A self-contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in coordinate and momentum space, Lippmann–Schwinger integral equations of scattering for the scattering wavefunction and the transition operator, optical theorem, and the unitarity relation for the transition operator. The present definition of the scattering amplitude in terms of the asymptotic wavefunction differs from the usual definition. The present definition has certain advantages, for example, in writing the optical theorem and in studying the analytical properties of the scattering amplitude.
In conventional fluids, acoustic excitations are described by the propagation of pressure and density fluctuations obeying the linearized Euler equations. In contrast, a superfluid—being a macroscopic quantum state with zero viscosity—supports excitations that are governed by the Gross–Pitaevskii equation and exhibit a linear dispersion at low momenta. Here we propose a novel type of excitation in superfluids, which we term "quantum pressure waves" (QPWs). A QPW is a localized, solitonic modulation of the quantum pressure term that appears in the Madelung formulation of the Gross–Pitaevskii equation. Unlike ordinary sound, a QPW does not carry a classical pressure gradient but instead represents a self‑consistent rearrangement of the quantum potential that propagates without dissipation. We develop an analytic theory of QPWs, derive their dispersion relation, and discuss their stability properties. The existence of QPWs suggests a new class of quantum fluid dynamics and points toward possible experimental observation in ultracold atomic gases.
The scattering of waves in classical physics and quantum scattering theory have many dissimilarities, but also many things in common. Many of the modern developments in classical wave theory have their origin in quantum scattering despite the later development of quantum physics. Although each field has diverged from the other over time, there are many analogies between the two disciplines and much in one area may enhance the other. In this work an outline is given of some aspects of the classical scattering theory of waves which have some relation with quantum theory. In addition, some numerical techniques are presented that may be of use in both areas.
We provide an exact solution of the scattering problem for the potentials of the form $v(x,y)={\ensuremath{\chi}}_{a}(x)[{v}_{0}(x)+{v}_{1}(x){e}^{i\ensuremath{\alpha}y}]$, where ${\ensuremath{\chi}}_{a}(x):=1$ for $x\ensuremath{\in}[0,a]$, ${\ensuremath{\chi}}_{a}(x):=0$ for $x\ensuremath{\notin}[0,a]$,
vj(x) are real or complex-valued functions, ${\ensuremath{\chi}}_{a}(x){v}_{0}(x)$ is an exactly solvable scattering potential in one dimension, and $\ensuremath{\alpha}$ is a positive real parameter. If $\ensuremath{\alpha}$ exceeds the wave number
k of the incident wave, the scattered wave does not depend on the choice of
v1(x). In particular,
v(x,y) is invisible if
v0(x)=0 and $k<\ensuremath{\alpha}$. For $k>\ensuremath{\alpha}$ and ${v}_{1}(x)\ensuremath{\ne}0$, the scattered wave consists of a finite number of coherent plane-wave pairs ${\ensuremath{\psi}}_{n}^{\ifmmode\pm\else\textpm\fi{}}$ with wave vector: ${\mathbf{k}}_{n}=(\ifmmode\pm\else\textpm\fi{}\sqrt{{k}^{2}\ensuremath{-}{[n\ensuremath{\alpha}]}^{2}},n\ensuremath{\alpha})$, where $n=0,1,2,...<k/\ensuremath{\alpha}$. This generalizes to the scattering of wave packets and suggests means for generating quantum states with a quantized component of momentum and pairs of states with an entangled momentum. We examine a realization of these potentials in terms of certain optical slabs. If $k=N\ensuremath{\alpha}$ for some positive integer
N, ${\ensuremath{\psi}}_{N}^{\ifmmode\pm\else\textpm\fi{}}$ coalesce and their amplitude diverge. If
k exceeds $N\ensuremath{\alpha}$ slightly, ${\ensuremath{\psi}}_{N}^{\ifmmode\pm\else\textpm\fi{}}$ have a much larger amplitude than ${\ensuremath{\psi}}_{n}^{\ifmmode\pm\else\textpm\fi{}}$ with
n<N. This marks a resonance effect that arises for the scattered waves whose wave vector makes a small angle with the faces of the slab.
We study numerically the reconnection of quantized vortices and the concurrent acoustic emission by the analysis of the Gross-Pitaevskii equation. Two quantized vortices reconnect following the process similar to classical vortices; they approach, twist themselves locally so that they become anti-parallel at the closest place, reconnect and leave separately.The investigation of the motion of the singular lines where the amplitude of the wave function vanishes in the vortex cores confirms that they follow the above scenario by reconnecting at a point. This reconnection is not contradictory to the Kelvin's circulation theorem, because the potential of the superflow field becomes undefined at the reconnection point. When the locally anti-parallel part of the vortices becomes closer than the healing length, it moves with the velocity comparable to the sound velocity, emits the sound waves and leads to the pair annihilation or reconnection; this phenomena is concerned with the Cherenkov resonance. The vortices are broken up to smaller vortex loops through a series of reconnection, eventually disappearing with the acoustic emission. This may correspond to the final stage of the vortex cascade process proposed by Feynman. The change in energy components, such as the quantum, the compressible and incompressible kinetic energy is analyzed for each dynamics. The propagation of the sound waves not only appears in the profile of the amplitude of the wave function but also affects the field of its phase, transforming the quantum energy due to the vortex cores to the kinetic energy of the phase field.