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American Journal of Physics

Asymptotic form of the penetration probability of the quantum harmonic oscillator into the classically forbidden region

D. E. Atems, J. M. Wadehra

Published May 1, 1995• 1 citation
Publisher

Abstract

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.

DOI
10.1119/1.17908

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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 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 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.
In this paper we suggest a simple mathematical procedure to derive the classical probability density of quantum systems via Bohr’s correspondence principle. Using Fourier expansions for the classical and quantum distributions, we assume that the Fourier coefficients coincide for the case of large quantum number. We illustrate the procedure by analyzing the classical limit for the quantum harmonic oscillator and the particle in a box, although the method is quite general. We find, in an analytical fashion, the classical distribution arising from the quantum one as the zeroth order term in an expansion in powers of Planck’s constant. We interpret the correction terms as residual quantum effects at the microscopic-macroscopic boundary.
This comparison, although well meant, is unsuitable for demonstrating in introductory quantum mechanics the convergence of classical and quantum predictions for large quantum numbers, for it really tries to match ‘‘classical apples’’ with ‘‘quantum pears’’! First, it compares a classical oscillatory state with a quantum stationary state, instead of comparing it to the corresponding quantum nonstationary state. Second, it compares probabilities of different quality, namely, a classical probability of the classical oscillator (which could be circumvented by specifying the initial conditions precisely) with a quantum probability of the quantum oscillator (which cannot be avoided by any means). However, it is shown that both defects can be nicely corrected, illuminating the fact that—if properly formulated— classical and quantum oscillators have more in common than the textbooks are trying to make the beginner believe. The associated quantum position probability density, because it is now truly analogous to the classical situation, agrees much better with the classical curve than the textbook one does, and its analytical form also explains why the textbook quantum curve, derived from an unwarranted comparison, nevertheless resembles the classical curve in some sense.
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.