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Ordering of boson operator functions by the Hausdorff similarity transform
Physics Letters A 334 (2005) 140–143 www.elsevier.com/locate/pla
Ordering of boson operator functions by the Hausdorff similarity transform W. Witschel Department of Theoretical Chemistry, University of Ulm, D-89069 Ulm, Germany Received 7 September 2004; received in revised form 5 November 2004; accepted 8 November 2004 Available online 20 November 2004 Communicated by P.R. Holland
Abstract A generating function technique making use of the Hausdorff similarity transform is used to give arbitrary products of bosons in normal or antinormal ordered form. Operator commutator formulas and ordering in terms of Stirling numbers are also discussed. 2004 Published by Elsevier B.V.
1. Ordered operator products Katriel [1] showed that products of boson operators can be expressed in terms of Stirling numbers. Michailov [2] extended these calculations by defining generalized Stirling numbers. Recently Blasiak, Penson and Salomon [3] published an interesting article where they expressed ns + k k + r s n + n(r−s) = a Sr,s (n, k) a a a a k=s
(1) in normal ordered form. The Sr,s (n, k) are new generalized Stirling numbers of the second kind. They made E-mail address: [email protected] (W. Witschel). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.11.018
use of advanced combinatorics to exploit the properties of the Sr,s (n, k). The present Letter aims at a simple straightforward technique to give arbitrary products of boson operators in normal or antinormal ordered form where the operators a + are standing right (a) or left (n) of the a operators. Use is made of the Hausdorff similarity transform which was used previously on related problems [4]. Another technique is the application of commutation formulas for the normal ordering of boson operators [6]. By repeated application of these formulas one can also give arbitrary ordered products of boson operators. Finally a technique for products of the type {(a + )r a r }{(a + )s a s } · · · {(a + )k a k } will be used to show that for these operator products the generalized Stir-
W. Witschel / Physics Letters A 334 (2005) 140–143
ling numbers [3] can be expressed by standard Stirling numbers of the first and second kind.
ˆ (n) : ordered form G ˆ (n)
G
=a
+ k1
kj j −1 + k2 + a + α1 · · · a + αm
2. Ordered products of boson operators by the Hausdorff transform
141
× exp
j
m=1
αm a
(7)
m=1
It was shown previously that operator orderings can be performed by introducing generating functions and the Hausdorff transform [4]. The Hausdorff formula states: e
α Aˆ
ˆ Be
−α Aˆ
2 ˆ [A, ˆ B]] ˆ + ··· ˆ B] ˆ + α [A, = Bˆ + α[A, 2! ∞ αm ˆ m ˆ ˜ = (2) A , B = B(α) m! m=0
ˆ B] ˆ = Aˆ Bˆ − Bˆ Aˆ and where [A, m m−1 ˆ Aˆ Aˆ , Bˆ = A, , Bˆ is the repeated commutator. ˆ ˆ With eα A e−α A = Iˆ, where Iˆ is the identity operator Eq. (2) is generalized for ˆ ˆ ˆ ˆ −α Aˆ α Aˆ ˆ k−1 −α Aˆ e B e eα A Bˆ k e−α A = eα A Be k α Aˆ ˆ k−1 −α Aˆ ˜ ˜ = B(α)e B e = B(α)
(3)
and ˆ ˆ ˆ ˜ eα A eB e−α A = exp B(α) .
Wˆ = a
a a
a ···a
m=1
× a+ −
j
k2 αm
m=1
k · · · a + − αj j .
(8)
m=2
The noncommutative part is disentangled and one is left with simple tedious expansions, multiplications and comparison of powers of the ordering parameters. Application of symbolic math programs like M APLE or M ATHEMATICA is useful. 2 Example. Normal ordering of Wˆ = (a + a 2 )2 .
ˆ (n) = a + 2 eα1 a a + 2 eα2 a G 2 2 = a + a + + α1 e(α1 +α2 )a ; expansion and comparison leads to + 2 2 2 4 3 2 a a = a + a 4 + 4a + a 3 + 2a + a 2 .
