ii,
=
where n is a unit vector in the direction (0, 0). Now we come to the final stage of our treatment where we can evaluate the quantum mechanical cxpectation values of the spin operators in the case of the EPR correlated state. Using eqs. (1), (9) and (16) one can readily see that for, say, particle 1 the expectation values are given by (J~a> = ~/i7 dQ1 dQ2
J J
X flj ~aWEPR (0k, +
( ~><~) 1(It>K~ I)~(ID <~1)2
—(ID<~I)2],
=\/i7~~
91’ 0~,92 )
J J
dQ~ dQ2n1
a(
1 —3n~‘~2 ) =0.
(10)
(17)
which can be recast in the forms involving the sums of the product of the angular momentum operators, —J’+J~ 2~ —J’_J2÷) (11) PEPR = ~(~ —2J~’~J~
The last result follows from the fact that the cornponents of any unit vector when integrated over the full solid angle yield zero, i.e., from 5 dQ~n 1 =0. Similarly one obtains for the second particle
.
It is therefore clear that to find the Wigner function corresponding to PEPR we have to evaluate Wigner functions for J~,J~and J_ respectively. Using eqs. (3)—(6) it can be shown that Wigner functions corresponding to J~,J+ and J_ are given by W~=~(l/~)cos0,
(12)
W~=~Ji7~sinoexp(iØ),
(13)
W_=~J~7i~sin0exp(—iØ).
(14)
Usingeqs. (l2)—(14)ineq. (ll),theWignerfunction corresponding to PEPR is found to be as given by eq. (11). As mentioned earlier WEPR can be negative, for example, when n1 and n2 are parallel to one another, WEPR= 1 /47t. Furthermore we have to evaluate the Wigner function corresponding to the total spin operator I given by —
.‘+ + ~ JJX.JYY+JZZ_
2
~+ ~
—
j
= 0. (18) The correlation between the two particles is calculated as follows, <(J1~a)(J2~b)>
XJdQIJdQ2(nl.a)(n2.b)WEPR(0I~9I,02~92)
3 j~dQJdQ(na)(nb)(nn) (8~t)(8ir) =
—
~ab.
(19)
It is therefore established that the description of the EPR correlated state in terms of Wigner functions as outlined above reproduces exactly the quantum mechanical results It is of particular interest that expressions (17) and ~
(2) are structurally identical to the corresponding expressions for ensemble probabilities given in a
2 (15)
For footnote see next page.
361
Volume 170, number S
PHYSICS LETTERS A
probabilistic local hidden variable theory as weighted averages of individual probabilities: a) > d21 dA 2P(At, 2 2)p (a, A ~) (20) —~
JJ
,
K (J(1 1 a) (/12)~ b)>
~~*j j dA1dA2p(2,,22)p(a,A1)q(b,22),
variable model reproducing the quantum mechanical results with the proviso that the hidden variable probability distribution function can become negative. This unphysical feature once again illustrates the deep rooted incompatibility between quantum mechanics and local realism.
(21)
where A~and 22 are hidden variables associated with particles I and 2 respectively,p(21, 22) is the hidden variable distribution function corresponding to the wave function of the system of two particles 1 and 2, and p’s and q’s are individual probabilities at the “realist” level (as opposed to probabilities at the “statistical” level). The locality condition is contamed in expressions (20) and (21) through (a) factorizability of the joint probability written as a product of p(a, A ~)and q(b, 22) in expression (21), and (b) assuming that neitherp(21, 22) nor the domains of 21 and 22 depend on the measurement apparatus settings along a and b. Expressions (1) and (17) therefore have the same “local” character because both WEPR and the range of values of (01, 02) and (01, 02) do not depend on the direction a or b. The directions (0, Oi) and (02, 02) can be interpreted as “intrinsic” directions associated with individual particles 1 and 2 characterised by their hidden variables A~and 22 respectively. The formalism in terms of Wignerfunctions as discussed in this paper is therefore interpretable as aform oflocal hidden ~ For the sake of completeness it should be noted that for the triplet state of two spin 1/2 systems given by the state vector ~ is ofthe form W= (1 /8a) (1 + 3n~ n2), where n~is a unit vector along the direction (it—01, Ø~).It should also be noted that the Wigner function correspondingto the reduced density matrix for a single particleof the two particle system is given by the uniform distribution W1 (0k, ~ ) = 1 /,,/~. This holds for both singlet and triplet states.
362
16 November 1992
Two of us (GSA, DH) are grateful to Professors V. Singh and L. Mandel for useful discussions and thank the Department of Science and Technology, Government of India for supporting this collaborative work. GSA also thanks Professor H. Walther for invitation to work at the Max Planck Institute for Quantum Optics, Garching which initiated this collaboration.
