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European Journal of Physics

Study of correspondence principle in 1-dimension through examples

Susobhan Paul, Satadal Bhattacharyya, Joydip Mitra

Published Jul 6, 2023• 1 citation
Publisher

Abstract

Abstract We present several examples both for non-perturbative and perturbative systems to illustrate Bohr’s correspondence principle which is not usually treated in common text-books. In particular, the importance of the limit n → ∞ and h → 0 in understanding of the classical limit of quantum theory has been discussed explicitly in each case.

DOI
10.1088/1361-6404/ace502

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Abstract We present several examples both for non-perturbative and perturbative systems to illustrate Bohr’s correspondence principle which is not usually treated in common text-books. In particular, the importance of the limit n → ∞ and h → 0 in understanding of the classical limit of quantum theory has been discussed explicitly in each case.
Abstract Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.
Two-dimensional central potentials leading to the identical classical and quantum motions are derived and their properties are discussed. Some of zero-energy states in the potentials are shown to cancel the quantum correction $Q=(\ensuremath{-}{\ensuremath{\Elzxh}}^{2}/2m)\ensuremath{\Delta}R/R$ to the classical Hamilton-Jacobi equation. The Bohr's correspondence principle is thus fulfilled exactly without taking the limits of high quantum numbers, of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0,$ or of the like. In this exact limit of Q=0,Q=0,Q=0, classical trajectories are found and classified. Interestingly, many of them are represented by closed curves. Applications of the found potentials in many areas of physics are briefly commented.
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.
Scope of this book Quantum mechanics is a difficult subject, and this book is intended to help the reader overcome the main difficulties in the way to understanding it. The first part of the book, Chs. 2–16, contains a systematic presentation of the basic principles of quantum theory, along with a number of examples which illustrate how these principles apply to particular quantum systems. The applications are, for the most part, limited to toy models whose simple structure allows one to see what is going on without using complicated mathematics or lengthy formulas. The principles themselves, however, are formulated in such a way that they can be applied to (almost) any nonrelativistic quantum system. In the second part of the book, Chs. 17–25, these principles are applied to quantum measurements and various quantum paradoxes, subjects which give rise to serious conceptual problems when they are not treated in a fully consistent manner. The final chapters are of a somewhat different character. Chapter 26 on decoherence and the classical limit of quantum theory is a very sketchy introduction to these important topics along with some indication as to how the basic principles presented in the first part of the book can be used for understanding them. Chapter 27 on quantum theory and reality belongs to the interface between physics and philosophy and indicates why quantum theory is compatible with a real world whose existence is not dependent on what scientists think and believe, or the experiments they choose to carry out. The Bibliography contains references for those interested in further reading or in tracing the origin of some of the ideas presented in earlier chapters.