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Physics, Mechanics & Astronomy
. Article .
November 2013 Vol. 56 No. 11: 2042–2046
doi: 10.1007/s11433-013-5311-2
Dirac multimode ket-bra operators’ Q-ordered and P-ordered
integration theory and general squeezing operator
FAN HongYi & LOU SenYue*
Department of Physics, Ningbo University, Ningbo 315211, China
Received March 13, 2013; accepted May 23, 2013; published online October 8, 2013
We develop quantum mechanical Dirac ket-bra operator’s integration theory in Q-ordering or P-ordering to multimode case, where
Q-ordering means all Qs are to the left of all Ps and P-ordering means all Ps are to the left of all Qs. As their applications,
we derive Q-ordered and P-ordered expansion formulas of multimode exponential operator e−iPl Λlk Qk . Application of the new
formula in finding
new general
squeezing operators is demonstrated. The general exponential operator for coordinate representation
⎞ ⎛ ⎞
⎛⎜
⎛⎜ ⎞⎟
⎜⎜ A B ⎟⎟⎟⎟ ⎜⎜⎜⎜ q1 ⎟⎟⎟⎟
⎜⎜ q1 ⎟⎟⎟
⎜
⎜
transformation ⎜⎝ ⎟⎠ → ⎜⎝
⎟⎠ ⎜⎝ ⎟⎠ is also derived. In this way, much more correpondence relations between classical
C D q2
q2
coordinate transformations and their quantum mechanical images can be revealed.
integration theory in Q-ordering or P-ordering, Q-ordered and P-ordered expansion formulas, multimode exponential
operator
PACS number(s): 03.65 Ca, 42.50 Dv
Citation:
Fan H Y, Lou S Y. Dirac multimode ket-bra operators’ Q-ordered and P-ordered integration theory and general squeezing operator. Sci China-Phys
Mech Astron, 2013, 56: 2042–2046, doi: 10.1007/s11433-013-5311-2
1 Introduction
The physical essence of quantum mechanics is characterized of [Q, P] = i, (we set = 1 for convenience), where
Q, P are the coordinate and momentum operator respectively
[1]. This fundamental commutative relation is the basis of
other complicated operators’ ordering. Often encountered
operator ordering problems arise in dealing with quantization and in calculating various matrix elements of operators.
In the preceding paper [2] we have respectively introduced
the theory of integration within Q-ordering (all Qs are on
the left of all Ps) and P-ordering (all Ps are on the left of
all Qs) of operators, because the order of operators Q and
P within the symbol Q(P) can be permuted, for instance,
Q(QP) = Q(PQ) =QP, Q(P)-ordered products can be integrated or differentiated with respect to c-number provided the
*Corresponding author (email: [email protected])
c Science China Press and Springer-Verlag Berlin Heidelberg 2013
integration is convergent. We can derive some new fundamental operator identities about their mutual re-ordering, for
example, for
1
δ (q − Q) δ (p − P) = |q q| p p| = √ |q p| eipq , (1)
2π
where |q , |p are the coordinate and momentum eigenvector,
respectively, we have
∞
dudv i(q−Q)u i(p−P)v
δ (q − Q) δ (p − P) =
e
e
4π2
−∞
∞
dudv i(p−P)v−iuv iu(q−Q)
=
e
e
4π2
∞−∞
du
δ (p − P − u) eiu(q−Q)
=
2π
−∞
∞
du
δ (p − P − u) eiu(q−Q)
=P
−∞ 2π
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Fan H Y, et al.
Sci China-Phys Mech Astron
November (2013)
Vol. 56
2043
No. 11
1 i(p−P)(q−Q) ,
(2)
P e
2π
this is the P-ordered expansion of δ (q − Q) δ (p − P) . Similarly, the Q-ordered expansion of δ (p − P) δ (q − Q) is
1
1 −i(q−Q)(p−P) δ (p − P) δ (q − Q) = √ |p q| e−ipq =
Q e
.
