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Journal of physics. A, Proceedings of the Physical Society. General

Variational trial functions in quantum theory. I. Bound states

T. L. John, K L Williams

Published Aug 1, 1972• 2 citations
Publisher

Abstract

The authors give a method of calculating variational trial functions in quantum theory. Applications are made to the discrete spectra of some simple one and two particle systems.
DOI
10.1088/0305-4470/5/8/008

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For Pt.I see ibid., vol.5, 1152 (1972). A method given in Part I for the calculation of variational trial functions is applied to the continuous spectra of some simple one and two particle systems.

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American Journal of Physics•Jan 1, 1977•22 citations
In applying the variational method, we demonstrate that, with careful study of the general features of the trial wave functions, we can achieve systematic improvements in calculating the eigenvalues of quantum mechanical systems. We apply five different sets of trial wave functions to calculate the ground state and first excited state energies of weakly bound potentials, such as V(x) = ∣x∣, √∣x∣ and ln (∣x∣). We demonstrate that accurate results can be obtained through a thorough understanding of the asymptotic behaviours. With proper adjustments, we show how to improve trial wave functions. Finally, by calculating to the fifth excited states, we compare the exact and variational results with those obtained from the semi-classical method, and discuss the merits of the different approaches.
The variational method can be used to construct an approximate ground state eigenfunction. In some cases the method can be used for finding approximately the energy of first few discrete states of a bound system. The perturbation theory uses a set of unperturbed states and the perturbation Hamiltonian H to construct the eigenfunctions and eigenvalues through highly formalized approximations. In contrast, the variational method is based on a guess for the state – a trial state is used to determine the corresponding energy. Any well behaved trial state introduced will yield an appropriate energy, although some trial states will be better than other, but whatever the guess, we can always make systematic improvements.