Variational Methods based on the Density Matrix

A method of setting up variational density matrices which satisfy the necessary subsidiary conditions is discussed, and some explicit variational forms are presented. Each variational matrix is defined essentially in terms of the density matrix for a known problem, for example a system of free particles in a box, using either stationary wave or progressive wave solutions. The latter form is shown, by suitable transformation along lines suggested by earlier work of Macke, to lead very directly to a method intimately connected with that proposed by von Weizsäcker, and a completely quantum-mechanical variational basis for this method is thereby provided. One- and three-dimensional cases for particles moving in a common potential are considered explicitly, and ways of achieving greater accuracy than is possible with von Weizsacker's scheme are clear, especially in the three-dimensional case. A simple example is worked out in order to compare the results obtained from the various approximate variational forms with the exact density matrix for a linear harmonic oscillator potential, and the results are encouraging.