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Note on the convergence of normal and antinormal ordering expansions of the boson exponential operator exp(-μâ+â)
Volume 92A, number 3
PHYSICS LETTERS
1 November 1982
NOTE ON THE CONVERGENCE OF NORMAL AND ANTINORMAL ORDERING EXPANSIONS OF THE BOSON EXPONENTIAL OPERATOR exp(-~th~â) R. BALTIN Department of Theoretical Chemistry, University of (Jim, (Jim, Fed. Rep. Germany Received 27 July 1982
f(a+â)
For the boson exponential operator = exp(_pñ+â) it is shown by an elementary direct calculation using occupation number eigenstates that for a certain domain of the complex p-plane the formally defined antinormally ordered expansion off yields states ofinfinite norm when applied to an occupation number eigenstate. The normally ordered expansion, however, is well defined for all p.
Normal and antinormal ordering of functions of boson operators ~ play an important role in the description of systems of harmonic oscillators and of quantized fields. For a detailed presentation see refs. [1,2]. The concept of ordering of operator functions f(â, is based upon a series expansion of the cnumber functionf(a, a*) (a complex), followed by a formal replacement of a, a* by ã~,respectively, and finally by a rearrangement of the a and á~by use of their commutation relation
all the expressions (2), (3), and (4) are supposed to represent the same operator: (5) f(a,a)=f”1’(a,a )f~aJ(a a)=F.
~
Usually, in performing these ordering procedures no attention is payed to questions of existence and convergence of the operator expansions (3) and (4). However, even for rather elementary, well-behaved c-number functionsf one easily encounters divergences in the case of antinormal ordering as has been shown by Cahill and Glauber [11] in the framework of a somewhat sophisticated formalism using coherent state displacement operators.
a,
a+)
a,
I-- ~+1 = La, a
‘~-‘
such that, in the case of normal ordering (antinormal ordering) all a stand to the right (left) of all â~. Thus the operator ~ ‘2~ —J ka, a ~ “ ~ can be written in normally ordered form
f~)~â â~)=
E
n, m0
~
~3)
~â+)flãm
nm
‘
or, alternatively, in antinormally ordered form
f(a)(a, â~)=
dnm am ~â+)n
(4)
n,m0
where the notation of ref. [2] has been used. Clearly, with respect to their effect on any boson state I i,L’), 110
-
-
-
For certain simple classes of functions f, the ordering can be achieved in closed form [3]. In a lot of papers this problem is solved by special techniques including coherent state formalism [2,4—6],parameter differentiation [1,2,7—9]and phase-space mapping [10].
When the operator function f especially depends only on the number operator a+a, but not on a, separately, one can get insight into the convergence properties of f(nl) orf(a) in a rather elementary direct way using the complete set {Im)} of occupation number eigenstates [2] à~â~m)mIm) (mc,l,2,...). (6) When eq. (5) is rewritten for this special class of functions and is applied to a state rn) we should 0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
Volume 92A, number
3
PHYSICS LETFERS
1 November 1982
expect f(ã~á)Im)=fQ’)(â~d)lrn) f(a)(â+ã)Irn) =f(m)Irn) (5a) .
(e_Pa~a)(~i)lrn) = ~ (m) (e—P ~,=o \~ m
As concerns the antinormally ordered expression, however, the relation
f(a)(a+a)Im)
=
=
=
(7)
~‘
(convergent for all p and all z) is formally replaced by a+a, the originating operator expansion e’~
~
-~-~-
v0
(8)
(d+~)V
V!
is well defined since the action of this expression on any basis state jm) is due to eqs. (6) and (7)
E
‘~ (e~— 1)~’Irn)= (1 + e—~— 1)~rn)
Vi
rnvIm)~~ e~mIrn).
(9)
(13)
.