References [11 R.P. Feynman, mt. J. Theor. Phys. 21(1982) 467. [2lJ.F. Clauser and A. Shimony, Rep. Prog. Phys. 41(1978) 1881,Scully, R. Shea and T. McCullen, Phys. Rep. 43 (1978) MO. 500; W. de Baere, Adv. Electron. Electron Phys. 68 (1986) 245; F. Selleri, ed., Quantum mechanics versus local realism (Plenum, New York, 1988). 13] W. MUckenheim, Lett. NuovoCimento 35 (1982) 300. [4]J.D. Ivanovic, Lett. Nuovo Ctmento 22 (1978) 14. (5] I.V. Polubarinov, Communication of the Joint Institute for Nuclear Research, Dubna, E2-88-80. [6] D. Home, V.L. Lepore and F. Selleri, Phys. Lett. A 158 (1991) 357; A.O. Baint and M. Bozic, Nuovo Cimento 101(1988) 595. [7] MO. Scully, Phys. Rev. D 28 (1983) 2477; M.O. Scully and L. Cohen, Ann. NY Acad. Sci. 480 (1986) 115; L. Cohen and MO. Scully, Found. Phys. 16 (1986) 295. [8] J. Six, Phys. Lett. A 150 (1990) 243. [9] G.S. Agarwal, Phys. Rev. A 24 (1981) 2889, section III. [10] J.C. Varilly and J.N. Gracia-Bondia, Ann. Phys. 190 (1989) 107. [111K. Wodkiewicz, Phys. Lett. A 129 (1988)1.
Physics Letters A 170 (1992) 359—362 North-Holland
Einstein—Podolsky—Rosen correlation parallelism between the Wigner function and the local hidden variable approaches —
G.S. Agarwal School of Physics, University of Hyderabad, Hyderabad 500 134, India
D. Home Department of Physics, Bose Institute, Calcutta 700 009, India
and W. Schleich Abteilung Theoretische Physik, III, Universitat Ulm, 79 Ulm, Germany Received 9 June 1992; accepted for publication 9 September 1992 Communicated by J.P. Vigier
We show that by using Wigner functions one can develop a treatment of the Einstein—Podolsky—Rosen correlated state oftwo spin 1/2 systems in a form similar to that of a local hidden variable model. The quantum mechanical results are exactly reproduced at the cost of allowing the probability distribution function to become negative.
Feynman [1] has one commented that the essential difference between a probabilistic classical world and the quantum world “is that somehow or other it appears as if the probabilities would have to go negative”. In the present note we seek to corroborate this contention in the context of the Einstein—Podolsky—Rosen (EPR) type correlated state of two spin 1/2 systems by using Wigner functions. We show that in terms of Wigner functions the quantum mechanical treatment can be cast in a form which is structurally similar to the local hidden variable description with the angular variables in the Wigner function interpreted as hidden variables. This form reproduces all the quantum mechanical correlation functions, but with negative probability distribution functions. This is consistent with the fact that in the proofs of Bell’s inequality one necessarily assumes that the relevant probabilities are positive definite, If this condition is relaxed, Bell’s inequality cannot be deduced anymore and the prohibition to reproduce the quantum mechanical correlation functions
in a form similar to local hidden variable theory does not hold well. Before giving the details of our treatment it should be useful to comment briefly on the previous relevant studies on this issue and place our present work in a proper perspective (for relevant reviews on the EPR paradox and Bell’s theorem, see for example ref. [2]). MUckenheim [31was the first to show by using a specific model that one can reproduce the quantum mechanical results for correlated spin measurements on the EPR-type state by allowing for negative probabilities. Variants of such models were later discussed by Ivanovic [4] and Polubarinov [5]. Generalisations of these models have been discussed by Home, Lepore and Selleri [61. However, such models are, in general, ad hoc. In contrast, our treatment stems naturally from first principles and serves to underscore the point that equivalence between the mathematical structures of quantum mechanics and local hidden variable theory can be achieved at the expense of introducing a nonphysical negative prob-
0375-960 1/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
359
Volume 170, number 5
PHYSICS LETTERS A
ability distribution [7]. This realisation is particularly important in the context of clarifying confusion generated by claims to have reproduced quantum mechanical results for the EPR-type correlated systems by using some form of local hidden variable theory, such as a recent claim by Six [8]. Moreover, it should be interesting to see the instructive role Wigner functions play in this context. Before we present our analysis we summarize the main results of the paper. In phase space formulation each state (density matrix) of a quantum mechanical system is represented by a distribution function which does not necessarily possess the properties of a classical distribution function. Representing by (0, ~) the phase space variables [9] ~ of a spin 1/2 system, we find that the singlet state of a system of two spin 1/2 particles can be represented in phase space by the Wigner function W(0
16 November 1992
the Wigner function is negative in a certain range of (0, ~,) values t2 The mathematical treatment of this paper will be based on the definition of the Wigner function for spins in the form discussed by Agarwal [9,10]. Following Agarwal, we define the Wigner function corresponding to an arbitrary operator G as follows. We first expand the given operator G in terms of the multipole operators TKQ defined by ~
(— I
I —m
K
I
Q
m’) urn> Kim’
TKQ = ><
(
mm
where
(_~, ~
)~“(2k+1)1/2
)
is the Wigner
(3) —
3j symbol, the pa-
rameter Q ranges from IKI to + K~in steps of unity and K ranges from 0 to 2j in steps of one. The operators TKQ form a complete set and hence we write the operator G as —
(1)
1,91,02,92)_—~—(l—3n1~n2),
G=~GKQTKQ,
(4)
KQ
where n1 is the unit vector in the direction (Os, ~“i) corresponding to particle 1 and n2 is that for the partide 2 in the direction (02, 92). It is important to note that W can be negative for example when n1 and n2 are parallel. This is how the quantum correlations enter the distribution function. The spin vector J in the phase space is represented by the vector ,~/i7~n. All quantum 2~> canmechanical be obtainedexpectation in terms of valthe ues like function K J,Y J~ (1) as Wigner
=
~
f dQ1 $ dQ2 ~
~l)
8it ~
where GKQ =
Tr ( GT~Q).
