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arXiv:hep-th/9406106 · Physical Review A

Non-Hermitian techniques of canonical transformations in quantum mechanics

Haewon Lee, Won Sik L’YI

Published Feb 1, 1995• 24 citations
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Abstract

The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.

DOI
10.1103/physreva.51.982

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The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.
The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.
A canonical transformation changes variables such as coordinates and momenta to new variables preserving either the Poisson bracket or the commutation relations depending on whether the problem is classical or quantal respectively. Classically canonical transformations are well established as a powerful tool for solving differential equations. Quantum canonical transformations have been defined and used relatively recently because of the non-commutativeness of the quantum variables. Three elementary canonical transformations and their composite transformations have quantum implementations. Quantum canonical transformations have been mostly used in time-independent Schrodinger equations and a harmonic oscillator with time-dependent angular frequency is probably the only time-dependent problem solved by these transformations. In this work, we apply quantum canonical transformations to a harmonic oscillator in which both angular frequency and equilibrium position are time-dependent.
Hamilton-Jacobi theory is a fundamental subject of classical mechanics and has also an important role in the development of quantum mechanics. Its conceptual framework results from the advantages of transformation theory and, for this reason, relates to the features of the generating function of canonical transformations connecting a specific Hamiltonian H to a new Hamiltonian K chosen to simplify Hamilton's equations of motion. Usually, the choice is between K=0 and a cyclic K depending on the new conjugate momenta only. Here, we investigate more general representations of the new Hamiltonian. Furthermore, it is pointed out that the two common alternatives of K=0 and a cyclic K should be distinguished in more detail. An attempt is made to clearly discern the two regimes for the harmonic oscillator and, not surprisingly, some correspondences to the quantum harmonic oscillator appear. Thanks to this preparatory work, generalized coordinates and momenta associated with the Hamilton's principal function S (i.e., K=0) are used to determine dependences in the Hamilton's characteristic function W (i.e., cyclic K). The procedure leads to the Schrodinger's equation where the Hamilton's characteristic function plays the role of the Schrodinger's wave function, whereas the first integral invariant of Poincare' takes the place of the reduced Planck constant. This finding advances the pedagogical value of the Hamilton-Jacobi theory in view of an introduction to Schrodinger's wave mechanics without the tools of quantum mechanics.