(9)
(10)
(4)
The mathematical properties of the Hausdorff formula are discussed thoroughly by Fröhlich [5]. Boson operators with the commutator [a, ˆ aˆ + ] = 1 ˆ ˆ are a special case of [A, B] = c, where c is a real or complex number. In the following, use is made of the commutator property. Domain, range and convergence of the operator formulas are not discussed. To simplify the notation a and a + are written without the operator hatchet. Consider the boson operator product Wˆ : + k1 l1 + k2 l2
ˆ (a) : and the antinormal ordered form G
j
k1 j (a) + ˆ = exp G αm a a − αm
+ kj lj
a .
(5)
ˆ is introduced: Instead of Wˆ the generating function G ˆ = a + k1 eα1 a a + k2 eα2 a · · · a + kj eαj a G
(6)
where the αm are c-number ordering coefficients. With the repeated application of Eq. (3) one gets the normal
3. Repeated application of operator commutation formulas For completeness it should be mentioned that boson operator products can be ordered by repeated application of commutator formulas. Wilcox [6] derived by parameter differentiation for ˆ = cIˆ the commutation operators Pˆ and Qˆ with [Pˆ , Q] formula Pˆ m Qˆ n =
[[m,n]] j =0
m!n!cj Qˆ n−j Pˆ m−j j !(m − j )!(n − j )!
(11)
[[m, n]] = min (in the original article the upper limit of the sum is missing). Similar formulas are derived by the coherent state technique [7]. The generating function technique of the preceding text gives the commutation in a simple form which
142
W. Witschel / Physics Letters A 334 (2005) 140–143
may be interesting for teaching applications. e
αa +
a k = (a − α)k e
αa +
Example. (12)
by Eq. (8)
expansion leads to ∞ ∞ αm + m k αm m a a = (a − α)k a + . m! m!
m=0
Comparison of the coefficients of equal powers of α leads to the commutation. 4. Ordering of products {a + a r }m {a + a s }k ··· s
For arbitrary products the preceding techniques can be used. The special products r m + s s k a a ··· Kˆ = a + a r (14) can be treated in a different way making use of Stirling numbers. Furthermore it is shown that for these special cases the generalized Stirling numbers of the second kind [3] can be expressed by multinomial coefficients, conventional Stirling numbers of the first and of the second kind. r For brevity the calculations are limited to (a + a r )m . r r + Katriel [1] showed that (a a ) can be expressed by Stirling numbers of the first kind s(r, j ):
r m + r r m j = s(r, j )N a a (15) j =1
where N = a + a with the commutators [N, a] = −a, N, a + = a + .
Nk N
(n)
k (a)
=
k k (n) j a+a = S(k, j )a + a j ,
k (a) = a+a = (−1)k
k j =0
with A1 = (s(2, 1))2 , A2 = 2s(2, 1)s(2, 2) and A3 = (s(2, 2))2 . Ordering with Eq. (17) leads to + 2 2 2 a a = A3 S(4, 1) + A2 S(3, 1) + A1 S(2, 1) a + a + A3 S(4, 2) + A2 S(3, 2) 2 + A1 S(2, 2) a + a 2 3 + A3 S(4, 3) + A2 S(3, 3) a + a 3 + A3 S(4, 4)a + a 4 4
= a + a 4 + 4a + a 3 + 2a + a 2 . 4
(17)
3
2
(20)
This result agrees with Table 1 of Ref. [3]: + 2 2 2 2 = S2,2 (2, 1)a +a + S2,2 (2, 2)a + a 2 a a + S2,2 (2, 3)a + a 3 + S2,2 (2, 4)a + a 4 . (21) For example the generalized Stirling number in Eq. (21) 2 S2,2 (2, 2) = s(2, 2) S(4, 2) 3
4
+ 2s(2, 1)s(2, 2)S(3, 2) 2 + s(2, 1) S(2, 2)
(16)
To the r.h.s. of Eq. (15) the multinomial formula is applied and equal powers of N k are collected. They are rewritten in terms of Stirling numbers of the second kind S(k, j ) for normal form {N k }(n) and antinormal form {N k }(a) :
(19)
(13)
m=0
r
2 2 + 2 2 2 = s(2, 1)a + a + s(2, 2) a + a a a 2 3 4 = A1 a + a + A2 a + a + A3 a + a
(22)
is expressed by binomial coefficients, Stirling numbers r of the first and second kind. The ordering of (a r a + )m follows the same lines. It is: r−1 r a r a + = a r−1 a + a + 1 a + r−1 + a a+r . = a r−1 a + (23) Repeated application of the commutator leads to r a r a + = a +a + 1 a + a + 2 · · · a +a + r
j =1
j +j
(−1)j S(k + 1, j + 1)a a
. (18)
(a + a)r+1 (24) a +a with the Pochhammer symbol (X)n which can be ex=
W. Witschel / Physics Letters A 334 (2005) 140–143
pressed by Stirling numbers of the first kind [8] ar a+ = r
=
As an independent control we can check this result with the symmetry relation [3, Eq. (47)] and the numerical results of Table 1 [3] S˜r,s (n, k) = Sr,s (n + 1, k + s).
