Download Deriving new operator identities by alternately using normally

SCIENCE CHINA
Physics, Mechanics & Astronomy
. Research Paper .
September 2010 Vol. 53 No. 9: 1626–1630
doi: 10.1007/s11433-010-4071-5
Deriving new operator identities by alternately using normally,
antinormally, and Weyl ordered integration technique
FAN HongYi1, YUAN HongChun1 & JIANG NianQuan2*
1 Department
2 College
of Physics, Shanghai Jiao Tong University, Shanghai 200030, China;
of Physics and Electric Information, Wenzhou University, Wenzhou 325035, China
Received December 21, 2009; accepted April 1, 2010; published online July 21, 2010
Dirac’s ket-bra formalism is the language of quantum mechanics. We have reviewed how to apply Newton-Leibniz integration
rules to Dirac’s ket-bra projectors in previous work. In this work, by alternately using the technique of integration within normal,
antinormal, and Weyl ordering of operators we not only derive some new operator ordering identities, but also deduce some
new integration formulas regarding Laguerre and Hermite polynomials. This may open a new route of directly deriving some
complicated mathematical integration formulas by virtue of the quantum mechanical operator ordering technique, without really
performing the integrations in the ordinary way.
operator ordering identities, Laguerre and Hermite polynomials, the IWOP technique
PACS: 03.65.-w, 02.30.Cj
1 Introduction
In the foreword of the book The Principles of Quantum Mechanics, Dirac wrote: “ The symbolic method, · · · enables one
to express the physical law in a neat and concise way, and
will probably be increasingly used in the future as it becomes
better understood and its own special mathematics gets developed [1].” Following his expectation, the technique of integration within an ordered product (IWOP) of operators was
invented which can directly apply the Newton-Leibniz integration rule to ket-bra projective operators [2,3]. The essence
of the IWOP technique is to convert these ket-bra operators
into certain ordered products (normal ordering, antinormal
ordering, and Weyl ordering) of bosonic creation and annihilation operators such that they can be treated as ordinary
parameters while performing the integrations, but operators’
essence will not be lost in this approach [4]. For example,
using the IWOP technique [5,6] we have converted the completeness relation of the coordinate eigenvector |x into a pure
Gaussian integration within the normal ordering : : ,
∞
∞
2
dx
dx |x x| =
(1)
√ : e−(x−X) : = 1,
π
−∞
−∞
*Corresponding author (email: [email protected])
c Science China Press and Springer-Verlag Berlin Heidelberg 2010
√
†
where
X = (a + a )/ 2 is the coordinate operator with
a, a† = 1. It turns out that performing the ket-bra integra ∞ x tion −∞ dx
μ μ x| leads to the single-mode squeezing operator. Recently, Li’s group has constructed some useful entangled state representations by virtue of the IWOP technique
[7,8].
In this work we shall demonstrate how to derive some new
operator ordering identities and new integration formulas regarding Laguerre polynomial Ln(α) (x) and Hermite polynomial Hn (x) by alternately using the technique of integration
within normal, antinormal, and Weyl ordering of operators.
This may open a new route of directly deriving some complicated mathematical integration formulas by virtue of the
quantum mechanical operator ordering technique, even without really performing the integrations in the ordinary way.
2 Antinormally ordered expansion of the operator X n and H n (X)
To begin with, using the Baker-Hausdorff formula
1
1
eA+B = eA eB e− 2 [A,B] = eB eA e− 2 [B,A]
phys.scichina.com
(2)
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FAN HongYi, et al. Sci China Phys Mech Astron September (2010) Vol. 53 No. 9
and the generating function of Hn (x)
=
∞ n
t
2
Hn (x) = e2tx−t ,
n!
n=0
where
Hn (x) =
[n/2]
l=0
(3)
(−1)l n!
(2x)n−2l ,
l!(n − 2l)!
(4)
.
1 2.
√λ a+a†
eλX = e 2 ( ) = ..eλX− 4 λ .. =
n
n=0
λ
2
n!
.
..
.Hn (X) ..,
(5)
..
where .. .. stands for the antinormal ordering. Comparing eq.
(5) with the same power of λ in the expansion of eλX =
∞ n
λ
n
n! X , we have the neat formula
n=0
.
.
X n = 2−n ..Hn (X) ..,
(6)
which is just the antinormal ordering expansion of X n . Taking
n = 2 for example,
.
. 1. 1 ..
.H2 (X) .. = .. 4X 2 − 2 ..
4
4
1 2
=
a + 2aa† + a†2 − 1 = X 2 ,
2
(8)
(9)
From the relation
d
Hn (x) = 2nHn−1 (x) ,
dx
(10)
and eq. (6) we have
.
.
. . d
.
d ..
.Hn (X) .. = 2n nX n−1 = 2n..Hn−1 (X) .. = .. Hn (X) ..,
dX
dX
(11)
namely,
. . d
.
d ..
(12)
.Hn (X) .. = .. Hn (X) ...
dX
dX
.
.
This is another property of ..Hn (X) ...
