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American Journal of Physics

Classically forbidden behavior of the quantum harmonic oscillator for large quantum numbers

James J. Diamond

Published Oct 1, 1992• 5 citations
Publisher

Abstract

The probability that the quantum harmonic oscillator in quantum level n will penetrate into its classically forbidden region, has been calculated, via an algorithm based on a quadrature formula for the probability density, for values of the quantum number n ranging up to 250. These results have been analyzed numerically. The conjecture is made that the probability P(n) of the position of the oscillator in quantum state n being measured in the classically forbidden region follows the approximate formula P(n)≊A(n+1/2)−1/3+O(n+1/2)−1 where the constant A has the value A=0.1339706...; this asymptotic formula has a relative error of less than 2% for quantum numbers 250≳n≳7. The consequences of this formula in regard to the rate of approach to the classical limit are discussed.

DOI
10.1119/1.17013

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The probability that the quantum harmonic oscillator in quantum level n will penetrate into its classically forbidden region, has been calculated, via an algorithm based on a quadrature formula for the probability density, for values of the quantum number n ranging up to 250. These results have been analyzed numerically. The conjecture is made that the probability P(n) of the position of the oscillator in quantum state n being measured in the classically forbidden region follows the approximate formula P(n)≊A(n+1/2)−1/3+O(n+1/2)−1 where the constant A has the value A=0.1339706...; this asymptotic formula has a relative error of less than 2% for quantum numbers 250≳n≳7. The consequences of this formula in regard to the rate of approach to the classical limit are discussed.
The penetration probability of the quantum harmonic oscillator into the classically forbidden region is examined in the limit of large quantum numbers using analytical methods. An expression is derived for the asymptotic form of the penetration probability, and is shown to be in good agreement with an earlier result obtained by numerical methods. The asymptotic form of the probability density in the region between the classical turning points is also presented and found to have a simple physical interpretation.
The first application of quantum ideas to a system with a nonuniform potential energy is to the harmonic oscillator: a pendulum, or a mass on a spring. The result is a quantization of energy with equally spaced energy levels. The wave function is localized in the potential well but can do quantum tunneling beyond the turning points into classically forbidden regions. A wave function can be a negative quantity, but the related position probability is always positive. The wave function shows the particle is not in any single place; in a sense it is in many places simultaneously. The fact that the lowest energy quantum state does not vanish is an example of zero-point energy. States of very high energy show a strong analogy to classical oscillator behavior, an example of the Bohr correspondence principle.
Hall et al. (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds." A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall et al. (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may arise as distributional fixed points in a new generalization of Stein's method These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.
At large quantum numbers, the probability densities for particle-in-a-box or simple harmonic oscillator converge to the classical result upon coarse-graining the quantum mechanical probability densities by introducing a finite resolution in the measurement of the particle's position. This resolution in the position can be related to the resolution of the secondary total angular momentum quantum number (mmm) when interpreting the probabilistic outcomes of the Stern--Gerlach-type thought experiments for large values of the angular momentum quantum numbers (jjj).
The harmonic oscillator is defined by a quadratic potential. Many important problems in physics reduce to that of a quantum harmonic oscillator because the leading term of a potential near its minimum is quadratic. It is a general and noteworthy feature of quantum systems that the lowest energy is not zero. Calculate the probability that the quantum oscillator is found beyond the classical limit. Agreement with the uncertainty principle may be checked by calculating the expectation values of x2 and p2 and then taking the square root of the product. For large quantum numbers, the solution must approach that of the classical harmonic oscillator.
Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.