- Email: [email protected]
The azimuthal angle operator for angular momentum and phase operators of oscillators
6 November
& *H -_
1995
__
PHYSICS
LETTERS
A
El!3 ELSEWIER
Physics Letters A 207 (1995) 243-249
The azimuthal angle operator for angular momentum and phase operators of oscillators Tapan K. Kar, Debajyoti Bhaumik Theoretical
Received
Physics Group. Saha Institure
of Nuciear
Physics. I / AF, Bidhannagar.
’ Calculra
16 June 1995; revised manuscript received 24 August 1995: accepted for publication Communicated by P.R. Holland
700 064, India 5 September
1995
Abstract
We discuss the relationship between the azimuthal angle operators corresponding to a quantum mechanical angular momentum “vector” and the phase operators associated with the two oscillators from which one can compose the angular momentum operators 21la Schwinger.
Proper quantization of classical phase has been the subject of many recent investigations [ 1,2]. Attempts have been made to find a Hermitian phase operator for the quantum harmonic oscillator (or a single mode of the electromagnetic field) - a delicate problem since the phases of harmonic oscillators depend both on coordinates and momenta. Recently Pegg and Barnet! (PB) [2] were able to construct a Hermitian phase operator 4, and a unitary exponential phase operator exp(i+), in a truncated state space with a finite number of oscillator quanta. The PB prescription consists in evaluating all relevant physical quantities in the finite dimensional space before taking the dimensionality of the state space to infinity to arrive at the results obtainable in ordinary quantum mechanics. Similar to the notion of a phase, we frequently encounter another quantity, the angle variable which also has the periodicity 27r. However, in trying to define a Hermitian angle operator corresponding to these angle variables not much difficulty is faced, basically because the angles specifying the orientation in space of a rotating body are periodic variables that are fully determined by generalized coordinates. Nienhuis and van Enk [3] have discussed the quantum operators corresponding to the spherical angles of an angular momentum operator i Two angles (polar and azimuthal) are necessary to describe the orientation of a vector in the spherical polar system of coordinates. Since the three components of angular momentum (J,, J,, JZ> do not commute with one another, it is expected that the operators representing polar and azimuthal angles should be non-commuting. The z-component of the angular momentum operator & is uniquely related to the operator 6 for the polar angle of the vector J by the defining relation cos 8 = j,/ \15^2(p rovided we assign its eigenvalues to the interval [0, n-1). The standard angular momentum basis states 1j, m), which are the simultaneous
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T. K. Kar, D. Bhaumik/
Physics Letfers A 207 f 1995) 2-13-249
eigenstates of j2 and 1, belonging to the eigenvalues j( j + 1) and m respectively, are also the eigenstates of & (repres_ented by 10,)) with corresponding eigenvalues 0, = arccos[ m/ \ij(j+1>]. The azimuthal angle operator @, on the other hand, was defined by Nienhuis and van Enk in such a way that the action of exp(i&) changed I j, m) to I j, m + 1) and was made Hermitian by allowing its action on the state I j, j) to map it onto I j, -j), and consequently the eigenstates of @ were found to be 1Qr,> corresponding to the discrete eigenvalues Gr = @a + 27rr/(2 j + 11, @,, being a constant and r = 0, 1,. . . ,2 j. In the present article we discuss the relationship of the unitary azimuthal angle operator ei6 with the phase operators corresponding to the oscillators from which we can construct the SU(2) generators of the rotation group. Following Schwinger 141, the bosonic operators 8, and 2: (r = + , - >, representing two oscillators, and obeying the commutation relations
may be employed
to construct
4+=$+&,
the angular momentum
operators through the bilinears
2z = @+a+-a+;_),
~_=i;+_~+,
(14
which satisfy the Lie algebra for the rotation group, and the corresponding
p=;(
+ ii++ii++ at_ii_ + )i
Casimir operator may be written as
81 ii +ii+_ii_ 2
(lb)
+1. 1
It is well known that the angular momentum states I j, m) may be built out of the “vacuum” of the boson creation operators iit+ and tit_, namely,
(a++)““(a+->‘-” 10
Ijm)= [(
j+m)!(j-m)!]l’*
o_)=
+’
In+,
n_=2j-n
> +
’
by the operation
(2)
where, in writing Eq. (2), we have used n++ n_ -=. 2
II+-- nj7
(3)
2=m.