(N)r+1 N r+1 (−1)r+1 s(r + 1, m)(−N)m . N
143
(25)
m=1
5. Discussion Example. 2 2 + 2 2 = s(3, 1) − s(3, 2)N + s(3, 3)N 2 . (26) a a With the multinomial formula [8] the abbreviations 2 B0 = s(3, 1) , B1 = −2s(3, 1)s(3, 2),
2 B2 = 2s(3, 1)s(3, 2) + s(3, 2) ,
B3 = −2s(3, 2)s(3, 3), 2 B4 = s(3, 3) ,
(27)
and with Eq. (17) follows: 2 + 2 2 = B0 a a + B4 S(4, 1) + B3 S(3, 1)
Kˆ =
+ B2 S(2, 1) + B1 S(1, 1) a + a + B4 S(4, 2) + B3 S(3, 2) 2 + B2 S(2, 2) a + a 2 3 + B4 S(4, 3) + B3 S(3, 3) a + a 3 + B4 S(4, 4)a + a 4 . 4
For the general operator product of boson operators k k k a + , a, Wˆ = a + 1 a l1 a + 2 a l2 · · · a + j a lj normal and antinormal ordered expressions are derived by introducing a generating function and by making use of the Hausdorff transformation formula. As the derivation is simple it may be useful for teaching applications. For completeness it was pointed out that commutation formulas which are given in the literature or can easily be derived by the generating function technique of the preceding paragraph can also be used for the ordering of the product. Finally products of the form
a+ ar r
m + s s l a a ···
were ordered in terms of Stirling numbers. It was shown that the generalized Stirling and anti-Stirling numbers introduced by Blasiak et al. [3] can be expressed by conventional Stirling numbers of the first and second kind. (28)
The Bj contain multinomial coefficients and Stirling numbers of the first kind. The result is 2 + 2 2 4 3 2 a a = a + a 4 + 12a + a 3 + 38a + a 2 + 32a + a + 4 4 3 = S˜2,2 (2, 4)a + a 4 + S˜2,2 (2, 3)a + a 3 + S˜2,2 (2, 2)a + a 2 + S˜2,2 (2, 1)a + a + S˜2,2 (2, 0) (29) 2
where the S˜r,s (n, k) 0 k ns are the generalized anti-Stirling numbers of the second kind [3].
References [1] J. Katriel, Lett. Nuovo Cimento 10 (1974) 565. [2] V.M. Michailov, J. Phys. A: Math. Gen. 18 (1985) 231. [3] P. Blasiak, K.A. Penson, A.I. Solomon, Phys. Lett. A 309 (2003) 198. [4] W. Witschel, J. Phys. A: Math. Gen. 8 (1975) 143. [5] J. Fröhlich, Commun. Math. Phys. 54 (1977) 135. [6] R.M. Wilcox, J. Math. Phys. 8 (1967) 962. [7] D. Shalitin, Y. Tikochinsky, J. Math. Phys. 20 (1979) 1676. [8] J. Spanier, K. Oldham, An Atlas of Functions, Springer, Berlin, 1987.