Thus the normally ordered expansion (10) is well defined in the whole Hubert space spanned by the basis {Im>} since the series terminates and therefore is, of course, convergent and the sum equals f(rn) = exp(—j.un) for all p and all rn = 0, 1, 2, ... in accordance with (Sa). On the other hand, (e_~~~)(t2)Irn) = e~is fl7. (p)Irn>
where
5m(J2)
,
(rn+w~(l_e~)v
=~
2~ !~::~&~ v0
1)”Irn)
(~
=e_MmIrn)
f(m)Irn)
is not always valid. It may happen that the action of the operatorf(a) on Irn) yields an infinite eigenvalue or, if finite, another eigenvalue than f(rn). It is the purpose of this note to demonstrate this inconsistency for the case of the exponential operator exp {—p~a}for certain values of the complex number p. For completeness let us discuss the case of norma!When ordering, too. the complex number z in the expansion
e~
~
v0
—
v0 ~
~
(14)
(15)
/
neither terminates for any rn nor is convergent for all p as is seen by the ratio test for infinite series. Since Sm ~5 a power series with respect to M (16) w(p)E 1 e —
its radius r of convergence in the complex w-plane is given by, see e.g. ref. [12] + v’~/(rn + V + r lim p-+= V ~ p+1
(
)
=
Fun
V+
1
1.
(17)
v0
The normally and antinormally ordered expansions of (8) are well known, see e.g. ref. [2], =
~ v=O
Since for equal wi = or1 all members the series series cannot (15) have modulus greater than of 1, the converge for wi = 1 and therefore the operator expansion (11) is well defined only for p satisfying
(eM
—
1) ~
(10)
V!
and (e~a)(~z)= eM ~ (1 ~=O
—
e~)
(11)
1 ~eAI <1.
(18)
In the complex p-plane the domain 0 of existence [ convergence of 5m (p)] of the antinormally ordered expansion is separated from the domain of divergence by the lines
V!
Let us apply these expansions to the state Lrn). From
II —e~I= 1 or
âIrn)V~irn— 1),
(12a)
Rep=ln[2cos(Imp)]
(12b)
Fig. 1 shows the disconnected domain D by hatched
â~irn) V~iTTIm+ we obtain
1)
areas.
.
(19)
111
Volume 92A, number 3
PHYSICS LETTERS
1 November 1982
Imp
m+X+1\ -
i II I
__
(1 —eP)srn+i(p)
x
= ~ (1 value eV)~’ wherex=0 the terms with A = 0 (having the 1) have been added and subtracted. Thus we obtain by corn-
I
—
x
-
I
-7
I
(m+1)
srn+1e I
-~
I~IIIjjjIj~ -
-~
— —
I-
-
Fig. 1. The domain 0 where the expansion of [exp(_pa+a)l(a), eq. (11), exists, consists of infinitely many disconnected parts constituting of these partsa are periodical shown by array hatched along the areas. imaginary The boundaries axis. Three of these areas are excluded from D.
It remains to prove that for all m E
=
0, 1, 2,
...
e_mPim)
(20)
.
From
~
=
~
=
~ v0
(rn
+
V +
1) (1
—
eP)”
geometric series s 0
=
~ v0
(1
eP)~’= e~
—
So we finally get 2)M Srn+1
=
e_(m+
,
or e~s m
=
e~°~
and eq. (20) follows from eq. (14). I wish to thank F. Röscheisen for the drawing.
References
[2] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973). [3] J.H. Marburger, J. Math. Phys. 7 (1966) 829. [4] C.L. Mehta, J. Phys. Al (1968) 385.
~‘
—
[(rn +A+
1)(rn+X)](l
we find after multiplication by 1
112
Since ii —ePI <1 forpED,wecansumupthe
[1] W.H. Louisell, Radiation and noise in quantum electronics (McGraw-Hill, New York, 1964).
1 =
s~.
D =
~m+i(P)
and
—
Sm+l=~’2Sm =e2~srn-i
-
-f
~
parison with the definition (15) of Sm (1 —eP)sm+i =5m+1 5rn
Rep2
~
for p
(
~(m+X)~~
—
eU)X~
e~
[5] C.L. Mehta,J. Math. Phys. 18 (1977) 404. [6] Nuovo [7] P. J. Gluck, Schwinger, in: Cimento Quantum8B(l972)256. theory of angular momentum, eds. J. Schwinger, L.C. Biedenharn and H. van Dam(AcademicPress,NewYork, l965)pp.274—276. [8] N.H. McCoy, Proc. Edinburgh Math. Soc. 3(1932) 118. [9] R.M. Wilcox, J. Math. Phys. 8 (1967) 962. [10] G.S. Agarwal and E. Wolf, Phys. Rev. D2 (1970) 2161. [11] K.E. Cahill, R.J. Glauber, Phys. Rev. 177 (1969) 1857. [12] E. Kreyszig, Advanced engineering mathematics (Wiley, New York, 1972).