In view of the above description of the angular momentum operators in terms of two bosonic oscillators it is natural to look for a relationship between the PB phases of the two oscillators and the azimuthal angle operator may be considered to represent the two modes of electromagnetic e i@. These two oscillators, incidentally, radiation and hence the following discussion applies equally well to the relevant combination of the PB phases of two electromagnetic waves. We can write the annihilation operators for the oscillators d, and a_ in the polar decomposition as . 2_=ei4A_ , a+=e i&+J I\j c 9 d h with N, the number operators $*a, for the two oscillators and 0 + are the corresponding the two oscillators. Their action on the number states I n + ) of the oscillators are fi* In,>
=n+
In,>,
e’&* In,)
= In*--
1).
(4) phase operators for
(5)
For the construction of the Hermitian phase operators $* of the individual oscillators, following PB, the infinite dimensional space of the number states of each individual oscillator is truncated to a finite dimension.
T.K. Kar, D. Bhaumik/Physics
Letters A 207 (1995) 243-249
245
However, as we are interested only in the azimuthal angle operator associated with the angular momentum operator (in the context of a state specified by j), and since there is nothing to distinguish between the two oscillators so far as the generators of the angular momentum are concerned, the dimensionality of the state space for both the oscillators has to be the same, namely 2j + 1, with j = 0, 1, 2, . . . . Also it will suffice to focus our attention on a particular j value. In terms of the 2 j + 1 number states the “truncated” exponential phase operators ei6* for each individual oscillator have the following unitary representation, 2j- 1
,i&*=
C 1n*)(n,+
1 I +ei(2j+1)+o* I n*=2j)(O,
(6)
I.
Here 4,, correspond
to an arbitrary reference phase angle. It is necessary to note here that the “truncated” phase operators differ from those in the Pegg-Barnett formalism in that the dimension of the underlying truncated Hilbert space is not made arbitrarily large for the calculation of physical quantities such as expectation values. These “truncated” unitary phase operators have eigenstates 14, f >,
with corresponding
eigenvalues
&,,, given by
27rm,
m*=O, 1, 2 ,...,
h,*= #Jo*+-2j+l’ The Hermitian
phase operator
4*
2j.
can thus be expressed
(7b) in terms of the “truncated
phase states”
I +, f > as
3j
6*=
c
(8)
4%* I #J,*wrn* 1.
m*=O
We can also write C$+ in terms of the number states I n * > using (7a). It is clear from (2) that while constructing angular momentum states with a particular j value we need to choose out of all the possible product states I rz,, II_ ) in the (2j + 1) X (2 j + 1) dimensional space only those states which obey the restrictions (3), namely n, + n_ = 2 j. This can be achieved easily through a projection operator &j, $SjE
E
n+=o
l n,, 2j-n+)(n+,
2j--n+
I,
(9)
whose action on a general product state I II,, n_ > will select only such states that correspond to angular momentum j. It may also be noticed here that if we confine our calculation within the 2 j + 1 states of angular momentum j and deal with operators which act only on these 2 j + 1 states, then &j is unitary (in fact it is then the identity operator 13. The necessity of introducing such a projection operator will become clear when one writes 3, in (la) using (4) as
where a different notation
246
T.K. Kar, D. Bhaumik/
Physics Letters A 207 (1995) 243-249
