A Variational Method for the Calculation of Particle Densities and Sums of Eigenvalues in Wave Mechanics

Adopting a recent proposal made by Macke, the problem of calculating approximate particle densities and sums of eigenvalues by a variational method is examined thoroughly for the case of particles moving along the x axis in a given potential. An Euler equation is derived, which can, in principle, be solved to obtain the best possible density in the sense of the restricted variational method employed. Connection is made with previous work of von Weizsacker, and it is pointed out that the solution of von Weizsacker's equation may often be used to obtain a good initial approximation to the optimum results of the present treatment. Illustrative calculations have been carried through for the linear harmonic oscillator with the first ten levels singly occupied, and the results for the particle density are extremely encouraging.