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CRC Press eBooks

Approximation Methods IV: Variational Approach

S. Rajasekar, R. Velusamy

Published Sep 29, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781003172178-16

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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.
As in classical mechanics the number of exactly soluble quantum mechanical problems is very limited and in most cases of practical interest numerical techniques and approximation methods are unavoidable. The availability of fast computers has turned quantum computation into a major industry. This chapter looks at the several approximation techniques. In seeking to solve the Schrodinger equation with the perturbed Hamiltonian, it is convenient to work with series expansions by introducing a dimensionless parameter λ. In many cases the system of interest differs from an exactly soluble system by only a small disturbance, enabling an approximation to be made by expanding in powers of a smallness parameter. The Variational Method provides a very simple method of approximating energy eigenvalues. It is most useful for computing ground state energies and its success depends on our being able to make a reasonable guess for the form of the ground state wave function.
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 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.
We present here a systematic scheme for improving the variational wave functions and corresponding energy levels for quantum systems. By expanding the wave function around a variational parameter value, a family of independent functions can be systematically generated. The eigenstates are then obtained by diagonalizing the Hamiltonian matrix within the basis and optimized with respect to the variational parameter. As a test, the ground state of the quartic anharmonic oscillator has been investigated, and it is found that this scheme converges more rapidly than the conventional Lanczos method and yields better approximations of the energy levels than other variational methods. The effectiveness of this scheme for larger systems remains to be seen.
The steepest descent perturbation theory is extended to the calculation of the energy and eigenfunction of the excited state of a quantum system. In case of the orthogonality of the trial function for the excited state to those for lower-energy state or ground state in the same symmetry class is preserved, the variational collapse to lower energy state can be avoided in this proposal. An iterative procedure is given for generating better eigenvalue and eigenfunction of the excited state without requiring an infinite summation over reference states as in conventional perturbation theory. This new perturbation method can be applied to calculate the excitation energy and wave function of excited states for any many-body quantum system to a high degree of accuracy without so much computational effort as in conventional method.