are identical in the two cases. Expressing subspace{I j, terms of the product states of the oscillators 1II,, n_ > one obtains m>l,=
exp[
-j.,,,,j
_i(E+-
‘E’ 111++ 1, n_>(n+,
&)I =
n-+
exp[ - i<$+ - & )] with the help of (6) in
l I
n,, fl_=O
exp[ -i(2j
+
2j-
+
+ I)(+,+-
40_)]IO,, 2jWj,
O_ I
I
C
In++
1,2j)(n
+, O_ I exp[i(2j
+
1)&l
n+=O 2j+
1
C
IO,, n_)Qj,
n_+
1 I exp[ -i(2j+
l)&,+]
(11)
n_=O
It is now manifest that the third and fourth term in (11) involves states which are outside the 2 j + 1 dimensional space of angular momentum states 1j, m) under consideration. Since .?+ is an operator whose action on the I j, m) states keeps them within the 2 j + 1 dimensional space of angular momentum it is necessary to select that part of exp[ - i( 4, - & >] which confines its action within this space only. It ca? easily be verified that the operator [&. exp[ - i(J+ - c$_)]&~] would satisfy this requirement. Incidentally yj commutes with N,, N_ and exp[ - ii$+ - $_>I as can be checked using the definition of &j in (9). Thus J, when written in terms of phase operators 0 * using (10) has the form
The operator represented /(j - m)( j operator has
sj first projects only the I j, m) states from all possible states ((2 j + 1) X (2 j + 1) in number) by I n,, n_ > and then !+ acts on them raising them to I j, m + 1) with a factor f m + 1) . Denoting exp[-i($+$_)]gj by ei’ we can at once see from (11) that this unitary a representation in terms of oscillators as Zj-
I
eib = ,,;, +
1n++ 1, n_=2j-n+-
l)(n+,
2j-n,
I+e-i(2j+1x~o+-90-)IO+,
2j)(2j,
O_ I.
(13)
Since $o+- (be_ is the difference of the reference phases for the (-t > and (-1 oscillators and is a constant, expression (13) is the same as that obtained by Nienhuis _and van Enk [3]. That & represents the azimuthal phase operator of the angzar momentum raising operator J, can be seen with a little manipulation using (13) as the representation of e , ~+=~ei~~=2’~1&7i~~In,+l,2j-.+_1~(~+,2j-fr+l n+=O I C In++1,2j-n+-l)(n+,2j-n+l n+=O 2j-
=dm = /m(ei” = \im
-
exp[ -i(2j+
l)(+o+-
$,_)I
IO+, 2j)(2jV
O- I>
ei’.
The azimuthal angle operator thus defined is similar to the phase difference operattr discussed by Luis and Sanchez-Soto [5]. Since the operator ei’ commutes with the total number of quanta N++ g_ preserving the j value by virtue of (3), it was not necessary to invoke the limiting procedure prescribed by Pegg and Barnett for
T.K. Kar, D. Bhumik /
Physics
Letters
A 207 (1995)
243-249
247
the dimensionality of the state space of the individual oscillators to obtain physical results. Indeed this azimuthal angle operator does reproduce all the standard results regarding the spherical representation of the J. We have thus established the relationship between the oscillator phase operators and the azimuthal angle operator. The azimuthal angle operator e” thus defined has eigenstates 1‘jp,) given by
with corresponding Qr=Qo+-
eigenvalues 25-r
r=O,
2jfl’
1,2 ,...,
2j,
where we have written (2 ’ + l)(&+$e_) = (2j + l)@,. n The connection of e’ I with the phase difference operator exp[ -X4+ - $_)I in the truncated space is indicative of the possibility that the states 1Qr,> can be expressed in terms of the “truncated” phase states ) 4,,,+) and 1$,,_) of the individual oscillators with appropriate Clebsch-Gordan coefficients. Indeed inverting Eq. (7a) to express I n, ) and I n_ > in terms of I +,+ > and 1&,_ > and substituting in (14a) we have
l@J =
d
1
E g exp[in+(@, (2j + 1)3 ,1+=0 m*=O
;i&
=
t
m,
r-
1 (2j+
1)
exp( -2ijL)I
4,+3 +m-)
exp(-W+m+-r) I k+, 4m+-r) r 1
exP(
C +
- +*++ $,_)I
-2ij~2j+m+-m_+,)I~m+)
(‘5)
hj+l+m+-r).
