Exact quantum states of a general time-dependent quadratic system from classical action

Abstract

A generalization of driven harmonic oscillator with time-dependent mass and frequency, by adding total time-derivative terms to the Lagrangian, is considered.The generalization which gives a general quadratic Hamiltonian system does not change the classical equation of motion. Based on the observation by Feynman and Hibbs, the propagators (kernels) of the systems are calculated from the classical action, in terms of solutions of the classical equation of motion: two homogeneous and one particular solutions. The kernels are then used to find wave functions which satisfy the Schrödinger equation. One of the wave functions is shown to be that of a Gaussian pure state. In every case considered, we prove that the kernel does not depend on the way of choosing the classical solutions, while the wave functions depend on the choice. The generalization which gives a rather complicated quadratic Hamiltonian is simply interpreted as acting an unitary transformation to the driven harmonic oscillator system in the Hamiltonian formulation.

Unitary relation for the time-dependentSU(1,1)systems

“…and u(t), v(t) as the two real, linearly independent solutions of the second-order differential oscillator [4]. By defining ρ(t) and a time-constant Ω, which are positive, as…”

“…Since β(t) can be understood as a result of a simple unitary transformation which does not depend on the generators (see, e.g., Ref. [4]), from now on we will take β(t) = 0. As an extension of the unitary relation in the quadratic systems [5,9], we will give the unitary transformation which relates the system of H and the system described by…”

“…The evolution operator and transition probabilities of the harmonic oscillator with a time-dependent frequency have been known in terms of classical solutions of the oscillator [5]. The wave functions of the quadratic systems [4,5], if the centers of probability distributions of the functions remain at the origin of the space coordinate, are closely related to the SU(1, 1) coherent states of Perelomov [6] which are obtained by applying displacement-type elements of the group on a fiducial vector in a representation space.…”

“…Through a unitary transformation method, the complete set of wave functions for a general quadratic system has been given in terms of the classical solutions of the system [4,9], and the fact that wave functions are described by the classical solutions can be clearly understood from the path integral approach for this system [4]. On the other hand, it turns out that the unitary transformation for a time-dependent quadratic system can be used for the same quadratic system with an inverse-square potential to give the wave functions [9].…”

“…The time-dependent quadratic system (a generalized harmonic oscillator) [4] is a realization of particular representations of the SU(1, 1) group. The evolution operator and transition probabilities of the harmonic oscillator with a time-dependent frequency have been known in terms of classical solutions of the oscillator [5].…”