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European Journal of Physics

A variational calculation exhibiting both minima and maxima

Mingli Zhou, Kenneth R. Brownstein

Published Nov 1, 1999• 0 citations
Publisher

Abstract

A simple example of a variational calculation for the quantum mechanical energies of a system is presented. The trial function employed has only one (nonlinear) variational parameter. We point out a rather unexpected result for the energy extrema associated with this parameter.
DOI
10.1088/0143-0807/20/6/318

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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.
Certain problems connected with the choice of trial variational functions for the quantum many-body problem are discussed. It is convenient to make use of the reduced-density-matrix formulation developed by L\"owdin and Mayer, since a system whose Hamiltonian contains only 1- and 2-body interactions is characterized by the two simplest reduced density matrices ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. It is known that symmetry, Hermiticity, and normalizability conditions are not sufficient to ensure a physically realizable choice of the functions ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. However an additional nontrivial restriction is imposed by the fact that such functions must be possible variational extrema. The implications of this condition are investigated in some detail and it turns out that all the nonphysical choices of ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$ which have been exhibited by various authors are eliminated. The condition is essentially nonquantum inasmuch as an analogous argument should be possible for a classical system. Moreover one of the consequences of this condition is that the system (classical or quantum) may exhibit macroscopic behavior, i.e., order-disorder transitions. The analysis leads to apparently reasonable choices of trial forms for ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$, although it is still not known whether any further restriction must be imposed to complete the sufficiency argument.
Abstract The optimization of parameters in trial wavefunctions by minimizing the variance in local energy was used for some of the earliest variationala and diffusionb QMC calculations, and it has been used ever since by most practitioners of QMC. Perhaps the best known examples are given in the paper by Schmidt and Moskowitzc for optimizing their nine-parameter Jastrow functions for first-row atoms. An alternative method of optimization, leading to lower variational energies for these atoms, is reported in this paper.
When applying the variational method, it is demonstrated that the systematic improvements in calculating the eigen‐energies of quantum mechanical systems can be achieved by careful study of the general features of the trial wave functions. Two sets of trial wave functions were used for the harmonic oscillator potential. In the first example, it is shown, even with a crude initial guess, that accurate results will be obtained by adjusting the wave function without introducing any additional variational parameters. Even though the second example requires more algebraic manipulation, it is shown that it converges to the exact result much faster. It is suggested that these calculations be incorporated in computer algebra software packages for the teaching of quantum mechanics.