nl,=o
Thus if we write
with r an integer modulo 2j + 1, then the Clebsch-Gordan C:I + .m_= CL+,,+-,=
coefficients
1 Jm
exp( - 2ij+,+
-r)
1 =C~~+,(2j+l)+m+-r=
gipT
exP(-2ij~((2j+l)+m+-r)
CL+ ,m_ have the following for 2 j 2 m, 2 r,
for r-
1 >m+>O.
values, ( 16a)
( ‘6b)
The phase state I Gr) as defined in (14a) are 2 j + 1 in number having discrete eigenvalues (14b); and are similar to the standard angular momentum states 1j, m). This similarity may be extended to the angular momentum coherent states as discussed by Radcliffe 161 in 1971 if we can construct [7] a coherent superposition of 1Qr) states. To this end let us construct creation and annihilation operators of the “phase quanta” of angles 6’ and 6 such that & = &$,
( ‘7)
T. K. Kar. D. Bhaumik/
248
Physics Letters A 207 (I 9951 243-249
and whose action on 1Gr) is as follows, 4’ IGr,,> = fir+,
lGr+,>,
WJ
= @.I@~_,>.
(18)
These operators satisfy the commutator relations simjlar t,” the $_and have the following explicit expressions in terms of N,, N_ and @,
4’
= &)I/2
exp ji(N’+-ii_+Zj)& i
= z 1
8, of the single mode operators.
I +KI@lJ)(@?j
#IcP,><@r-i
1.
r=l
They
(19)
The phase creation and annihilation operators are not Hermitian and hence they themselves are not observables. Nevertheless they offer instructive and calculational advantage in their use for description of the phase states of the harmonic oscillator. To construct a unique coherent state superposing 1Dr,> states, let us choose @a = 0 so that ~0~= 2rrr/ (2j+ 11, r=O, 1,2 ,... ,2 j. This will then ensure that
In fact it can be easily seen that for k = 1,. . . ,2j. We can now construct the coherent state I j, cy) superposing different operating eaJ+ on 1G0 = 0), CYbeing a continuous parameter
(47mT)n
*j
Ij,a)=Nea2J+l@o)=_N~
fi
n=O
I~) n
I Gr,>states in the usual manner [6,7,81
(2’)
7
where J1”=
(lal/wpF-iy)2n fz.1 ( n=0
l/2
5
I
.
This coherent state is analogous to the Radcliffe state [6] for the angular momentum. One can easily calculate the expectation values of the angular momentum operators and take the appropriate classical limit. The probability of finding the azimuthal angle cD~= 2rrm/(2 j + 1) in the state I j, Q > is given by
P(Qm) = I(QmI j, a>l’=
[ I a l@qgTiy2m/m! Ego[ I crl~~]*“/n!
*
(22)
In the limit of large j and m the distribution corresponds to the classical results. To conclude we have used the projection operator technique to discuss the quantum operators for the spherical angles of a three dimensional angular momentum vector J^ constructed out of the phase operators of two single mode oscillators. For a given quantum number j the dimensionality of the number state space of the two oscillators required to construct the spherical angle operators was 2 j + I. Thus our result shows how a quantum version of the spherical angle operator for angular momentum can be constructed out of the operators for the “constituent” oscillators.
T. K. Kar, D. Bhoumik / Physics Letters A 207 (1995) 243-249
The authors would like to thank Professor reading of the manuscript.
Binayak
Dutta-Roy
for helpful
249
discussions
and for a critical
References [I] L. Susskind and J. Glogower, Physics I (1964) 49; P. Carruthers and M.M. Nieto, Rev. Mod. Phys. 40 (1968) 441; M.M. Nieto, Phys. Ser. T48 (1993) 5; Le-Man Kuang and Xi Chen, Phys. Rev. A 50 (1994) 4228. [2] D.T. Pegg and S.M. Barnett Phys. Rev. A 39 (1989) 1665; 43 (1991) 2579; S.M. Bamett and D.T. Pegg, Phys. Rev. A 41 (1990) 3427; D. Ellinas, J. Math. Phys. 32 (1991) 135; A. Luis and L.L. Sanchez-Soto. Phys. Rev. A 47 (1993) 1492. [3] G. Nienhuis and S.J. van Enk, Phys. Ser. T48 (1993) 87. [4] J. Schwinger. Quantum theory of angular momentum, eds. L.C. Biedenharn and H. van Dam (Academic 229-279. [S] A. Luis and L.L. Sanchez-Soto, Phys. Rev. A 48 (1993) 4702; D.T. Pegg and J.A. Vaccaro, Phys. Rev. A 51 (1995) 859; A. LUIS and L.L. Sanchez-Soto. Phys. Rev. A 51 (1995) 861. [6] J.M. Radcliffe, J. Phys. A 4 (1971) 313. [7] V. Buzek, A.D. Wilson-Gordon, P.L. Knight and W.K. Lai, Phys. Rev. A 45 (1992) 8079. [8] D. Bhaumik. T. Nag and B. Dutta-Roy, J. Phys. A 8 (1975) 1868.