Moreover, from eqs. (3) and (2) we see
∞ n
.
t
2
2
2.
Hn (λX) = e2tλX−t = ..e2tλX−(λ +1)t ..
n!
n=0
which is different from eq. (6). For n = 2 case,
2
.
.
X . . 2..H2 √ .. = ..2 a + a† − 4..
2
= 2 a2 + 2aa† + a†2 − 4
2
a + a†
=4 √
−2
2
= H2 (X) .
(15)
(16)
By analogy to the above derivation we can deduce the normal
ordering expansion of X n
X n = (2i)−n : Hn (iX) : ,
Hn (X) = 2n : X n : .
(7)
which implies the following new relation
. .
..
.
..
.Hn+m (X) .. = ..Hn (X) ....Hm (X) ...
Comparing the same power of t on the two sides we obtain a
new operator identity
⎞
⎛
n
⎟⎟⎟ ..
.. ⎜⎜⎜⎜
2
λX
⎟⎟⎠ ..
λ + 1 .Hn ⎜⎝ Hn (λX) =
(14)
λ2 + 1
(17)
and the normally ordered form of Hn (X)
as expected. It then follows from eq. (6) that
.
.
X n+m = 2−n−m ..Hn+m (X) .. = X n X m ,
n=0
n
⎞
⎛
λ2 + 1 t . ⎜⎜⎜
⎟⎟⎟ ..
..H ⎜⎜ λX
⎟⎟⎠ .. (13)
n⎝ 2
n!
λ +1
Especially, when λ = 1, eq. (14) reduces to
√ n . X .
2 ..Hn √ ..,
Hn (X) =
2
we easily obtain
∞
∞
1627
(18)
3 New integration formulas about Hermite
polynomials
Using the above results we can derive new integration formulas regarding Hermite polynomials Hn (x). Recalling the
P-representation in the coherent state |β basis [9]
2
d β
ρ=
P (β) |β β| , β ≡ β1 + iβ2 ,
(19)
π
where
|β|2
|β = exp −
+ βa† |0
2
is the coherent state satisfying
a |β = β |β .
(20)
(21)
Utilizing eqs. (6) and
(19) as well as the vacuum projector
|0 0| = : exp −a† a : and considering (17), we have
2
β + β∗
d β
Hn √
X n = 2−n
|β β|
π
2
2
d β √ Hn 2β1
= 2−n
π
: exp − |β|2 + βa† + β∗ a − a† a :
= (2i)−n : Hn (iX) : ,
(22)
1628
FAN HongYi, et al. Sci China Phys Mech Astron September (2010) Vol. 53 No. 9
from this identity, we can immediately know the n-order moment of the quadrature X in the coherent state
√ β| X n |β = (2i)−n Hn i 2β1 .
(23)
Since within the normal ordering symbol a and a† commute,
they can be treated as c-numbers. So we can set a → x and
a† → y. In this way we obtain
d2 β √ Hn 2β1 exp − |β|2 + yβ + xβ∗
π
x + y xy
−n
= i Hn i √ e ,
2
ρ=
.
d β ..
. −β| ρ |β exp[|β|2 + β∗ a − βa† + a† a]..,
π
2
(25)
provided that the integral is convergent, which shows that
when ρ is in normal ordering, its coherent state matrix element −β| ρ |β can be immediately obtained. So we can get
ρ’s antinormally ordered expansion by just performing the
integral of eq. (25). According to eqs. (17) and (25) and
considering eq. (6) again, we obtain the following equation
−n
d2 β √ Hn − 2β2 exp[− |β|2 + xβ∗ − βy]
π
x + y −xy
n
= i Hn √
e ,
2
n+1
(30)
Due to
:
:
Hn (X) = : Hn (X) : ,
we have a new integration formula
2
d β n
β2 exp 2 −|β|2 + xβ∗ − yβ
π
x + y −2xy
−n − 3n+2
= i 2 2 Hn √ e
,
2
without really performing this integration.
Finally, in the similar way for the momentum P =
(26)
(27)
which is different from eq. (24).
On the other hand, in ref. [11], by considering the Prepresentation of an operator ρ and using the Weyl ordering
of the coherent state projector
:
:
|β β| = 2 : exp −2 β∗ − a† (β − a) : ,
d2 β :
−β| : X n : |β
π :
:
× exp 2 β∗ a − a† β + a† a :
2
3n+2
d β : n
= in 2 2
β
π : 2
:
× exp 2 −|β|2 + β∗ a − a† β + a† a : .
Hn (X) = 2
Thus, one immediately gets another new integration formula
:
†
: stands for the Weyl ordering, a and a commute
:
: . By using eqs. (18) and (29), we obtain
(31)
(32)
1
√
(a −
i 2
a† ), we easily obtain some integration formulas as follows
2
d β √ Hn 2β2 exp − |β|2 + yβ + xβ∗
π
x − y xy
−n
= i Hn √ e ,
(33)
2
d2 β ..
X = (2i)
. −β| : Hn (iX) : |β
π
.
