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Science in China. Series G, Physics, mechanics & astronomy/Science in China. Series G, Physics, Mechanics & Astronomy

Dirac multimode ket-bra operators' Q-ordered and P-ordered integration theory and general squeezing operator

Fan Hong

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

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 iPlΛlkQk. Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation q1q2 → A B C D q1q2 is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.

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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 iPlΛlkQk. Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation q1q2 → A B C D q1q2 is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.
Studying bi-partite entangled state representations via the integration over ket-bra operators in Q\mathfrak{Q}Q -ordering or P\mathfrak{P}P -ordering
For two particles’ relative position and total momentum we have introduced the entangled state representation |η〉, and its conjugate state |ξ〉. In this work, for the first time, we study them via the integration over ket-bra operators in Q\mathfrak{Q}Q -ordering or P\mathfrak{P}P -ordering, where Q\mathfrak{Q}Q -ordering means all Qs are to the left of all Ps and P\mathfrak{P}P -ordering means all Ps are to the left of all Qs. In this way we newly derive P\mathfrak{P}P -ordered (or Q\mathfrak{Q}Q -ordered) expansion formulas of the two-mode squeezing operator which can show the squeezing effect on both the two-mode coordinate and momentum eigenstates. This tells that not only the integration over ket-bra operators within normally ordered, but also within P\mathfrak{P}P -ordered (or Q\mathfrak{Q}Q -ordered) are feasible and useful in developing quantum mechanical representation and transformation theory.
It is known that exp [iλ ( Q 1 P 1 – i/2)] is a unitary single-mode squeezing operator, where Q 1 , P 1 are the coordinate and momentum operators, respectively. In this paper we employ Dirac's coordinate representation to prove that the exponential operator , is an n -mode squeezing operator which enhances the standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive S n 's normally ordered expansion and obtain new n -mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.
In this paper, we introduce a new way to obtain the Q — P (P — Q) ordering of quantum mechanical operators, i.e., from the classical correspondence of Q — P (P — Q) ordered operators by replacing q and p with coordinate and momentum operators, respectively. Some operator identities are derived concisely. As for its applications, the single (two-) mode squeezed operators and Fresnel operator are examined. It is shown that the classical correspondence of Fresnel operator's Q — P (P — Q) ordering is just the integration kernel of Fresnel transformation. In addition, a new photo-counting formula is constructed by the Q — P ordering of operators.
It is known that exp[i\lamda(Q_1P_1-i/2)] is a unitary single-mode squeezing operator, where Q_1,P_1 are the coordinate and momentum operators, respectively. In this paper we employ Dirac's coordinate representation to prove that the exponential operator S_{n}=Exp[i\lamda sum_{i=1}^{n}](Q_{i}P_{i+1}+Q_{i+1}P_{i}))], (Q_{n+1}=Q_1P_{n+1}=P_1), is a n-mode squeezing operator which enhances the standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive S_{n}'s normally ordered expansion and obtain new n-mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.
Dirac's ket-bra formalism is the language of quantum mechanics and quantum field theory. In Refs.(Fan et al, Ann. Phys. 321 (2006) 480; 323 (2008) 500) we have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors. 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 useful integration formulas regarding to Laguerre and Hermite polynomials. This opens a new route of deriving mathematical integration formulas by virtue of the quantum mechanical operator ordering technique.