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Bell numbers and coherent states
21 August 2000
Physics Letters A 273 Ž2000. 159–161 www.elsevier.nlrlocaterpla
Bell numbers and coherent states J. Katriel Department of Chemistry, Technion – Israel Institute of Technology, Haifa 32000, Israel Received 19 June 2000; accepted 12 July 2000 Communicated by P.R. Holland
Abstract The combinatorial properties of the bell numbers are shown to follow from the algebraic properties of the boson operators in an extremely simple way. This treatment sheds further light on the ‘hidden’ combinatorial significance of the boson operators and of the identities they satisfy, and allows a consistent introduction of q-deformations. q 2000 Elsevier Science B.V. All rights reserved. PACS: 02.10.Eb; 42.50.Ar Keywords: Stirling numbers; Bell numbers; Coherent states; q-deformations
The normally-ordered expansion of an integral power of the number operator a†a in terms of the boson operators a and a†, that satisfy the commutation relation w a,a† x s 1, can be written in the form w1x k
k
Ž a†a . s Here,
k
l k Ý l Ž a† . a l . ls0
½ l5
½5
Ž 1.
are the Stirling numbers of the second
kind w2x. They have a well-known combinatorial interpretation, being equal to the number of partitions of a set of k objects into l non-empty subsets. Many of the properties of the Stirling numbers and
many of the identities they satisfy can be obtained in terms of the relation formulated in Eq. Ž1. w1x. In the present Letter the boson operator version of the theory of the Stirling numbers is extended by examining expectation values with respect to coherent states. This yields in an extremely simple way some of the fundamental properties of the Stirling and Bell numbers. The number-states < m:, that satisfy a†a < m: s m < m:, ² m < m: s 1, and the coherent states < g :, that satisfy a < g : s g < g :, ²g < g : s 1, are the two most important sets of states within the boson-operator Fock space. They are related by the well-known expression
ž
< g : s exp y E-mail address: [email protected] ŽJ. Katriel..
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 8 8 - 6
`
gm
/Ý' ms0
m!
< m: .
Ž 2.
J. Katrielr Physics Letters A 273 (2000) 159–161
160
Evaluating the expectation value of both sides of Eq. Ž1. with respect to the number operator eigenstate < m: we obtain k
mk s
k
`
½5
Ý l
ls0
ml ,
f Ž a . s exp Ž e a y 1 . s
k
k
²g < Ž a a . < g : s
½5
Ý ls0 l
ak
` †
f Ž a . ' ²g
Ý ks0
s
Ý ks0
ak
k
k!
½5
Ý l
k! ls0
Ea
s
`
l
Ž e y 1. s l ! Ý ks l
s
Ý Ž k y 1. ! b Ž k . .
ks0
On the other hand, from Eq. Ž4.
k
k
k
k
Ž 3.
Ž 4.
bŽ k . .
l Ý kl b Ž k . .
ž/
Evaluating the left-hand side of Eq. Ž5. using the representation of the coherent state in terms of the number states, Eq. Ž2., we obtain bŽ k . s
m ky 1
`
1
, Ý e ms1 Ž m y 1 . !
which is arguablly the simplest derivation of the celebrated Ž‘remarkable’ w3x. Dobinski formula. For the q-boson operators, that satisfy w5x w a,a† x q s aa† y qa†a s 1, Eq. Ž1. becomes w6x k
k
Ž a†a . s
l k Ý l q Ž a† . a l . ls0
½5
Ž 6.
The q-coherent states 1
< g : s exp q Ž < g < 2 .
y 2
`
gm
Ý w mx ! ( q
ms0
satisfy a < g : s g < g :. Here,
w mx q s sbŽ k . ,
k!
ks0
ks0
½5
½5
Ý
b Ž l q 1. s
ak k . k! l
Ý ls0 l
ak
`
s ea
Setting < g < s 1 we obtain ²g < Ž a†a . < g : s
bŽ k . .
Equating these two expressions for E f Ea, expanding the exponential and equating powers of a we obtain the recurrence relation w4x
which, along with the initial value f Ž0. s 1, can be integrated into f Ž a . s expw < g < 2 Že a y 1.x Expanding f Ž a . with respect to < g < 2 and equating coefficients of equal powers of the latter with those in Eq. Ž3. we obtain the generating function w2x a
Ea
Ea
where the dependence of f Ž a . on g is not explicitly indicated. Eq. Ž3. means that f Ž a . is a generating function for the Stirling numbers of the second kind. To obtain an explicit expression for f Ž a . we differentiate with respect to a , and use the identity † † e a a aa†a s a†e a Ž a aq 1. a. This yields
EfŽa.
k!
a ky 1
`
Ef
²g < Ž a†a . < g :
k < <2 l g ,
ak
Differentiating with respect to a we obtain
Ef
k < <2 l g .