2π
2π
(3)
We have also obtained the Q-ordered expansin formula of the
Wigner operator in ref. [2]
1 Δ (q, p) = Q e−2i(q−Q)(p−P) ,
(4)
π
where the original definition of the Wigner operator is [3]
∞
dudv i(q−Q)u+i(p−P)v
Δ (q, p) =
e
.
(5)
4π2
2 New Q-ordered and P-ordered expansion
formulas of the multimode exponential operator
e−iPl Λl k Q k
Similarly, we have deduced the P-ordered form of Wigner
operator
1 Δ (q, p) = P e2i(q−Q)(p−P) .
(6)
π
Then using the Weyl correspondence rule of any operator
H (Q, P) [4]
∞
(7)
H (Q, P) =
dqdph (q, p) Δ (q, p) ,
which is the Q-ordered expansion.
Proof of eq. (10): Using the Baker-Hausdorff formula we
have
=
=
In this section we shall prove that for n-mode P
(P1 , P2 , . . . , Pn ) and Q = (Q1 , Q2 , . . . , Qn ) and a real n × n
real matrix Λ the following operator identity holds
e−iPΛQ ≡ e(−iP)l Λlk Qk = P exp{(−iP)l eΛ − 1 Qk
lk
= Pe
(eΛ −1)Q
−iP
,
(10)
which is the P-ordered expansion; and
e−iPΛQ ≡ e(−iP)l Λlk Qk = QeiP(e
−Λ
−1)Q
,
(11)
−∞
e−iPi Λi j Q j Qk eiPi Λi j Q j = Qk + −iPi Λi j Q j , Qk
1
+ −iPi Λi j Q j , −Λkl Ql + · · ·
2
1
= Qk − Λk j Q j + Λki Λi j Q j + · · ·
2
−Λ
= e
Q j.
(12)
−∞
where h (q, p) is the classical correspondence function of
H (Q, P) , we have
∞
1
H (Q, P) =
dqdph (q, p) Q e−2i(q−Q)(p−P) , (8)
π
kj
Then under the transformation
of e−iPi Λi j Q j the n-mode coor dinate eigenvector q should become
−∞
or
e−iPi Λi j Q j q = eΛ q ,
1
dqdph (q, p) P e2i(q−Q)(p−P) . (9)
H (Q, P) =
π
Once the classical correspondence h (q, p) of H (Q, P) is
known, the Q-ordered or P-ordered form of H (Q, P) can be
obtained by doing the integration in eq. (8) or eq. (9) respectively, which brings much convenience for arranging operators as either in Q-ordering or in P-ordering. In this work
we shall develop Dirac ket-bra operators’ Q-ordered and Pordered integration theory to multimode case, which can
reveal much more correpondence relations between classical coordinate transformations and their quantum mechanical
images (operators). In sect. 2 we derive two new Q-ordered
and P-ordered expansion formulas for multimode exponential operator e(−iP)l Λlk Qk . They are very useful in obtaining
general squeezing operators, say, for example, in sect. 3 the
one- and two- mode combinatorial squeezing operator in exponential form is derived to corresponds to the transformation
(q1 , q2 ) → (q1 cosh λ + q2 e−r sinh λ, q2 cosh λ + q1 er sinh λ).
The explicit exponential
operator
representa for coordinate
q
A
B
q
1
1
tion transformation q
→ C D q
is also derived
2
2
in sect. 4. In sect. 5 we make some remarks on Q- and Pordering of operators.
qT = (q1 , q2 , . . . , qn ) ,
(13)
here the superscript T means transpose operation, this can be
checked by using eq. (12)
Qk q = Qk eiPi Λi j Q j eΛ q = eiPi Λi j Q j e−iPi Λi j Q j Qk eiPi Λi j Q j eΛ q
= eiPi Λi j Q j e−Λ Q j eΛq
kj
= eiPi Λi j Q j e−Λ
(14)
eΛ q eΛ q = qk q .
kj
j
Further, using the completeness relation
co ∞ of both
ordinate and momentum eigenvectors, −∞ dn q q q =
∞
1, −∞ dn p p p = 1, and the overlap
1
eipq , pq ≡ p · q,
q p = n
(2π)
as well as eqs. (13) and (3), we obtain
(−iP)l Λlk Qk
e
dn qe(−iP)l Λlk Qk q q
−∞
∞
∞
n
n Λ
=
d q
d p p p e q q
=
∞
−∞
−∞
(15)
2044
Fan H Y, et al.