PHYSICS LETTERS
1 November 1982
NOTE ON THE CONVERGENCE OF NORMAL AND ANTINORMAL ORDERING EXPANSIONS OF THE BOSON EXPONENTIAL OPERATOR exp(-~th~â) R. BALTIN Department of Theoretical Chemistry, University of (Jim, (Jim, Fed. Rep. Germany Received 27 July 1982
f(a+â)
For the boson exponential operator = exp(_pñ+â) it is shown by an elementary direct calculation using occupation number eigenstates that for a certain domain of the complex p-plane the formally defined antinormally ordered expansion off yields states ofinfinite norm when applied to an occupation number eigenstate. The normally ordered expansion, however, is well defined for all p.
Normal and antinormal ordering of functions of boson operators ~ play an important role in the description of systems of harmonic oscillators and of quantized fields. For a detailed presentation see refs. [1,2]. The concept of ordering of operator functions f(â, is based upon a series expansion of the cnumber functionf(a, a*) (a complex), followed by a formal replacement of a, a* by ã~,respectively, and finally by a rearrangement of the a and á~by use of their commutation relation
all the expressions (2), (3), and (4) are supposed to represent the same operator: (5) f(a,a)=f”1’(a,a )f~aJ(a a)=F.
~
Usually, in performing these ordering procedures no attention is payed to questions of existence and convergence of the operator expansions (3) and (4). However, even for rather elementary, well-behaved c-number functionsf one easily encounters divergences in the case of antinormal ordering as has been shown by Cahill and Glauber [11] in the framework of a somewhat sophisticated formalism using coherent state displacement operators.
a,
a+)
a,
I-- ~+1 = La, a
‘~-‘
such that, in the case of normal ordering (antinormal ordering) all a stand to the right (left) of all â~. Thus the operator ~ ‘2~ —J ka, a ~ “ ~ can be written in normally ordered form
f~)~â â~)=
E
n, m0
~
~3)
~â+)flãm
nm
‘
or, alternatively, in antinormally ordered form
f(a)(a, â~)=
dnm am ~â+)n
(4)
n,m0
where the notation of ref. [2] has been used. Clearly, with respect to their effect on any boson state I i,L’), 110
-
-
-
For certain simple classes of functions f, the ordering can be achieved in closed form [3]. In a lot of papers this problem is solved by special techniques including coherent state formalism [2,4—6],parameter differentiation [1,2,7—9]and phase-space mapping [10].
When the operator function f especially depends only on the number operator a+a, but not on a, separately, one can get insight into the convergence properties of f(nl) orf(a) in a rather elementary direct way using the complete set {Im)} of occupation number eigenstates [2] à~â~m)mIm) (mc,l,2,...). (6) When eq. (5) is rewritten for this special class of functions and is applied to a state rn) we should 0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
Volume 92A, number
3
PHYSICS LETFERS
1 November 1982
expect f(ã~á)Im)=fQ’)(â~d)lrn) f(a)(â+ã)Irn) =f(m)Irn) (5a) .
(e_Pa~a)(~i)lrn) = ~ (m) (e—P ~,=o \~ m
As concerns the antinormally ordered expression, however, the relation
f(a)(a+a)Im)
=
=
=
(7)
~‘
(convergent for all p and all z) is formally replaced by a+a, the originating operator expansion e’~
~
-~-~-
v0
(8)
(d+~)V
V!
is well defined since the action of this expression on any basis state jm) is due to eqs. (6) and (7)
E
‘~ (e~— 1)~’Irn)= (1 + e—~— 1)~rn)
Vi
rnvIm)~~ e~mIrn).
(9)
(13)
.