× exp[|β|2 + β∗ a − βa† + a† a]..
2
d β .. √ −n
= (2i)
.Hn − 2β2
π
.
exp[− |β|2 + β∗ a − βa† + a† a]..
.
.
= 2−n ..Hn (X) ...
n
:
where :
:
within :
(24)
which is a new integration formula which is not easily obtained just by using ordinary mathematical integration. Here
we derive it even without really performing the integration in
eq. (24). This is an obvious advantage of the IWOP technique.
Next, the antinormally ordered expansion of an arbitrary
bosonic operator in the coherent state |β basis is [10]
we have derived the useful formula which can convert ρ into
its Weyl ordered form
2
d β :
−β| ρ |β
ρ=2
π :
:
(29)
× exp 2 β∗ a − a† β + a† a : ,
(28)
and
d2 β √ Hn 2β1 exp − |β|2 + xβ∗ − yβ
π
x − y −xy
n
= i Hn √ e ,
i 2
d2 β n
β1 exp 2 −|β|2 + β∗ a − a† β
π
3n+2
x−y
= (−i)−n 2− 2 Hn √ e−2xy .
i 2
(34)
(35)
4 Operator identities about Laguerre polynomials
Using eq. (1) and the IWOP technique, we have the normally ordered expansion formula
FAN HongYi, et al. Sci China Phys Mech Astron September (2010) Vol. 53 No. 9
e
λX 2
=
∞
−∞
dxe
λx2
= (1 − λ)−1/2
∞
=2
2
dx
2
√ eλx e−(x−X) :
π
|x x| = :
−∞
−λX 2
:.
: exp
λ−1
(36)
Comparing eq. (36) with the generating function of the Laguerre polynomials
(1 − t)−α−1 exp
∞
xt Lαn (x) tn ,
=
t−1
n=0
(37)
we see when α = −1/2,
:
∞
2
L−1/2
−X 2 λn : = eλX ,
n
(38)
n=0
which is a new operator identity. Then comparing eq. (38)
λn 2n
2
with eλX = ∞
n=0 n! X , we see
X 2n = n! : L−1/2
−X 2 : .
(39)
n
Due to X 2n = (2i)−2n : H2n (iX) : in eq. (17), we obtain the relation connecting the Hermite polynomial and Laguerre polynomial
H2n (iX) = (−1)n 22n n!L−1/2
−X 2 ,
(40)
n
this relation still holds when iX → x, i.e.,
x2 ,
H2n (x) = (−1)n 22n n!L−1/2
n
(41)
which coincides with ref. [12]. Since H2n (X) = 22n : X 2n: : in
eq. (18), we also have
: X 2n : = (−1)n n!L−1/2
X2 .
(42)
n
Considering eqs. (25) and (39), we derive
2
d β ..
−X 2 : |β
. −β| : L−1/2
X 2n = n!
n
π
.
× exp[|β|2 + β∗ a − βa† + a† a]..
2
d β .. −1/2 2 .L
2β2
= n!
π n
.
× exp[− |β|2 + β∗ a − βa† + a† a]...
−2n
H2n
x + y −xy
√ e .
2
1629
(45)
In a similar manner, by considering eqs. (29) and (42), we
can obtain
2
d β :
2
(−1)n n!L−1/2
−β| : X 2n : |β
X
=
2
n
π :
:
× exp 2 β∗ a − a† β + a† a :
2
d β : 2n
= (−1)n 2n+1
β
π : 2
:
× exp 2 − |β|2 + β∗ a − a† β + a† a : .
(46)
From the above equation, due to L−1/2
X2
=
n
: −1/2 2 :
X
: , we see that
: Ln
2
:
d β 2n :
n+1
2
β2 : exp 2 − |β|2 + β∗ a − a† β + a† a :
π
:
:
= n! : L−1/2
(47)
X2 : .
n
It then follows that the new integration formula is
2
d β 2n
n+1
β2 exp 2 − |β|2 + xβ∗ − yβ
2
π
−1/2 1
2 −2xy
(x + y) e
= n!Ln
.
(48)
2
In summary, we have introduced an effective approach ot
deriving operator identities and new integration formulas by
alternately using normally, antinormally, and Weyl ordered
integration techniques. This opens a new route of deriving
mathematical integration formulas by virtue of the quantum
mechanical operator ordering technique, even without really
performing the integrations in the ordinary way.
This work was supported by the National Natural Science Foundation of
China (Grant Nos. 10775097 and 10874174) and the Special Funds of the
National Natural Science Foundation of China (Grant No. 10947017/A05).
(43)
It then follows from eq. (6) that
2
.
d β −1/2 2 ..
L
n!
2β2 . exp[− |β|2 + β∗ a − βa† + a† a]..
π n
.
.
= 2−2n ..H2n (X) ..,
(44)
which is a new identity. Further, when a → x and a† → y,
we deduce a new integration formula as follows
2
d β −1/2 2 L
2β2 exp[− |β|2 + xβ∗ − yβ]
n!
π n
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