Ordering of boson operator functions by the Hausdorff similarity transform W. Witschel Department of Theoretical Chemistry, University of Ulm, D-89069 Ulm, Germany Received 7 September 2004; received in revised form 5 November 2004; accepted 8 November 2004 Available online 20 November 2004 Communicated by P.R. Holland
Abstract A generating function technique making use of the Hausdorff similarity transform is used to give arbitrary products of bosons in normal or antinormal ordered form. Operator commutator formulas and ordering in terms of Stirling numbers are also discussed. 2004 Published by Elsevier B.V.
1. Ordered operator products Katriel [1] showed that products of boson operators can be expressed in terms of Stirling numbers. Michailov [2] extended these calculations by defining generalized Stirling numbers. Recently Blasiak, Penson and Salomon [3] published an interesting article where they expressed ns + k k + r s n + n(r−s) = a Sr,s (n, k) a a a a k=s
(1) in normal ordered form. The Sr,s (n, k) are new generalized Stirling numbers of the second kind. They made E-mail address: [email protected] (W. Witschel). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.11.018
use of advanced combinatorics to exploit the properties of the Sr,s (n, k). The present Letter aims at a simple straightforward technique to give arbitrary products of boson operators in normal or antinormal ordered form where the operators a + are standing right (a) or left (n) of the a operators. Use is made of the Hausdorff similarity transform which was used previously on related problems [4]. Another technique is the application of commutation formulas for the normal ordering of boson operators [6]. By repeated application of these formulas one can also give arbitrary ordered products of boson operators. Finally a technique for products of the type {(a + )r a r }{(a + )s a s } · · · {(a + )k a k } will be used to show that for these operator products the generalized Stir-
W. Witschel / Physics Letters A 334 (2005) 140–143
ling numbers [3] can be expressed by standard Stirling numbers of the first and second kind.
ˆ (n) : ordered form G ˆ (n)
G
=a
+ k1
kj j −1 + k2 + a + α1 · · · a + αm
2. Ordered products of boson operators by the Hausdorff transform
141
× exp
j
m=1
αm a
(7)
m=1
It was shown previously that operator orderings can be performed by introducing generating functions and the Hausdorff transform [4]. The Hausdorff formula states: e
α Aˆ
ˆ Be
−α Aˆ
2 ˆ [A, ˆ B]] ˆ + ··· ˆ B] ˆ + α [A, = Bˆ + α[A, 2! ∞ αm ˆ m ˆ ˜ = (2) A , B = B(α) m! m=0
ˆ B] ˆ = Aˆ Bˆ − Bˆ Aˆ and where [A, m m−1 ˆ Aˆ Aˆ , Bˆ = A, , Bˆ is the repeated commutator. ˆ ˆ With eα A e−α A = Iˆ, where Iˆ is the identity operator Eq. (2) is generalized for ˆ ˆ ˆ ˆ −α Aˆ α Aˆ ˆ k−1 −α Aˆ e B e eα A Bˆ k e−α A = eα A Be k α Aˆ ˆ k−1 −α Aˆ ˜ ˜ = B(α)e B e = B(α)
(3)
and ˆ ˆ ˆ ˜ eα A eB e−α A = exp B(α) .
Wˆ = a
a a
a ···a
m=1
× a+ −
j
k2 αm
m=1
k · · · a + − αj j .
(8)
m=2
The noncommutative part is disentangled and one is left with simple tedious expansions, multiplications and comparison of powers of the ordering parameters. Application of symbolic math programs like M APLE or M ATHEMATICA is useful. 2 Example. Normal ordering of Wˆ = (a + a 2 )2 .
ˆ (n) = a + 2 eα1 a a + 2 eα2 a G 2 2 = a + a + + α1 e(α1 +α2 )a ; expansion and comparison leads to + 2 2 2 4 3 2 a a = a + a 4 + 4a + a 3 + 2a + a 2 .
