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Chinese Journal of Quantum Electronics

Applying invariant eigen-operator method to derive quantization spectrum of charged harmonic oscillator in a uniform electric field

Feng Jin-fu

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Invariant eigen-operator method was applied to derive energy-level gap of charged harmonic oscillator in a uniform electric field.The integral transform method was employed to derive ground state energy and wave function.The quantization spectra of charged harmonic oscillator in a uniform electric field are obtained by simple analysis.

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The quantum states for a time-dependent charged oscillator are investigated by an invariant operator method and a unitary transformation approach. The wave functions are derived precisely in both cases of discrete and continuous quantum spectra. By choosing time functions of the Hamiltonian in different forms, our theory can be applied to diverse types of quantum dynamical systems. In the limit of a simple case, we confirmed that our result associated with a discrete spectrum recovers to that of Storchak.
The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.