Quantum Dynamics of a Particle in a Linear Potential: Invariant Operator Approach and Discrete Spectrum Solutions

• URL: http://arxiv.org/abs/2606.02112v1
• Date: Mon, 01 Jun 2026 11:44:09 GMT
• Title: Quantum Dynamics of a Particle in a Linear Potential: Invariant Operator Approach and Discrete Spectrum Solutions
• Abstract summary: We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method.<n>Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained.<n>The resulting formalism provides an exact quantum description of a particle under a constant force.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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