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International Journal of Mathematical Education in Science and Technology

Demonstration of systematic im provements in application of the variational method to weakly bound potentials

W. N. Mei

Published Jul 1, 1999• 0 citations
Publisher

Abstract

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.

DOI
10.1080/002073999287798

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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.
In applying the variational method, six different sets of trial wave functions are used to calculate the ground state and first excited state energies of the strongly bound potentials, i.e. V(x) = x 2 m , where m = 4, 5 and 6. It is shown that accurate results can be obtained from thorough analysis of the asymptotic behaviour of the solutions. Consequently, it was found that there is a progressive qualitative change in the wave functions from the harmonic to anharmonic and then strongly bounded region, which explains why using a harmonic oscillator wave function together with the conventional perturbative approach to calculate the energy correction, was unsuccessful.
Three sets of trial wave functions were applied to calculate the ground state and first excited state energies of the x 4 and x 6 anharmonic oscillator potentials. We demonstrate that, even with a rather crude initial guess, results can be improved by proper adjustment of the trial wave functions after careful study of their general features. Meanwhile, we found some interesting and unexpected results when applying the second set of trial wave functions. Once again, we suggest using together with computer algebra software packages, these examples when teaching the variational method in a quantum mechanics course.
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.
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.
In chapters 13 and 14 we focused on perturbation theories for time-independent and time-dependent perturbations. For time-independent perturbations we obtained first-order and second-order corrections to energy eigenvalues and eigenfunctions. For time-dependent perturbations we obtained the probability for a system to make a transition from an ith initial state to fth final state. We have also discussed how to study the effect of adiabatic and sudden perturbations. In the present chapter we consider the variational method. 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(1) 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 which 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. We can construct a reasonably accurate eigenfunction often by a physical insight and ingenuity.