(9)
(10)
(4)
The mathematical properties of the Hausdorff formula are discussed thoroughly by Fröhlich [5]. Boson operators with the commutator [a, ˆ aˆ + ] = 1 ˆ ˆ are a special case of [A, B] = c, where c is a real or complex number. In the following, use is made of the commutator property. Domain, range and convergence of the operator formulas are not discussed. To simplify the notation a and a + are written without the operator hatchet. Consider the boson operator product Wˆ : + k1 l1 + k2 l2
ˆ (a) : and the antinormal ordered form G
j
k1 j (a) + ˆ = exp G αm a a − αm
+ kj lj
a .
(5)
ˆ is introduced: Instead of Wˆ the generating function G ˆ = a + k1 eα1 a a + k2 eα2 a · · · a + kj eαj a G
(6)
where the αm are c-number ordering coefficients. With the repeated application of Eq. (3) one gets the normal
3. Repeated application of operator commutation formulas For completeness it should be mentioned that boson operator products can be ordered by repeated application of commutator formulas. Wilcox [6] derived by parameter differentiation for ˆ = cIˆ the commutation operators Pˆ and Qˆ with [Pˆ , Q] formula Pˆ m Qˆ n =
[[m,n]] j =0
m!n!cj Qˆ n−j Pˆ m−j j !(m − j )!(n − j )!
(11)
[[m, n]] = min (in the original article the upper limit of the sum is missing). Similar formulas are derived by the coherent state technique [7]. The generating function technique of the preceding text gives the commutation in a simple form which
142
W. Witschel / Physics Letters A 334 (2005) 140–143
may be interesting for teaching applications. e
αa +
a k = (a − α)k e
αa +
Example. (12)
by Eq. (8)
expansion leads to ∞ ∞ αm + m k αm m a a = (a − α)k a + . m! m!
m=0
Comparison of the coefficients of equal powers of α leads to the commutation. 4. Ordering of products {a + a r }m {a + a s }k ··· s
For arbitrary products the preceding techniques can be used. The special products r m + s s k a a ··· Kˆ = a + a r (14) can be treated in a different way making use of Stirling numbers. Furthermore it is shown that for these special cases the generalized Stirling numbers of the second kind [3] can be expressed by multinomial coefficients, conventional Stirling numbers of the first and of the second kind. r For brevity the calculations are limited to (a + a r )m . r r + Katriel [1] showed that (a a ) can be expressed by Stirling numbers of the first kind s(r, j ):
r m + r r m j = s(r, j )N a a (15) j =1
where N = a + a with the commutators [N, a] = −a, N, a + = a + .
Nk N
(n)
k (a)
=
k k (n) j a+a = S(k, j )a + a j ,
k (a) = a+a = (−1)k
k j =0
with A1 = (s(2, 1))2 , A2 = 2s(2, 1)s(2, 2) and A3 = (s(2, 2))2 . Ordering with Eq. (17) leads to + 2 2 2 a a = A3 S(4, 1) + A2 S(3, 1) + A1 S(2, 1) a + a + A3 S(4, 2) + A2 S(3, 2) 2 + A1 S(2, 2) a + a 2 3 + A3 S(4, 3) + A2 S(3, 3) a + a 3 + A3 S(4, 4)a + a 4 4
= a + a 4 + 4a + a 3 + 2a + a 2 . 4
(17)
3
2
(20)
This result agrees with Table 1 of Ref. [3]: + 2 2 2 2 = S2,2 (2, 1)a +a + S2,2 (2, 2)a + a 2 a a + S2,2 (2, 3)a + a 3 + S2,2 (2, 4)a + a 4 . (21) For example the generalized Stirling number in Eq. (21) 2 S2,2 (2, 2) = s(2, 2) S(4, 2) 3
4
+ 2s(2, 1)s(2, 2)S(3, 2) 2 + s(2, 1) S(2, 2)
(16)
To the r.h.s. of Eq. (15) the multinomial formula is applied and equal powers of N k are collected. They are rewritten in terms of Stirling numbers of the second kind S(k, j ) for normal form {N k }(n) and antinormal form {N k }(a) :
(19)
(13)
m=0
r
2 2 + 2 2 2 = s(2, 1)a + a + s(2, 2) a + a a a 2 3 4 = A1 a + a + A2 a + a + A3 a + a
(22)
is expressed by binomial coefficients, Stirling numbers r of the first and second kind. The ordering of (a r a + )m follows the same lines. It is: r−1 r a r a + = a r−1 a + a + 1 a + r−1 + a a+r . = a r−1 a + (23) Repeated application of the commutator leads to r a r a + = a +a + 1 a + a + 2 · · · a +a + r
j =1
j +j
(−1)j S(k + 1, j + 1)a a
. (18)
(a + a)r+1 (24) a +a with the Pochhammer symbol (X)n which can be ex=
W. Witschel / Physics Letters A 334 (2005) 140–143
pressed by Stirling numbers of the first kind [8] ar a+ = r
=
As an independent control we can check this result with the symmetry relation [3, Eq. (47)] and the numerical results of Table 1 [3] S˜r,s (n, k) = Sr,s (n + 1, k + s).