Press, New York, 1965) pp.
& *H -_
1995
__
PHYSICS
LETTERS
A
El!3 ELSEWIER
Physics Letters A 207 (1995) 243-249
The azimuthal angle operator for angular momentum and phase operators of oscillators Tapan K. Kar, Debajyoti Bhaumik Theoretical
Received
Physics Group. Saha Institure
of Nuciear
Physics. I / AF, Bidhannagar.
’ Calculra
16 June 1995; revised manuscript received 24 August 1995: accepted for publication Communicated by P.R. Holland
700 064, India 5 September
1995
Abstract
We discuss the relationship between the azimuthal angle operators corresponding to a quantum mechanical angular momentum “vector” and the phase operators associated with the two oscillators from which one can compose the angular momentum operators 21la Schwinger.
Proper quantization of classical phase has been the subject of many recent investigations [ 1,2]. Attempts have been made to find a Hermitian phase operator for the quantum harmonic oscillator (or a single mode of the electromagnetic field) - a delicate problem since the phases of harmonic oscillators depend both on coordinates and momenta. Recently Pegg and Barnet! (PB) [2] were able to construct a Hermitian phase operator 4, and a unitary exponential phase operator exp(i+), in a truncated state space with a finite number of oscillator quanta. The PB prescription consists in evaluating all relevant physical quantities in the finite dimensional space before taking the dimensionality of the state space to infinity to arrive at the results obtainable in ordinary quantum mechanics. Similar to the notion of a phase, we frequently encounter another quantity, the angle variable which also has the periodicity 27r. However, in trying to define a Hermitian angle operator corresponding to these angle variables not much difficulty is faced, basically because the angles specifying the orientation in space of a rotating body are periodic variables that are fully determined by generalized coordinates. Nienhuis and van Enk [3] have discussed the quantum operators corresponding to the spherical angles of an angular momentum operator i Two angles (polar and azimuthal) are necessary to describe the orientation of a vector in the spherical polar system of coordinates. Since the three components of angular momentum (J,, J,, JZ> do not commute with one another, it is expected that the operators representing polar and azimuthal angles should be non-commuting. The z-component of the angular momentum operator & is uniquely related to the operator 6 for the polar angle of the vector J by the defining relation cos 8 = j,/ \15^2(p rovided we assign its eigenvalues to the interval [0, n-1). The standard angular momentum basis states 1j, m), which are the simultaneous
’ E-mail: [email protected]. 0375-9601/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00684-2
144
T. K. Kar, D. Bhaumik/
Physics Letfers A 207 f 1995) 2-13-249
eigenstates of j2 and 1, belonging to the eigenvalues j( j + 1) and m respectively, are also the eigenstates of & (repres_ented by 10,)) with corresponding eigenvalues 0, = arccos[ m/ \ij(j+1>]. The azimuthal angle operator @, on the other hand, was defined by Nienhuis and van Enk in such a way that the action of exp(i&) changed I j, m) to I j, m + 1) and was made Hermitian by allowing its action on the state I j, j) to map it onto I j, -j), and consequently the eigenstates of @ were found to be 1Qr,> corresponding to the discrete eigenvalues Gr = @a + 27rr/(2 j + 11, @,, being a constant and r = 0, 1,. . . ,2 j. In the present article we discuss the relationship of the unitary azimuthal angle operator ei6 with the phase operators corresponding to the oscillators from which we can construct the SU(2) generators of the rotation group. Following Schwinger 141, the bosonic operators 8, and 2: (r = + , - >, representing two oscillators, and obeying the commutation relations
may be employed
to construct
4+=$+&,
the angular momentum
operators through the bilinears
2z = @+a+-a+;_),
~_=i;+_~+,
(14
which satisfy the Lie algebra for the rotation group, and the corresponding
p=;(
+ ii++ii++ at_ii_ + )i
Casimir operator may be written as
81 ii +ii+_ii_ 2
(lb)
+1. 1
It is well known that the angular momentum states I j, m) may be built out of the “vacuum” of the boson creation operators iit+ and tit_, namely,
(a++)““(a+->‘-” 10
Ijm)= [(
j+m)!(j-m)!]l’*
o_)=
+’
In+,
n_=2j-n
> +
’
by the operation
(2)
where, in writing Eq. (2), we have used n++ n_ -=. 2
II+-- nj7
(3)
2=m.
