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Chinese Physics

Generalized path-integral solution and coherent states of the harmonic oscillator in D -dimensions

Ma Yu-Quan, Jin Zhang, Chen Yong-Kang, Dai Hong

Published Oct 31, 2005• 1 citation
Publisher

Abstract

We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time-dependent forced harmonic oscillator in D ( D ⩾ 1) dimensions. The coherent states, defined as the eigenstates of annihilation operator, of the D -dimensional harmonic oscillator are derived. These coherent states correspond to the minimum uncertainty states and the relation between them is investigated.

DOI
10.1088/1009-1963/14/11/005

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We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time-dependent forced harmonic oscillator in D ( D ⩾ 1) dimensions. The coherent states, defined as the eigenstates of annihilation operator, of the D -dimensional harmonic oscillator are derived. These coherent states correspond to the minimum uncertainty states and the relation between them is investigated.
For a damped harmonic oscillator forced by a time-dependent field, the exact wave function is obtained by three different methods: (i) path integral, (ii) second quantization, and (iii) dynamical invariant. The explicit form of the dynamical invariant involves a solution to a corresponding auxiliary equation. The coherent states, defined as eigenstates of a new destruction operator, form a nonorthogonal, overcomplete set and correspond to the minimum uncertainty states. These coherent states give the exact classical motion of the damped driven harmonic oscillator.
Recent proposed conformable deformation of the quantum mechanics by a fractional parameter α∈(0,1] has been used to construct a conformable quantum harmonic oscillator which coincides with the standard quantum oscillator at α=1. We argue that there is a conformable generalization of the uncertainty principle and use it to define the coherent states along the general line of quantum mechanics. We focus on the coherent states of the conformable quantum harmonic oscillator in the xα-representation: we determine these states, compute their α-power of energy, their conformable time-dependent form and the conformable translation operator. Also, we show that the eigenvectors of the conformable annihilation operator are coherent states.
We first review the properties of the harmonic-oscillator coherent states which can be equivalently defined as (a) a specific subset of the $x\ensuremath{-}p$ minimum-uncertainty states, (b) eigenstates of the annihilation operator, or (c) states created by a certain unitary exponential displacement operator. Then we present a new method for finding coherent states for particles in general potentials. Its basis is the desire to find those states which most nearly follow the classical motion, but it is most nearly a generalization of the minimum-uncertainty method. The properties of these states are discussed in detail. Next we show that the annihilation operator and displacement operator methods, as heretofore defined, cannot be applied to general potentials (whose eigenvalues are not equally spaced). We define a generalization of these methods but show that the states so defined are not, in general, equivalent to the minimum-uncertainty coherent states. We discuss a number of properties of our coherent states and the procedures we have used.
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to find a complete set of wave functions for the oscillator. Harmonic oscillators with different (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) {\em invariance} under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.