A variational approach to the calculation of the two lowest eigenvalues of Schrödinger’s equation for the double-oscillator potential is presented. A family of two-parameter double-Gaussian trial functions provides an alternative to the exact series expansions for the energy eigenfunctions and yields energy estimates of high accuracy.
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.
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.
Abstract The problem of the harmonic oscillator with a centrally located delta function potential can be exactly solved in one dimension, where the eigenfunctions are expressed as superpositions of the Hermite polynomials or as confluent hypergeometric functions in general. The eigenfunctions obtained exactly are difficult to visualise and hence, to gain more insight, one can attempt to use model wave functions which are explicitly and simply expressed. Here, we apply the variational method to verify how closely one can approach the exact ground state eigenvalues using such trial wave functions. We obtain the estimates of the ground state energies, which are closer to the exact values in comparison to earlier approximate results for both the repulsive and attractive delta potentials.
An improved variational path integral approach is developed and applied to the quantum double-well potential, in which part of the quartic term of the potential is included in the trial action. The expression of the effective classical potential (ECP) under a non-Gaussian expectation is obtained. Here the frequency and fourth-order derivative of the potential are treated as two variational parameters, determined by the minimization of the ECP at each point. We calculate the ECP, the free energy and the level splitting of a symmetrical double-well potential. It is shown that the present results are better than those of the Feynman–Kleinert Gaussian variational method.
A system consisting of two protons in two double minimum potentials coupled by a bilinear interaction was studied. Explicit calculations were performed for local potentials described by a combination of harmonic oscillator and Gaussian terms. To determine eigenfunctions and -values variational procedures were employed which made use of numerically determined 1-particle solutions. Eigenvalues and (graphical representations of) 2d eigenfunctions are presented, classified according to their symmetry, and used to determine the time evolution of the system for given initial conditions.