Thus the normally ordered expansion (10) is well defined in the whole Hubert space spanned by the basis {Im>} since the series terminates and therefore is, of course, convergent and the sum equals f(rn) = exp(—j.un) for all p and all rn = 0, 1, 2, ... in accordance with (Sa). On the other hand, (e_~~~)(t2)Irn) = e~is fl7. (p)Irn>
where
5m(J2)
,
(rn+w~(l_e~)v
=~
2~ !~::~&~ v0
1)”Irn)
(~
=e_MmIrn)
f(m)Irn)
is not always valid. It may happen that the action of the operatorf(a) on Irn) yields an infinite eigenvalue or, if finite, another eigenvalue than f(rn). It is the purpose of this note to demonstrate this inconsistency for the case of the exponential operator exp {—p~a}for certain values of the complex number p. For completeness let us discuss the case of norma!When ordering, too. the complex number z in the expansion
e~
~
v0
—
v0 ~
~
(14)
(15)
/
neither terminates for any rn nor is convergent for all p as is seen by the ratio test for infinite series. Since Sm ~5 a power series with respect to M (16) w(p)E 1 e —
its radius r of convergence in the complex w-plane is given by, see e.g. ref. [12] + v’~/(rn + V + r lim p-+= V ~ p+1
(
)
=
Fun
V+
1
1.
(17)
v0
The normally and antinormally ordered expansions of (8) are well known, see e.g. ref. [2], =
~ v=O
Since for equal wi = or1 all members the series series cannot (15) have modulus greater than of 1, the converge for wi = 1 and therefore the operator expansion (11) is well defined only for p satisfying
(eM
—
1) ~
(10)
V!
and (e~a)(~z)= eM ~ (1 ~=O
—
e~)
(11)
1 ~eAI <1.
(18)
In the complex p-plane the domain 0 of existence [ convergence of 5m (p)] of the antinormally ordered expansion is separated from the domain of divergence by the lines
V!
Let us apply these expansions to the state Lrn). From
II —e~I= 1 or
âIrn)V~irn— 1),
(12a)
Rep=ln[2cos(Imp)]
(12b)
Fig. 1 shows the disconnected domain D by hatched
â~irn) V~iTTIm+ we obtain
1)
areas.
.
(19)
111
Volume 92A, number 3
PHYSICS LETTERS
1 November 1982
Imp
m+X+1\ -
i II I
__
(1 —eP)srn+i(p)
x
= ~ (1 value eV)~’ wherex=0 the terms with A = 0 (having the 1) have been added and subtracted. Thus we obtain by corn-
I
—
x
-
I
-7
I
(m+1)
srn+1e I
-~
I~IIIjjjIj~ -
-~
— —
I-
-
Fig. 1. The domain 0 where the expansion of [exp(_pa+a)l(a), eq. (11), exists, consists of infinitely many disconnected parts constituting of these partsa are periodical shown by array hatched along the areas. imaginary The boundaries axis. Three of these areas are excluded from D.
It remains to prove that for all m E
=
0, 1, 2,
...
e_mPim)
(20)
.
From
~
=
~
=
~ v0
(rn
+
V +
1) (1
—
eP)”
geometric series s 0
=
~ v0
(1
eP)~’= e~
—
So we finally get 2)M Srn+1
=
e_(m+
,
or e~s m
=
e~°~
and eq. (20) follows from eq. (14). I wish to thank F. Röscheisen for the drawing.
References
[2] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973). [3] J.H. Marburger, J. Math. Phys. 7 (1966) 829. [4] C.L. Mehta, J. Phys. Al (1968) 385.
~‘
—
[(rn +A+
1)(rn+X)](l
we find after multiplication by 1
112
Since ii —ePI <1 forpED,wecansumupthe
[1] W.H. Louisell, Radiation and noise in quantum electronics (McGraw-Hill, New York, 1964).
1 =
s~.
D =
~m+i(P)
and
—
Sm+l=~’2Sm =e2~srn-i
-
-f
~
parison with the definition (15) of Sm (1 —eP)sm+i =5m+1 5rn
Rep2
~
for p
(
~(m+X)~~
—
eU)X~
e~
[5] C.L. Mehta,J. Math. Phys. 18 (1977) 404. [6] Nuovo [7] P. J. Gluck, Schwinger, in: Cimento Quantum8B(l972)256. theory of angular momentum, eds. J. Schwinger, L.C. Biedenharn and H. van Dam(AcademicPress,NewYork, l965)pp.274—276. [8] N.H. McCoy, Proc. Edinburgh Math. Soc. 3(1932) 118. [9] R.M. Wilcox, J. Math. Phys. 8 (1967) 962. [10] G.S. Agarwal and E. Wolf, Phys. Rev. D2 (1970) 2161. [11] K.E. Cahill, R.J. Glauber, Phys. Rev. 177 (1969) 1857. [12] E. Kreyszig, Advanced engineering mathematics (Wiley, New York, 1972).