Sci China-Phys Mech Astron
∞
∞
1
Λ = dn q
dn pe−ipe q p q
n
−∞
(2π) −∞
∞
∞
1
Λ
dn q
dn peip(1−e )q p q e−ipq
= −∞
(2π)n −∞
∞
∞
Λ
δ q − Q
=
dn p
dn qeip(1−e )q δ p − P
−∞
−∞
(1−eΛ )Q
iP
=P e
,
(16)
thus eq. (10) is proved.
Proof of eq. (11):
Similarly, using
e−iPi Λi j Q j Pk eiPi Λi j Q j = Pk + −iPi Λi j Q j , Pk
1
+ −iPi Λi j Q j , Pl Λlk + · · ·
2
1
= Pk + Pi Λik + Pi Λi j Λ jk + · · ·
2
Λ
= eΛ̃ Pi ,
(17)
= Pi e
ik
we have
e
(−iP)l Λlk Qk
ki
−Λ̃ p = e p ,
(18)
then
e(−iP)l Λlk Qk =
=
=
=
dn pe(−iP)l Λlk Qk p p
−∞
∞
∞
n
d p
dnq q q e−Λ̃ p p
−∞
−∞
∞
∞
−Λ̃
1
n
d q
dn peiq e −1 p q p eipq
−∞
(2π)n −∞
∞
∞
−Λ̃
δ p − P
dn q
dn peiq e −1 p δ q − Q
∞
−∞
= Qe
−∞
iQ e−Λ̃ −1 P
= QeiP(e
−Λ
−1)Q
,
(19)
so eq. (11) is proved. Note that within Q-ordering or P−
ordering the momentum operator P is permutable with the
position oprator Q, this is in similar to the case that bosonic
creation operators are permutable with annihilation operators
within the normal ordering symbol.
3 One- and two- mode combinatorial squeezing operator
−∞
Vol. 56
(20)
which is a one- and two- mode combinatorial squeezing operator. The correponding squeezed state can exhibit stronger
No. 11
squeezing when tanh λ < 1/(1 + cosh r) [5]. We now perform
this ket-bra integration within P-ordering
dp1 dp2 |p1 , p2 p1 , p2 | |q1 cosh λ
+ q2 e−r sinh λ, q2 cosh λ + q1 er sinh λ q1 , q2 |
1
=
dp1 dp2 exp −ip1 (q1 cosh λ
dq1 dq2
2π
!
+ q2 e−r sinh λ − ip2 (q2 cosh λ + q1 er sinh λ)
S =
dq1 dq2
× |p1 , p2 q1 , q2 |
1
=
dp1 dp2 exp −i (p1 q1 + p2 q2 )
dq1 dq2
2π
"
!
× (cosh λ − 1) − i p2 q1 er + p1 q2 e−r sinh λ
× |p1 , p2 q1 , q2 | e−ip1 q1 −ip2 q2
=
dq1 dq2
dp1 dp2 exp −i (p1 q1 + p2 q2 )
"
!
× (cosh λ − 1) − i p2 q1 er + p1 q2 e−r sinh λ
×δ (p − P1 ) δ (p2 − P2 ) δ (q1 − Q1 ) δ (q2 − Q2 )
r
−r
= P e−i(P1 Q1 +P2 Q2 )(cosh λ−1)−i(P2 Q1 e +P1 Q2 e ) sinh λ
= Pe
⎞⎛
⎛
⎜⎜ cosh λ
−i(P1 P2 )⎜⎜⎜⎝ r
⎞
− 1 e−r sinh λ ⎟⎟⎟⎟⎟⎜⎜⎜⎜ Q1 ⎟⎟⎟⎟
e sinh λ cosh λ − 1 ⎠⎝ Q2 ⎠ ,
(21)
then using
cosh λ e−r sinh λ
0 λe−r
ln er sinh λ cosh λ = er λ 0 ,
(22)
and eq. (10) we have
⎛
⎜⎜ 0
−i(P1 P2 )⎜⎜⎜⎝ r
⎞⎛
⎞
λe−r ⎟⎟⎟⎟⎟⎜⎜⎜⎜ Q1 ⎟⎟⎟⎟
e λ 0 ⎠ ⎝ Q2 ⎠
S =e
"
!
