Applicability of Approximate Quantumhyphen;Mechanical Wave Functions Having Discontinuities in Their First Derivatives

摘要

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 egr; 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 egr; 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 dgr;Tmacr;12to the expectation value of the kinetic energy. For a onehyphen;dimensional problemdgr;Tmacr;12equals;minus;lpar;planck;2sol;2mrpar;psgr;*lpar;0rpar; lsqb;psgr;2prime;lpar;0rpar;minus;psgr;1prime;lpar;0rpar;rsqb;.Here psgr;(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and psgr;1prime;(0) and psgr;2prime;(0) are the first derivatives of psgr; taken from the left and from the right, respectively, at this point. Similarly, for anNhyphen;dimensional problem having a surfaceS12=0 over which the first derivatives of the approximate wave function are discontinuous,dgr;Tmacr;12equals;surfacepsgr;*lsqb;lpar;part;sol;part;nrpar; lpar;psgr;2minus;psgr;1rpar;rsqb;dS12.Here the part;/part;nis the normal derivative with the normal pointed from region 1 towards region 2.