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Physica Scripta

Inverted oscillator: quantum discrete spectrum

Nadir Chaabi, Aymen Bendjoudi, Mustapha Maamache

Published Oct 14, 2024• 0 citations
Publisher

Abstract

Abstract We use the invariant operator method to investigate the quantum characteristics of the inverted oscillator. We introduce a unitary transformation that maps a time-dependent general Hermitian linear invariant operator to a time-independent one, which describes a harmonic oscillator with a unit mass and a constant frequency. Our investigation includes three distinguished cases: negative frequency, zero frequency, and positive frequency. Our most interesting result concerns the last case. For the case of negative frequency, we propose a new inner product by the introduction of the new metric operator η. Coherent state for this case will be constructed.

DOI
10.1088/1402-4896/ad8684

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Abstract We use the invariant operator method to investigate the quantum characteristics of the inverted oscillator. We introduce a unitary transformation that maps a time-dependent general Hermitian linear invariant operator to a time-independent one, which describes a harmonic oscillator with a unit mass and a constant frequency. Our investigation includes three distinguished cases: negative frequency, zero frequency, and positive frequency. Our most interesting result concerns the last case. For the case of negative frequency, we propose a new inner product by the introduction of the new metric operator η. Coherent state for this case will be constructed.
We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.
For a harmonic oscillator with time-dependent (positive) mass and frequency, a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time-dependent frequency, as well as operators. For a driven harmonic oscillator, a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given, as a generalization of previous results, in terms of the solution of classical equation of motion of the driven system. Thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system, provided the quantum states of the corresponding system of unit mass are given.
We derive quantum solutions of a generalized inverted or repulsive harmonic oscillator with arbitrary time-dependent mass and frequency using the quantum invariant method and linear invariants, and write its wave functions in terms of solutions of a second-order ordinary differential equation that describes the amplitude of the damped classical inverted oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum, the associated uncertainty relation, and the quantum correlations between coordinate and momentum. As a particular case, we apply our general development to the generalized inverted Caldirola–Kanai oscillator.
The unitary operator that transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different timescale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.