= exp −iλ e−r P1 Q2 + er P2 Q1 ,
(23)
or the integration result
∞
−∞
dq1 dq2 q1 cosh λ + q2 e−r sinh λ,
q2 cosh λ + q1 er sinh λ q1 , q2 |
"
!
= exp −iλ e−r P1 Q2 + er P2 Q1 ,
(24)
this is the merit of introducing Q-ordered or P-ordered operator integration theory for Dirac Ket-bra operators.
4
General
A B q1
C D q2
As an application of sect. 2 we consider
∞
dq1 dq2 q1 cosh λ + q2 e−r sinh λ, q2 cosh λ
S =
+ q1 er sinh λ q1 , q2 | ,
November (2013)
squeezing
for
q1
q2
→
As a more complicated example, we consider what is the general squeezing operator for
q1 → A B q1 ,
(25)
C D q
q
2
2
Fan H Y, et al.
Sci China-Phys Mech Astron
where A, D, B, C are all real numbers. We construct the corresponding integration over the ket-bra and employ the Pordering integration technique to derive
∞
A B q
W ≡
dq1 dq2 C D q1 q1 , q2 |
2
−∞
=
dq1 dq2
dp1 dp2 |p1 , p2 p1 , p2 |
× |Aq1 + Bq2 , Dq2 + Cq1 q1 , q2 |
1
dq1 dq2
dp1 dp2 e−ip1 (Aq1 +Bq2 )−ip2 (Dq2 +Cq1 )
=
2π
× |p1 , p2 q1 , q2 |
1
=
dp1 dp2 eip1 q1 +ip2 q2
dq1 dq2
2π
× e−i(Ap1 q1 +Dp2 q2 +Cp2 q1 +Bp1 q2 ) |p1 , p2 q1 , q2 | e−ip1 q1 −ip2 q2
=
dq1 dq2
dp1 dp2 eip1 q1 +ip2 q2
× e−i(Ap1 q1 +Dp2 q2 )−i(Cp2 q1 +Bp1 q2 )
×δ (p1 − P1 ) δ (p2 − P2 ) δ (q1 − Q1 ) δ (q2 − Q2 )
= P e−i[(A−1)P1 Q1 +(D−1)P2 Q2 ]−i(P2 Q1 C+P1 Q2 B)
= Pe
⎛
⎜⎜
−i(P1 P2 )⎜⎜⎝
⎞⎛
⎞
A − 1 B ⎟⎟⎟⎟⎜⎜⎜⎜ Q1 ⎟⎟⎟⎟
C D − 1 ⎠⎝ Q2 ⎠ .
(26)
Using (10) we see
∞
A B q
dq1 dq2 C D q1 q1 , q2 |
2
−∞
=e
⎛
⎜⎜
−i(P1 P2 ) ln⎜⎜⎝
⎞⎛
⎞
A B ⎟⎟⎟⎟⎜⎜⎜⎜ Q1 ⎟⎟⎟⎟
C D ⎠⎝ Q2 ⎠ ,
(27)
where
⎛
⎜⎜⎜
A B
⎜
ln C D = ⎜⎜⎜⎜
⎝
(Δ−t) ln
s−Δ
s+Δ
2 +(Δ+t) ln 2
b
Δ
2Δ
c
Δ
ln
s+Δ
s−Δ
(Δ+t) ln
ln
s+Δ
s−Δ
s−Δ
s+Δ
2 +(Δ−t) ln 2
2Δ
⎞
⎟⎟⎟
⎟⎟⎟
⎟⎟⎠
(28)
and
Δ=
#
4BC + (A − D)2 , s = A + D, t = A − D.
