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Cambridge University Press eBooks

Approximation methods. Time-independent perturbation theory, variational method

Gordon Leslie Squires

Published Mar 16, 1995• 0 citations
Publisher
DOI
10.1017/cbo9781139168182.007

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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.
Chapter 13 is devoted to two important approximation methods used to solve time-independent problems, namely stationary perturbation theory and the variational method. Perturbation theory is developed for both non-degenerate states and degenerate states, with illustrative examples. The Stark effect is considered separately. The dielectric constant of a transparent medium as a function of frequency is shown to result from the modification of the wave function due to the perturbation of the radiation field. Two complementary topics are presented: the canonical transformation method and the Feynman-Hellman method. Finally, the possibilities of the variational method are illustrated by using it first to determine an upper bound for the energy of the stationary states of a quantum system, and then to study more complicated cases, including the partially shielded Coulomb potential produced by a neutral atom.