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arXiv:quant-ph/0608211 · Journal of Physics A Mathematical and General

On the exactness of the semi-classical approximation for non-relativistic one-dimensional propagators

I Semiz, K Düztaş

Published Nov 8, 2006• 0 citations
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Abstract

For one-dimensional non-relativistic quantum mechanical problems, we investigate the conditions for all the position dependence of the propagator to be in its phase, that is, for the semi-classical approximation to be exact. For velocity-independent potentials we find that: (i) the potential must be quadratic in space, but can have arbitrary time dependence, (ii) the phase may be made proportional to the classical action, and the magnitude (‘fluctuation factor’) can also be found from the classical solution and (iii) for the driven harmonic oscillator the fluctuation factor is independent of the driving term.

DOI
10.1088/0305-4470/39/47/011

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For one-dimensional non-relativistic quantum mechanical problems, we investigate the conditions for all the position dependence of the propagator to be in its phase, that is, for the semi-classical approximation to be exact. For velocity-independent potentials we find that: (i) the potential must be quadratic in space, but can have arbitrary time dependence, (ii) the phase may be made proportional to the classical action, and the magnitude (‘fluctuation factor’) can also be found from the classical solution and (iii) for the driven harmonic oscillator the fluctuation factor is independent of the driving term.
For one dimensional non-relativistic quantum mechanical problems, we investigate the conditions for all the position dependence of the propagator to be in its phase, that is, the semi-classical approximation to be exact. For velocity independent potentials we find that: (i) the potential must be quadratic in space, but can have arbitrary time dependence. (ii) the phase may be made proportional to the classical action, and the magnitude (``fluctuation factor'') can also be found from the classical solution. (iii) for the driven harmonic oscillator the fluctuation factor is independent of the driving term.
For the case of the quantization of the usual non-relativistic classical Lagrangian function quadratic in the velocity the validity is demonstrated of the non-canonical space-time formulation of quantum mechanics proposed recently by the author, which aims to evaluate, without appealing to the Schrödinger equation, the transformation function K(x,t′;y,t′) in the space representation on the basis of the composition rule K(x, t″; y, t′) = ∫K(x, t″; z, t)dzK(z, t; y, t′) (1) coupled with the supposition that it is approximated to zeroth order in the quantum of action h by the so-called semi-classical kernel Ko(x, t″; y, t′) = [(i/h)∂2S/∂x∂y]1/2 exp [(i/ℏ) S(x,t″; y, t′] (2) written in terms of the classical action S(x, t″; y, t′) alone. In the first place the action function corresponding to the above Lagrangian is expanded in power of the interval of time T = t″ − t′. Then the deviation of the semi-classical kernel (2) from the unitary transformation function is shown to be of the third order in T, and the corresponding correction term is evaluated by solving the integral equation (1). It is also shown that the semi-classical kernel is unitary for a free motion of a particle with its mass being a function in the space coordinate.
A generalization of a 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 solutions and one particular solution. The kernels are then used to find wave functions that satisfy the Schr\"odinger 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 applying unitary transformation to the driven harmonic oscillator system in the Hamiltonian formulation.
Abstract A recently proposed approach to relativistic quantum mechanics (Grave de Peralta, Poveda, Poirier in Eur J Phys 42:055404, 2021) is applied to the problem of a particle in a quadratic potential. The methods, both exact and approximate, allow one to obtain eigenstate energy levels and wavefunctions, using conventional numerical eigensolvers applied to Schrödinger-like equations. Results are obtained over a nine-order-of-magnitude variation of system parameters, ranging from the non-relativistic to the ultrarelativistic limits. Various trends are analyzed and discussed—some of which might have been easily predicted, others which may be a bit more surprising.