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

We develop a systematic approach to determine the large {pipe}p{pipe} behavior of the momentum-space wave function, φ{symbol}(p), of a one-dimensional quantum system for which the position-space wave function, ψ(x), has a discontinuous derivative at any order. We find that if the kth derivative of the potential energy function for the system has a discontinuity, there is a corresponding discontinuity in ψ^(k+2)(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wave function 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.