Having obtained the expansion coefficients GKQ, the Wigner function WG(O, ~) is defined as ~ YKQ(O, 9) GKQ, (6) KQ where ~KQ are the spherical harmonics. If G stands for the density matrix p so that WG(O, ø)
Tr(p)=1, n~
W( 0~,~
02, 92),
(5)
(7)
then the corresponding Wigner function is normalized as
(2) where the angular integrations are over the full solid angle. The similarity of (2) to local hidden variable theory is to be noticed. However, all quantum correlations are properly contained in the result (2) as
SI
Professor V. Singh has brought to our attention the work of Varilly and Gracia-Bondia 1101, which also discusses the Wigner functions for spins.
360
52
Phase space description for EPR correlations has been discussed before, Wodkiewicz 111] used the spin analogue of the P-function for a phase space description of the EPR correlations and noted that the underlying P-function becomes negative whichin turn leads to the violation of Bell’s inequalities. Scully and Cohen 171 have used Wigner functions extensively in the context ofthe EPR problem and hidden variable theories. However, the Wigner functions used by Scully and Cohen are defined differently from the one used by us. Our phase space description comes closest to the classical continuous description of spin with a fixed length.
Volume 170, number 5
PHYSICS LETTERS A
j. (0, ~,) dQ= 1 ~J~J~l
(8)
.
16 November 1992
Using eqs. (12)—(l4) and eq. (15), the Wigner function corresponding to J is found to be
w,=~fi7~ (sin0cosØ~+sin0sinØ9+cosO±)
The Wigner functions can be used to evaluate the quantum mechanical expectation values as follows, Tr(pG)= dQ W~(0,9)W0(o, ~) (9)
J
Now let us proceed to find the Wigner function for an EPR correlated singlet state of two spin 1/2 systems, labelled by indices 1 and 2. First, we observe that in this case the Wigner function for each spin is in a four-dimensional space and j= 1/2, K= 0,1. We then note that the density matrix corresponding to the EPR correlated state (1 /,f2) I t >1 ~ >2 I~>II~>2) is given by —
(16)
ii,
=
where n is a unit vector in the direction (0, 0). Now we come to the final stage of our treatment where we can evaluate the quantum mechanical cxpectation values of the spin operators in the case of the EPR correlated state. Using eqs. (1), (9) and (16) one can readily see that for, say, particle 1 the expectation values are given by (J~a> = ~/i7 dQ1 dQ2
J J
X flj ~aWEPR (0k, +
( ~><~) 1(It>
—(ID
=\/i7~~
91’ 0~,92 )
J J
dQ~ dQ2n1
a(
1 —3n~‘~2 ) =0.
(10)
(17)
which can be recast in the forms involving the sums of the product of the angular momentum operators, —J’+J~ 2~ —J’_J2÷) (11) PEPR = ~(~ —2J~’~J~
The last result follows from the fact that the cornponents of any unit vector when integrated over the full solid angle yield zero, i.e., from 5 dQ~n 1 =0. Similarly one obtains for the second particle
.
It is therefore clear that to find the Wigner function corresponding to PEPR we have to evaluate Wigner functions for J~,J~and J_ respectively. Using eqs. (3)—(6) it can be shown that Wigner functions corresponding to J~,J+ and J_ are given by W~=~(l/~)cos0,
(12)
W~=~Ji7~sinoexp(iØ),
(13)
W_=~J~7i~sin0exp(—iØ).