One readily obtains
`
Ý ks0
where m ls mŽ m y 1. PPP Ž m y l q 1.. This is the classical expression for the k’th power of the variable m in terms of the falling factorial m l. On the other hand, the evaluation of the expectation values of both sides of Eq. Ž1. with respect to the coherent state < g : yields †
where bŽ k . is the Bell number, that is equal to the number of partitions of a set of k objects into any number of subsets. In this case Eq. Ž3. yields
Ž 5.
qm y1 qy1
and exp q Ž x . s Ý`ms0 x m w m x q !.
< m:
J. Katrielr Physics Letters A 273 (2000) 159–161
Evaluating the expectation values of both sides of Eq. Ž6. with respect to < g : we obtain k
k
²g < Ž a†a . < g : s
k
½5
Ý ls0 l
to be the step that introduces the transcendental features required for the emergence of the generating functions and of the Dobinski formula.
Acknowledgements
Thus, k
161
k
½5
Ý ls0 l
q
< g < 2 l s exp q Ž < g < 2 . `
=
Ý ms1
2m
y1
1 w m x ky q . w m y ax q !
For < g < s 1 the left-hand side becomes the q-Bell number w7x and the right-hand side is the q-Dobinski formula. With the presently reported results the formulation of a virtually complete theory of the Stirling and Bell numbers in terms of the algebraic and Fock space properties of the boson operators can be declared to have been achieved. It is remarkable that the application of the boson coherent states turns out
This research was supported by the fund for the promotion of research at the Technion.
References w1x J. Katriel, Lett. Nouv. Cim. 10 Ž1974. 565. w2x M. Abramowitz, I.A. Stegun ŽEds.., Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. w3x J. Pitman, Amer. Math. Monthly 104 Ž1997. 201. w4x H.S. Wilf, Generatingfunctionology, Academic Press, Boston, 1990. w5x M. Arik, D.D. Coon, J. Math. Phys. 17 Ž1976. 524. w6x J. Katriel, M. Kibler, J. Phys. A 25 Ž1992. 2683. w7x S.C. Milne, Trans. Amer. Math. Soc. 245 Ž1978. 89.
Physics Letters A 273 Ž2000. 159–161 www.elsevier.nlrlocaterpla
Bell numbers and coherent states J. Katriel Department of Chemistry, Technion – Israel Institute of Technology, Haifa 32000, Israel Received 19 June 2000; accepted 12 July 2000 Communicated by P.R. Holland
Abstract The combinatorial properties of the bell numbers are shown to follow from the algebraic properties of the boson operators in an extremely simple way. This treatment sheds further light on the ‘hidden’ combinatorial significance of the boson operators and of the identities they satisfy, and allows a consistent introduction of q-deformations. q 2000 Elsevier Science B.V. All rights reserved. PACS: 02.10.Eb; 42.50.Ar Keywords: Stirling numbers; Bell numbers; Coherent states; q-deformations
The normally-ordered expansion of an integral power of the number operator a†a in terms of the boson operators a and a†, that satisfy the commutation relation w a,a† x s 1, can be written in the form w1x k
k
Ž a†a . s Here,
k
l k Ý l Ž a† . a l . ls0
½ l5
½5
Ž 1.
are the Stirling numbers of the second
kind w2x. They have a well-known combinatorial interpretation, being equal to the number of partitions of a set of k objects into l non-empty subsets. Many of the properties of the Stirling numbers and
many of the identities they satisfy can be obtained in terms of the relation formulated in Eq. Ž1. w1x. In the present Letter the boson operator version of the theory of the Stirling numbers is extended by examining expectation values with respect to coherent states. This yields in an extremely simple way some of the fundamental properties of the Stirling and Bell numbers. The number-states < m:, that satisfy a†a < m: s m < m:, ² m < m: s 1, and the coherent states < g :, that satisfy a < g : s g < g :, ²g < g : s 1, are the two most important sets of states within the boson-operator Fock space. They are related by the well-known expression
ž
< g : s exp y E-mail address: [email protected] ŽJ. Katriel..
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 8 8 - 6
`
gm
/Ý' ms0
m!
< m: .
Ž 2.
J. Katrielr Physics Letters A 273 (2000) 159–161
160
Evaluating the expectation value of both sides of Eq. Ž1. with respect to the number operator eigenstate < m: we obtain k
mk s
k
`
½5
Ý l
ls0
ml ,
f Ž a . s exp Ž e a y 1 . s
k
k
²g < Ž a a . < g : s
½5
Ý ls0 l
ak
` †
f Ž a . ' ²g
Ý ks0
s
Ý ks0
ak
k
k!
