Applicability of Approximate Quantum-Mechanical Wave Functions Having Discontinuities in Their First Derivatives

The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄₁₂ to the expectation value of the kinetic energy. For a one‐dimensional problem $${\mathrm{\delta T̄}}_{12}{\mathrm{=-(\hslash }}^2{\text{/2m)ψ}}^{*}{\text{(0) [ψ}}_2^{\prime }{\text{(0)−ψ}}_1^{\prime }\text{(0)].}$$ Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ₁′(0) and ψ₂′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N‐dimensional problem having a surface S₁₂=0 over which the first derivatives of the approximate wave function are discontinuous, $${\mathrm{\delta T̄}}_{12}={\displaystyle {\int }_{\mathrm{surface}}}{\psi }^{*}{\mathrm{[(\partial /\partial n)\; (\psi }}_2{\mathrm{-\psi }}_1{\mathrm{)]dS}}_{12}.$$ Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.