Using analogues of familiar image methods in electrostatics and optics, we show how to construct closed-form wave packet solutions of the two-dimensional free-particle Schrödinger equation in geometries restricted by two infinite wall barriers separated by an angle Θ N = π/ N . As an example, we evaluate probability densities and expectation values for a zero-momentum wave packet solution initially localized in a Θ 3 = π/3 = 60° wedge. We review the time development of zero-momentum wave packets placed near a single infinite wall barrier in the appendix.
Using analogues of familiar image methods in electrostatics and optics, we show how to construct closed-form wave packet solutions of the two-dimensional free-particle Schrödinger equation in geometries restricted by two infinite wall barriers separated by an angle Θ N = π/ N . As an example, we evaluate probability densities and expectation values for a zero-momentum wave packet solution initially localized in a Θ 3 = π/3 = 60° wedge. We review the time development of zero-momentum wave packets placed near a single infinite wall barrier in the appendix.
We consider time-dependent Gaussian wave packet solutions of the Schrödinger equation, with arbitrary initial central position, x 0 , and momentum, p 0 , for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets . We show how difference or mirror solutions of the form ψ( x , t ) − ψ(− x , t ) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate ⟨ p 2 ⟩ t explicitly which illustrates how the free-particle kinetic (and hence total energy) is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, ⟨ x ⟩ t , and momentum, ⟨ p ⟩ t , and their relation to the impulsive force during the `collision' with the wall. Finally, the x 0 , p 0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.
We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and Ntoinfty regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wavepackets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasi-classical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
Abstract This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.
We consider the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.
In this work, nondispersive wave packet solutions to linear partial differential equations are investigated. These solutions are characterized by infinite energy content; otherwise, they are continuous, nonsingular and propagate in free space without spreading out. Examples of such solutions are Berry and Balazs’ Airy packet, MacKinnon’s wave packet and Brittingham’s Focus Wave Mode (FWM). It is demonstrated in this thesis that the infinite energy content is not a basic problem per se and that it can be dealt with in two distinct ways. First these wave packets can be used as bases to construct highly localized, slowly decaying, time-limited pulsed solutions. In the case of the FWMs, this path leads to the formulation of the bidirectional representation, a technique that provides the most natural basis for synthesizing Brittingham-like solutions. This representation is used to derive new exact solutions to the 3-D scalar wave equation. It is also applied to problems involving boundaries, in particular to the propagation of a localized pulse in an infinite acoustic waveguide and to the launch ability of such a pulse from the opening of a semi-infinite waveguide. The second approach in dealing with the infinite energy content utilizes the bump-like structure of nondispersive solutions. With an appropriate choice of parameters, these bump fields have very large amplitudes around the centers, in comparison to their tails. In particular, the FWM solutions are used to model massless particles and are capable of providing an interesting interpretation to the results of Young’s two slit experiment and to the wave-particle duality of light. The bidirectional representation provides, also, a systematic way of deriving packet solutions to the Klein-Gordon, the Schrodinger and the Dirac equations. Nondispersive solutions of the former two equations are compared to previously derived ones, e.g., the Airy packet and MacKinnon's wave packet.