In view of the above description of the angular momentum operators in terms of two bosonic oscillators it is natural to look for a relationship between the PB phases of the two oscillators and the azimuthal angle operator may be considered to represent the two modes of electromagnetic e i@. These two oscillators, incidentally, radiation and hence the following discussion applies equally well to the relevant combination of the PB phases of two electromagnetic waves. We can write the annihilation operators for the oscillators d, and a_ in the polar decomposition as . 2_=ei4A_ , a+=e i&+J I\j c 9 d h with N, the number operators $*a, for the two oscillators and 0 + are the corresponding the two oscillators. Their action on the number states I n + ) of the oscillators are fi* In,>
=n+
In,>,
e’&* In,)
= In*--
1).
(4) phase operators for
(5)
For the construction of the Hermitian phase operators $* of the individual oscillators, following PB, the infinite dimensional space of the number states of each individual oscillator is truncated to a finite dimension.
T.K. Kar, D. Bhaumik/Physics
Letters A 207 (1995) 243-249
245
However, as we are interested only in the azimuthal angle operator associated with the angular momentum operator (in the context of a state specified by j), and since there is nothing to distinguish between the two oscillators so far as the generators of the angular momentum are concerned, the dimensionality of the state space for both the oscillators has to be the same, namely 2j + 1, with j = 0, 1, 2, . . . . Also it will suffice to focus our attention on a particular j value. In terms of the 2 j + 1 number states the “truncated” exponential phase operators ei6* for each individual oscillator have the following unitary representation, 2j- 1
,i&*=
C 1n*)(n,+
1 I +ei(2j+1)+o* I n*=2j)(O,
(6)
I.
Here 4,, correspond
to an arbitrary reference phase angle. It is necessary to note here that the “truncated” phase operators differ from those in the Pegg-Barnett formalism in that the dimension of the underlying truncated Hilbert space is not made arbitrarily large for the calculation of physical quantities such as expectation values. These “truncated” unitary phase operators have eigenstates 14, f >,
with corresponding
eigenvalues
&,,, given by
27rm,
m*=O, 1, 2 ,...,
h,*= #Jo*+-2j+l’ The Hermitian
phase operator
4*
2j.
can thus be expressed
(7b) in terms of the “truncated
phase states”
I +, f > as
3j
6*=
c
(8)
4%* I #J,*wrn* 1.
