Variable operator technique and the min-max theorem

Abstract

We investigate a variation method where the trial function is generated from the application of a variable operator on a reference function. Two conditions are identified, one for obtaining a maximum and another for a minimum. Although the conditions are easy to understand, the overall formulation is somewhat unusual as each condition gives rise to a two-step variation process. As an example, projection operators are used to form the variable operator, and by this tactics one obtains the new interpretation that the pseudopotential formalism is in fact equivalent to a minimax procedure. The two-step variational process is nevertheless more flexible than the pseudopotential formalism, for it can also be used when the variable operator is not manifestly expressed in terms of projectors. This is illustrated by a comparison of the two-step method with the variational solution of Dirac’s relativistic electron equation. The same comparison leads to an alternative proof that the process of maximizing energy by varying the u–l coupling operator eliminates all negative-energy contributions from a trial spinor. The latter observation is crucial for the derivation of the min-max theorem in relativistic quantum mechanics.