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American Journal of Physics

Hellmann-Feynman Theorems in Classical and Quantum Mechanics

William A. McKinley

Published Aug 1, 1971• 22 citations
Publisher

Abstract

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.

DOI
10.1119/1.1986322

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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 The generalized Hellmann–Feynman and curvature theorems are derived for the generalized Sturm–Liouville‐type eigenvalue equation which is widely encountered in many branches of physics. In classical mechanics, the Sturm–Liouville eigenvalue equation primarily arises from applying the theory of small oscillations to vibration problems while, in quantum mechanics, the most important example is the time‐independent Schrödinger equation. The generalized theorems, respectively, provide simple useful expressions for the first and second λ derivatives of the eigenvalue where λ is an arbitrary real parameter appearing in the eigenvalue equation. These results, which apply to both classical and quantal systems, include as special cases the well‐known quantum mechanical Hellmann–Feynman and curvature theorems for the time‐independent Schrödinger equation, thus unifying the formalism. This connection between classical and quantum mechanics is of interest per se, and also because classical eigenvalue equations find application in the treatment of some quantum chemical problems. Two derivations of both generalizations are presented, the first proof applying to exact and Rayleigh–Ritz (linear variational) solutions of the eigenvalue problem, and the second, more general one, to arbitrary optimal variational as well as exact solutions. Necessary and sufficient conditions for the validity of the results are discussed. To demonstrate the insight they afford, several applications of the theorems in conjunction with perturbation theory are described.
We present a simple derivation of the Hellmann–Feynman theorem at finite temperature. We illustrate its validity by considering three relevant examples, which can be used in quantum mechanics lectures: the one-dimensional harmonic oscillator, the one-dimensional Ising model, and the Lipkin model. We show that the Hellmann–Feynman theorem allows one to calculate expectation values of operators that appear in the Hamiltonian. This is particularly useful when the total free energy is available, but there is no direct access to the thermal average of the operators themselves.
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.
An external parameter is, by definition, a quantity which may be varied independently of the coordinate positions and of the momenta of the various particles composing the system. Thus, the Schr\"odinger equation, for a particular value of the external parameter, belongs to a family of differential equations obtained by varying the numerical value of the external parameter. An explicit example of an external parameter is an imposed magnetic field. The way in which the energy changes throughout the family of Schr\"odinger equations is given by the Hellmann-Feyman theorem. An exact integral relationship, involving a pair of external parameters, is derived from the Hellmann-Feynman theorem using assumptions which are not unduly restrictive. This relationship is used to arrive at trial (variational) wave functions for quantum mechanical problems which are not exactly soluble. The method is demonstrated in two sample problems. A technique for progressively improving variational calculations is discussed.
Some exact equations are derived which clarify some potentials that are hidden in the classical theorems such as the Hellmann–Feynman (H–F) and the integral Hellmann–Feynman (I-H–F) theorems. The differential form of the density equation given previously includes not only the classical force operator but also the force operator associated with the quantum-mechanical potential introduced by Bohm. The latter arises essentially only from the noncommuting property of coordinates and momenta in quantum mechanics. However, after integration only the classical force term survives and results in the H–F theorem. The important role of the quantum force term is completely hidden in the H–F theorem. This fact would be closely related to the nondeterminicity and the classical interpretation of the H–F theorem. A similar role of the quantum potential is also shown for the I-H–F theorem. We have also investigated the origin of force density (the integrand of the H–F theorem) and isolated the roles of the generalized exchange and correlation effects which are also hidden in the H–F theorem.