(N)r+1 N r+1 (−1)r+1 s(r + 1, m)(−N)m . N
143
(25)
m=1
5. Discussion Example. 2 2 + 2 2 = s(3, 1) − s(3, 2)N + s(3, 3)N 2 . (26) a a With the multinomial formula [8] the abbreviations 2 B0 = s(3, 1) , B1 = −2s(3, 1)s(3, 2),
2 B2 = 2s(3, 1)s(3, 2) + s(3, 2) ,
B3 = −2s(3, 2)s(3, 3), 2 B4 = s(3, 3) ,
(27)
and with Eq. (17) follows: 2 + 2 2 = B0 a a + B4 S(4, 1) + B3 S(3, 1)
Kˆ =
+ B2 S(2, 1) + B1 S(1, 1) a + a + B4 S(4, 2) + B3 S(3, 2) 2 + B2 S(2, 2) a + a 2 3 + B4 S(4, 3) + B3 S(3, 3) a + a 3 + B4 S(4, 4)a + a 4 . 4
For the general operator product of boson operators k k k a + , a, Wˆ = a + 1 a l1 a + 2 a l2 · · · a + j a lj normal and antinormal ordered expressions are derived by introducing a generating function and by making use of the Hausdorff transformation formula. As the derivation is simple it may be useful for teaching applications. For completeness it was pointed out that commutation formulas which are given in the literature or can easily be derived by the generating function technique of the preceding paragraph can also be used for the ordering of the product. Finally products of the form
a+ ar r
m + s s l a a ···
were ordered in terms of Stirling numbers. It was shown that the generalized Stirling and anti-Stirling numbers introduced by Blasiak et al. [3] can be expressed by conventional Stirling numbers of the first and second kind. (28)
The Bj contain multinomial coefficients and Stirling numbers of the first kind. The result is 2 + 2 2 4 3 2 a a = a + a 4 + 12a + a 3 + 38a + a 2 + 32a + a + 4 4 3 = S˜2,2 (2, 4)a + a 4 + S˜2,2 (2, 3)a + a 3 + S˜2,2 (2, 2)a + a 2 + S˜2,2 (2, 1)a + a + S˜2,2 (2, 0) (29) 2
where the S˜r,s (n, k) 0 k ns are the generalized anti-Stirling numbers of the second kind [3].
References [1] J. Katriel, Lett. Nuovo Cimento 10 (1974) 565. [2] V.M. Michailov, J. Phys. A: Math. Gen. 18 (1985) 231. [3] P. Blasiak, K.A. Penson, A.I. Solomon, Phys. Lett. A 309 (2003) 198. [4] W. Witschel, J. Phys. A: Math. Gen. 8 (1975) 143. [5] J. Fröhlich, Commun. Math. Phys. 54 (1977) 135. [6] R.M. Wilcox, J. Math. Phys. 8 (1967) 962. [7] D. Shalitin, Y. Tikochinsky, J. Math. Phys. 20 (1979) 1676. [8] J. Spanier, K. Oldham, An Atlas of Functions, Springer, Berlin, 1987.