A Contribution to the Theory of the B. W. K. Method

TLDR: In this paper, the authors put Zwaan's scheme for deriving the Kramers connection formulas on a rigorous basis, differential equations are set up governing the variation in the coefficients used to fit a linear combination of the B. W. K. type approximation functions to an exact solution of Schrodinger's equation in one dimension.

Abstract: In order to put Zwaan's scheme for deriving the Kramers connection formulas on a rigorous basis, differential equations are set up governing the variation in the coefficients used to fit a linear combination of the B. W. K. type approximation functions to an exact solution of Schr\"odinger's equation in one dimension. Approximate solutions of the differential equations are worked out which lend themselves to the setting up of the connection formulas and give definite upper bounds to the errors involved in their use. The method is also used to set up the Sommerfeld phase integral quantum condition independently of the connection formulas. An upper limit to the error in the energy is worked out. A similar treatment of the problem of the transmission of matter waves through rounded potential barriers is formulated.