(29)
Eqs. (27) and (28) are new result about ket-bra integration,
which will have more applications in future.
Finally we point out that since Λ is a real matrix
†
e(−iP)l Λlk Qk = eiQk Λlk Pl = ei(Pl Qk +iδlk )Λlk = eiPl Λlk Qk −TrΛ , (30)
we can unitarilize e(−iP)l Λlk Qk as
e(−iP)l Λlk Qk +1/2TrΛ ≡ U,
†
−1
(31)
such that U = U . According to eq. (13) and the matrix
formula det Λ = exp (Tr ln Λ) , we have the unitary transformation
√
e(−iP)l Λlk Qk +1/2TrΛ q = det eΛ eΛq ,
(32)
November (2013)
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so
e
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No. 11
(−iP)l Λlk Qk +1/2TrΛ
√
Λ
= det e
∞
−∞
dnq eΛ q q .
(33)
When Λ is a symmetric real matrix, with the use of the integration method within the normally ordering [6–9] we have
√
e(−iP)l Λlk Qk +1/2TrΛ = det sechΛ exp a† tanh Λa†T /2
× exp a† ln sechΛaT
× exp −a tanh ΛaT /2 ,
(34)
√
+ iP
/ 2.
where a = Q
5 Some remarks
While we are recalling the theory of integration within normally ordered-, antinormally ordered- and Weyl orderedproduct of operators which respectively lead to normal ordering, antinormal ordering and Weyl ordering [10] of operators
after performing the corresponding integration over ket-bras,
this work is mainly concentrated on developing Dirac ketbra operator’s integration theory in Q-ordering or P-ordering
to multimode case and demenstrating how classical transformations can be mapped onto the ordered exponential operators through the integration, and the integration directly lead
to operators’ Q-ordering or P-ordering form. It is naturally
to mention an important paper by Wang et al. [11], these
authors also introduced some non-hermitian Hamiltonian for
constructing some kind of generalized squeezed state [12]. In
ref. [11] they provided an approach to functional forms for
the exponential quadratic operators in coordinate-momentum
space, their result (eq. (21) in ref. [11]) is elegant and useful.
Thus it will be benificial to readers to compare their approach
to putting operators in either Q-ordering or P-ordering with
ours. In fact, while WANG et al.’s approach is based on the
differential behaviour of the momentum operator in coordinate space and matrix manipulations, ours has the following
merits.
1. The method in our work is actually some kind of integration transformation with the pre-ordered (Q-ordered or
P-ordered) integration kernel, using the method of integration over ket-bra projecting operators of continuum variables.
2. Our method is making full use of the completeness of coordinate-momentum representations, Dirac’s Deltafunction and their mutual transformation.
3. Our method brings convenience for converting other
kinds of ordered operators into Q-ordering or P-ordering.
4. Our method can reveal much more correpondence relations between classical coordinate transformations and their
quantum mechanical images (operators).
In summary, except for the integration theory within normally ordered, or Weyl ordered, of operators [7–9] we can
develop Dirac ket-bra operator Q-ordered or P-ordered integration theory to multimode case, which can help us to find
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Fan H Y, et al.
Sci China-Phys Mech Astron
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much more correpondence relations between classical coordinate transformations and their quantum mechanical images,
in this way Dirac’s symbolic method is got enriched.
6 Fan H Y, Lu H L, Fan Y. Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations. Ann Phys, 2006, 321: 480–494
This work was supported by the National Natural Science Foundation of
7 Fan H Y. Operator ordering in quantum optics theory and the development of Dirac’s symbolic method. J Opt B-Quantum Semicalss Opt,
2003, 5: R147–R163
China (Grant No. 11175113). We thank the referee for bringing refs. [11,12]
to our attention.
8 Fan H Y, Yuan H C, Jiang N Q. Deriving new operator identities by
alternately using normally, antinormally, and Weyl ordered integration
technique. Sci China-Phys Mech Astron, 2010, 53: 1626–1630
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