m*=O
We can also write C$+ in terms of the number states I n * > using (7a). It is clear from (2) that while constructing angular momentum states with a particular j value we need to choose out of all the possible product states I rz,, II_ ) in the (2j + 1) X (2 j + 1) dimensional space only those states which obey the restrictions (3), namely n, + n_ = 2 j. This can be achieved easily through a projection operator &j, $SjE
E
n+=o
l n,, 2j-n+)(n+,
2j--n+
I,
(9)
whose action on a general product state I II,, n_ > will select only such states that correspond to angular momentum j. It may also be noticed here that if we confine our calculation within the 2 j + 1 states of angular momentum j and deal with operators which act only on these 2 j + 1 states, then &j is unitary (in fact it is then the identity operator 13. The necessity of introducing such a projection operator will become clear when one writes 3, in (la) using (4) as
where a different notation
246
T.K. Kar, D. Bhaumik/
Physics Letters A 207 (1995) 243-249
are identical in the two cases. Expressing subspace{I j, terms of the product states of the oscillators 1II,, n_ > one obtains m>l,=
exp[
-j.,,,,j
_i(E+-
‘E’ 111++ 1, n_>(n+,
&)I =
n-+
exp[ - i<$+ - & )] with the help of (6) in
l I
n,, fl_=O
exp[ -i(2j
+
2j-
+
+ I)(+,+-
40_)]IO,, 2jWj,
O_ I
I
C
In++
1,2j)(n
+, O_ I exp[i(2j
+
1)&l
n+=O 2j+
1
C
IO,, n_)Qj,
n_+
1 I exp[ -i(2j+
l)&,+]
(11)
n_=O
It is now manifest that the third and fourth term in (11) involves states which are outside the 2 j + 1 dimensional space of angular momentum states 1j, m) under consideration. Since .?+ is an operator whose action on the I j, m) states keeps them within the 2 j + 1 dimensional space of angular momentum it is necessary to select that part of exp[ - i( 4, - & >] which confines its action within this space only. It ca? easily be verified that the operator [&. exp[ - i(J+ - c$_)]&~] would satisfy this requirement. Incidentally yj commutes with N,, N_ and exp[ - ii$+ - $_>I as can be checked using the definition of &j in (9). Thus J, when written in terms of phase operators 0 * using (10) has the form
The operator represented /(j - m)( j operator has
sj first projects only the I j, m) states from all possible states ((2 j + 1) X (2 j + 1) in number) by I n,, n_ > and then !+ acts on them raising them to I j, m + 1) with a factor f m + 1) . Denoting exp[-i($+$_)]gj by ei’ we can at once see from (11) that this unitary a representation in terms of oscillators as Zj-
I
eib = ,,;, +
1n++ 1, n_=2j-n+-
l)(n+,
2j-n,
I+e-i(2j+1x~o+-90-)IO+,
2j)(2j,
O_ I.
(13)
Since $o+- (be_ is the difference of the reference phases for the (-t > and (-1 oscillators and is a constant, expression (13) is the same as that obtained by Nienhuis _and van Enk [3]. That & represents the azimuthal phase operator of the angzar momentum raising operator J, can be seen with a little manipulation using (13) as the representation of e , ~+=~ei~~=2’~1&7i~~In,+l,2j-.+_1~(~+,2j-fr+l n+=O I C In++1,2j-n+-l)(n+,2j-n+l n+=O 2j-
=dm = /m(ei” = \im
-
exp[ -i(2j+
l)(+o+-
$,_)I
IO+, 2j)(2jV
O- I>
ei’.
The azimuthal angle operator thus defined is similar to the phase difference operattr discussed by Luis and Sanchez-Soto [5]. Since the operator ei’ commutes with the total number of quanta N++ g_ preserving the j value by virtue of (3), it was not necessary to invoke the limiting procedure prescribed by Pegg and Barnett for
T.K. Kar, D. Bhumik /
Physics
Letters
A 207 (1995)
243-249
247
the dimensionality of the state space of the individual oscillators to obtain physical results. Indeed this azimuthal angle operator does reproduce all the standard results regarding the spherical representation of the J. We have thus established the relationship between the oscillator phase operators and the azimuthal angle operator. The azimuthal angle operator e” thus defined has eigenstates 1‘jp,) given by
with corresponding Qr=Qo+-
eigenvalues 25-r
r=O,
2jfl’
1,2 ,...,
2j,
where we have written (2 ’ + l)(&+$e_) = (2j + l)@,. n The connection of e’ I with the phase difference operator exp[ -X4+ - $_)I in the truncated space is indicative of the possibility that the states 1Qr,> can be expressed in terms of the “truncated” phase states ) 4,,,+) and 1$,,_) of the individual oscillators with appropriate Clebsch-Gordan coefficients. Indeed inverting Eq. (7a) to express I n, ) and I n_ > in terms of I +,+ > and 1&,_ > and substituting in (14a) we have
l@J =
d
1
E g exp[in+(@, (2j + 1)3 ,1+=0 m*=O
;i&
=
t
m,
r-
1 (2j+
1)
exp( -2ijL)I
4,+3 +m-)
exp(-W+m+-r) I k+, 4m+-r) r 1
exP(
C +
- +*++ $,_)I
-2ij~2j+m+-m_+,)I~m+)
(‘5)
hj+l+m+-r).
