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International Journal of Theoretical Physics

Tomita representations of quantum and classical mechanics in a bra/ket formulation

Hans Herlof Grelland

Published Jun 1, 1993• 8 citations
Publisher
DOI
10.1007/bf01215298

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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.
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.