½5
Ý l
k! ls0
Ea
s
`
l
Ž e y 1. s l ! Ý ks l
s
Ý Ž k y 1. ! b Ž k . .
ks0
On the other hand, from Eq. Ž4.
k
k
k
k
Ž 3.
Ž 4.
bŽ k . .
l Ý kl b Ž k . .
ž/
Evaluating the left-hand side of Eq. Ž5. using the representation of the coherent state in terms of the number states, Eq. Ž2., we obtain bŽ k . s
m ky 1
`
1
, Ý e ms1 Ž m y 1 . !
which is arguablly the simplest derivation of the celebrated Ž‘remarkable’ w3x. Dobinski formula. For the q-boson operators, that satisfy w5x w a,a† x q s aa† y qa†a s 1, Eq. Ž1. becomes w6x k
k
Ž a†a . s
l k Ý l q Ž a† . a l . ls0
½5
Ž 6.
The q-coherent states 1
< g : s exp q Ž < g < 2 .
y 2
`
gm
Ý w mx ! ( q
ms0
satisfy a < g : s g < g :. Here,
w mx q s sbŽ k . ,
k!
ks0
ks0
½5
½5
Ý
b Ž l q 1. s
ak k . k! l
Ý ls0 l
ak
`
s ea
Setting < g < s 1 we obtain ²g < Ž a†a . < g : s
bŽ k . .
Equating these two expressions for E f Ea, expanding the exponential and equating powers of a we obtain the recurrence relation w4x
which, along with the initial value f Ž0. s 1, can be integrated into f Ž a . s expw < g < 2 Že a y 1.x Expanding f Ž a . with respect to < g < 2 and equating coefficients of equal powers of the latter with those in Eq. Ž3. we obtain the generating function w2x a
Ea
Ea
where the dependence of f Ž a . on g is not explicitly indicated. Eq. Ž3. means that f Ž a . is a generating function for the Stirling numbers of the second kind. To obtain an explicit expression for f Ž a . we differentiate with respect to a , and use the identity † † e a a aa†a s a†e a Ž a aq 1. a. This yields
EfŽa.
k!
a ky 1
`
Ef
²g < Ž a†a . < g :
k < <2 l g ,
ak
Differentiating with respect to a we obtain
Ef
k < <2 l g .
One readily obtains
`
Ý ks0
where m ls mŽ m y 1. PPP Ž m y l q 1.. This is the classical expression for the k’th power of the variable m in terms of the falling factorial m l. On the other hand, the evaluation of the expectation values of both sides of Eq. Ž1. with respect to the coherent state < g : yields †
where bŽ k . is the Bell number, that is equal to the number of partitions of a set of k objects into any number of subsets. In this case Eq. Ž3. yields
Ž 5.
qm y1 qy1
and exp q Ž x . s Ý`ms0 x m w m x q !.
< m:
J. Katrielr Physics Letters A 273 (2000) 159–161
Evaluating the expectation values of both sides of Eq. Ž6. with respect to < g : we obtain k
k
²g < Ž a†a . < g : s
k
½5
Ý ls0 l
to be the step that introduces the transcendental features required for the emergence of the generating functions and of the Dobinski formula.
Acknowledgements
Thus, k
161
k
½5
Ý ls0 l
q
< g < 2 l s exp q Ž < g < 2 . `
=
Ý ms1
2m
y1
1 w m x ky q . w m y ax q !
For < g < s 1 the left-hand side becomes the q-Bell number w7x and the right-hand side is the q-Dobinski formula. With the presently reported results the formulation of a virtually complete theory of the Stirling and Bell numbers in terms of the algebraic and Fock space properties of the boson operators can be declared to have been achieved. It is remarkable that the application of the boson coherent states turns out
This research was supported by the fund for the promotion of research at the Technion.
References w1x J. Katriel, Lett. Nouv. Cim. 10 Ž1974. 565. w2x M. Abramowitz, I.A. Stegun ŽEds.., Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. w3x J. Pitman, Amer. Math. Monthly 104 Ž1997. 201. w4x H.S. Wilf, Generatingfunctionology, Academic Press, Boston, 1990. w5x M. Arik, D.D. Coon, J. Math. Phys. 17 Ž1976. 524. w6x J. Katriel, M. Kibler, J. Phys. A 25 Ž1992. 2683. w7x S.C. Milne, Trans. Amer. Math. Soc. 245 Ž1978. 89.