(14)
Usingeqs. (l2)—(14)ineq. (ll),theWignerfunction corresponding to PEPR is found to be as given by eq. (11). As mentioned earlier WEPR can be negative, for example, when n1 and n2 are parallel to one another, WEPR= 1 /47t. Furthermore we have to evaluate the Wigner function corresponding to the total spin operator I given by —
.‘+ + ~ JJX.JYY+JZZ_
2
~+ ~
—
j
XJdQIJdQ2(nl.a)(n2.b)WEPR(0I~9I,02~92)
3 j~dQJdQ(na)(nb)(nn) (8~t)(8ir) =
—
~ab.
(19)
It is therefore established that the description of the EPR correlated state in terms of Wigner functions as outlined above reproduces exactly the quantum mechanical results It is of particular interest that expressions (17) and ~
(2) are structurally identical to the corresponding expressions for ensemble probabilities given in a
2 (15)
For footnote see next page.
361
Volume 170, number S
PHYSICS LETTERS A
probabilistic local hidden variable theory as weighted averages of individual probabilities: a) > d21 dA 2P(At, 2 2)p (a, A ~) (20) —~
JJ
,
K (J(1 1 a) (/12)~ b)>
~~*j j dA1dA2p(2,,22)p(a,A1)q(b,22),
variable model reproducing the quantum mechanical results with the proviso that the hidden variable probability distribution function can become negative. This unphysical feature once again illustrates the deep rooted incompatibility between quantum mechanics and local realism.
(21)
where A~and 22 are hidden variables associated with particles I and 2 respectively,p(21, 22) is the hidden variable distribution function corresponding to the wave function of the system of two particles 1 and 2, and p’s and q’s are individual probabilities at the “realist” level (as opposed to probabilities at the “statistical” level). The locality condition is contamed in expressions (20) and (21) through (a) factorizability of the joint probability written as a product of p(a, A ~)and q(b, 22) in expression (21), and (b) assuming that neitherp(21, 22) nor the domains of 21 and 22 depend on the measurement apparatus settings along a and b. Expressions (1) and (17) therefore have the same “local” character because both WEPR and the range of values of (01, 02) and (01, 02) do not depend on the direction a or b. The directions (0, Oi) and (02, 02) can be interpreted as “intrinsic” directions associated with individual particles 1 and 2 characterised by their hidden variables A~and 22 respectively. The formalism in terms of Wignerfunctions as discussed in this paper is therefore interpretable as aform oflocal hidden ~ For the sake of completeness it should be noted that for the triplet state of two spin 1/2 systems given by the state vector ~ is ofthe form W= (1 /8a) (1 + 3n~ n2), where n~is a unit vector along the direction (it—01, Ø~).It should also be noted that the Wigner function correspondingto the reduced density matrix for a single particleof the two particle system is given by the uniform distribution W1 (0k, ~ ) = 1 /,,/~. This holds for both singlet and triplet states.
362
16 November 1992
Two of us (GSA, DH) are grateful to Professors V. Singh and L. Mandel for useful discussions and thank the Department of Science and Technology, Government of India for supporting this collaborative work. GSA also thanks Professor H. Walther for invitation to work at the Max Planck Institute for Quantum Optics, Garching which initiated this collaboration.
References [11 R.P. Feynman, mt. J. Theor. Phys. 21(1982) 467. [2lJ.F. Clauser and A. Shimony, Rep. Prog. Phys. 41(1978) 1881,Scully, R. Shea and T. McCullen, Phys. Rep. 43 (1978) MO. 500; W. de Baere, Adv. Electron. Electron Phys. 68 (1986) 245; F. Selleri, ed., Quantum mechanics versus local realism (Plenum, New York, 1988). 13] W. MUckenheim, Lett. NuovoCimento 35 (1982) 300. [4]J.D. Ivanovic, Lett. Nuovo Ctmento 22 (1978) 14. (5] I.V. Polubarinov, Communication of the Joint Institute for Nuclear Research, Dubna, E2-88-80. [6] D. Home, V.L. Lepore and F. Selleri, Phys. Lett. A 158 (1991) 357; A.O. Baint and M. Bozic, Nuovo Cimento 101(1988) 595. [7] MO. Scully, Phys. Rev. D 28 (1983) 2477; M.O. Scully and L. Cohen, Ann. NY Acad. Sci. 480 (1986) 115; L. Cohen and MO. Scully, Found. Phys. 16 (1986) 295. [8] J. Six, Phys. Lett. A 150 (1990) 243. [9] G.S. Agarwal, Phys. Rev. A 24 (1981) 2889, section III. [10] J.C. Varilly and J.N. Gracia-Bondia, Ann. Phys. 190 (1989) 107. [111K. Wodkiewicz, Phys. Lett. A 129 (1988)1.