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arXiv:1803.09088 · European Journal of Physics

Three theorems of quantum mechanics and their classical counterparts

Claude Semay

Published May 23, 2018• 6 citations
arXivHTMLPDF

Abstract

Abstract The Hellmann–Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is shown here that these three theorems are valid for classical mechanics with a nonstandard kinetic energy. This brings some information about the connections between the quantum and classical worlds. Constraints about the functional form of the kinetic energy are also discussed.

DOI
10.1088/1361-6404/aac74c

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Abstract The Hellmann–Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is shown here that these three theorems are valid for classical mechanics with a nonstandard kinetic energy. This brings some information about the connections between the quantum and classical worlds. Constraints about the functional form of the kinetic energy are also discussed.
Theorems giving exact expressions for the derivative of the energy with respect to a parameter in the Hamiltonian for cyclic classical mechanical problems in both the relativistic and nonrelativistic cases are derived. Comparison is made with similar well-known theorems in the quantum mechanical case. Applications of both classical and quantum derivative relations are made to deduction of parametric dependence of the energy in simple cases, deduction of a meaning to the expectation value of the Dirac operator β, and construction of a test of consistency for a well posed problem in the theory of a Dirac particle in a bound state.
Abstract Quantum mechanics was created mainly to account for microscopic phenomena that occur on a scale of order IO−10 m. It has turned out to be an astonishingly successful theory which applies on a scale down to at least 10−18 m. Quantum theory has become established as part of the essential core of physics. The purpose of this introductory chapter is twofold. Firstly, in Sections 1.1–1.7 we review the formalism of elementary nonrelativistic quantum mechanics. In this review we attempt to give enough of the terminology and basics of the subject so that a reader who is not already familiar with quantum mechanics will be able to follow the applications considered later in the book. For the reader who finds our discussion too brief, the literature on quantum mechanics is vast and we recommend in particular Refs 1–5. The second purpose of this chapter is to present the basic tools of operator methods as they are used in this book: thus in Sections 1.8–1.11 we discuss commutation relations, shift operators, and the virial, hypervirial, and Hellmann Feynman theorems.
Abstract The Work-Energy Theorem is a fundamental principle in mechanics, establishing a quantitative relationship between the net work done on a particle and its change in kinetic energy. In this study, we present a rigorous derivation of the theorem across classical, quantum, and relativistic frameworks. The analysis includes expressing kinetic energy in terms of de Broglie wavelength, introducing identities linking force, displacement, and wavelength, and extending the theorem to relativistic particles. We provide detailed mathematical derivations, highlight the transition between classical and quantum regimes, and discuss the limits of applicability, including decoherence effects and classical limits. This work demonstrates the universality of the Work-Energy Theorem and bridges diverse physical perspectives under a unified framework.