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

Abstract

We develop a systematic approach to determine the large |p| behavior of the momentum-space wavefunction, φ(p), of a one-dimensional quantum system for which the position-space wavefunction, 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 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.

Exact results for the infinite supersymmetric extensions of the infinite square well

“…) , consistent with theorems [25] connecting the discontinuities of ψ(x) (here encoded in the increasingly smooth x S+1 behaviour of the wave functions at the walls) very directly to the large momentum limit of f(p). There are likely many closed-form results waiting to be uncovered in the continued mathematical physics analysis of both the positionspace and momentum-space versions of this problem.…”

Universal Tan relations for quantum gases in one dimension

“…where X ij = (m σ x i + m σ ′ x j )/(m σ + m σ ′ ) and x ij = x i − x j are the center of mass and relative coordinates of the two particles. Due to the discontinuity the Fourier transform of the wave-functions at large momenta will have an 1/k 2 asymptotic behavior [67,68]. This fundamental observation will be used extensively throughout this paper and will constitute our main tool in deriving the Tan relations.…”

“…Subsequently, it was shown that they can be rederived using the Operator Product Expansion of local operators [6,7]. Here, we use what we consider to be the most powerful and transparent technique which uses the fact that the derivative of the wave-function of a system with contact interactions is discontinuous when the coordinates of two particles coincide [67,68]. This is the same method employed by Olshanii and Dunjko [5] in their study of the short distance, large momentum distribution of the Lieb-Liniger model and by Werner and Castin in their comprehensive papers on Tan relations for 2D and 3D mixtures [10,13].…”

The biharmonic oscillator and asymmetric linear potentials: from classical trajectories to momentum-space probability densities in the extreme quantum limit

“…Recently, two of us have developed a systematic description of the large p behavior of φ(p) for one-dimensional problems for which the potential energy function is less than perfect, that is, which has a discontinuous derivative at any order [11]. The general result can be expressed in the following way.…”

“…In section 4 we then use these two systems to exemplify new results [11] regarding the large p behavior of the momentum-space wavefunction φ(p) for potentials which are less than perfect, in the sense that they have a discontinuity in some derivative. Two of us have recently shown that if the potential energy function V (x) has a discontinuity in the kth derivative, then the φ(p) solutions exhibit a power-law 'tail', proportional to 1/p k+3 with a coefficient precisely determined by the details of the wavefunction at the location of the generalized 'kink'.…”