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Physical Review

Exact Integral Relationship in Quantum Mechanics: Application to the Variational Method

Edwin M. Roberts

Published Nov 1, 1962• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.128.1381

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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.
In most cases solving a quantum mechanical problem exactly is not always possible. In such a case, an analytic solution to the Schrödinger equation is appropriate. This results to numerical solutions that may not necessarily be the best approach. Interest sometimes may not necessarily be in the detailed eigenfunctions, but rather in the energy levels and the qualitative features of the eigenfunctions. So, the numerical solutions are usually less intuitively reasonable. It is possible to find the values of the energy levels to be slightly sensitive to the deviation of the wave function from its true form. So, the expectation value of the energy for an approximate wave function can be a very good estimate of the corresponding energy eigenvalue. Using an approximate wave function dependent on some set of variational parameters and minimizing its energy by these variational parameters yield such energy estimates. This technique is the so-called variational method due to this minimization procedure.
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.
A path-integral formulation of quantum mechanics is investigated which is closely related to that of Feynman. It differs from Feynman's formulation in that it involves the Hamiltonian function of the canonically conjugate coordinates and momenta. The classical limit yields the variational principle: $\ensuremath{\delta}ff(p\ifmmode\ddagger\else\textdaggerdbl\fi{}\ensuremath{-}N)dt=0$. A path-integral formula is also obtained for the energy eigenstate projection operator associated with the time-independent Schr\"odinger equation. The classical limit of the projection operator formula yields a modified form of the well-known variational principle for the phase-space orbit of given energy. Relativistically covariant Hamiltonian variational principles are analyzed and lead naturally to a relativistic scalar wave equation which involves a proper time variable which is canonically conjugate to the mass in the same manner as the ordinary time variable is conjugate to the energy in nonrelativistic quantum theory.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.