Less than perfect quantum wavefunctions in momentum-space: How phi(p) senses disturbances in the force

Abstract

We develop a systematic approach to determine the large |p| behavior of the momentum-space wavefunction, phi(p), of a one-dimensional quantum system for wich the position-space wavefunction, psi(x), has a discontinuous derivative at any order. We find that if the k-th derivative of the potential energy function has a discontinuity, there is a corresponding discontinuity in psi^{(k+2)}(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wavefunction proportional to 1/p^{k+3}. A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.