nl,=o
Thus if we write
with r an integer modulo 2j + 1, then the Clebsch-Gordan C:I + .m_= CL+,,+-,=
coefficients
1 Jm
exp( - 2ij+,+
-r)
1 =C~~+,(2j+l)+m+-r=
gipT
exP(-2ij~((2j+l)+m+-r)
CL+ ,m_ have the following for 2 j 2 m, 2 r,
for r-
1 >m+>O.
values, ( 16a)
( ‘6b)
The phase state I Gr) as defined in (14a) are 2 j + 1 in number having discrete eigenvalues (14b); and are similar to the standard angular momentum states 1j, m). This similarity may be extended to the angular momentum coherent states as discussed by Radcliffe 161 in 1971 if we can construct [7] a coherent superposition of 1Qr) states. To this end let us construct creation and annihilation operators of the “phase quanta” of angles 6’ and 6 such that & = &$,
( ‘7)
T. K. Kar. D. Bhaumik/
248
Physics Letters A 207 (I 9951 243-249
and whose action on 1Gr) is as follows, 4’ IGr,,> = fir+,
lGr+,>,
WJ
= @.I@~_,>.
(18)
These operators satisfy the commutator relations simjlar t,” the $_and have the following explicit expressions in terms of N,, N_ and @,
4’
= &)I/2
exp ji(N’+-ii_+Zj)& i
= z 1
8, of the single mode operators.
I +KI@lJ)(@?j
#IcP,><@r-i
1.
r=l
They
(19)
The phase creation and annihilation operators are not Hermitian and hence they themselves are not observables. Nevertheless they offer instructive and calculational advantage in their use for description of the phase states of the harmonic oscillator. To construct a unique coherent state superposing 1Dr,> states, let us choose @a = 0 so that ~0~= 2rrr/ (2j+ 11, r=O, 1,2 ,... ,2 j. This will then ensure that
In fact it can be easily seen that for k = 1,. . . ,2j. We can now construct the coherent state I j, cy) superposing different operating eaJ+ on 1G0 = 0), CYbeing a continuous parameter
(47mT)n
*j
Ij,a)=Nea2J+l@o)=_N~
fi
n=O
I~) n
I Gr,>states in the usual manner [6,7,81
(2’)
7
where J1”=
(lal/wpF-iy)2n fz.1 ( n=0
l/2
5
I
.
This coherent state is analogous to the Radcliffe state [6] for the angular momentum. One can easily calculate the expectation values of the angular momentum operators and take the appropriate classical limit. The probability of finding the azimuthal angle cD~= 2rrm/(2 j + 1) in the state I j, Q > is given by
P(Qm) = I(QmI j, a>l’=
[ I a l@qgTiy2m/m! Ego[ I crl~~]*“/n!
*
(22)
In the limit of large j and m the distribution corresponds to the classical results. To conclude we have used the projection operator technique to discuss the quantum operators for the spherical angles of a three dimensional angular momentum vector J^ constructed out of the phase operators of two single mode oscillators. For a given quantum number j the dimensionality of the number state space of the two oscillators required to construct the spherical angle operators was 2 j + I. Thus our result shows how a quantum version of the spherical angle operator for angular momentum can be constructed out of the operators for the “constituent” oscillators.
T. K. Kar, D. Bhoumik / Physics Letters A 207 (1995) 243-249
The authors would like to thank Professor reading of the manuscript.
Binayak
Dutta-Roy
for helpful
249
discussions
and for a critical
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