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arXiv:1212.4516 · Advances in Mathematical Physics

Wide Effectiveness of a Sine Basis for Quantum-Mechanical Problems in d Dimensions

Richard L. Hall, Alexandra Lemus Rodríguez

Published Jan 1, 2013• 3 citations
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Abstract

It is shown that the spanning set for L 2 ( [ 0,1 ] ) provided by the eigenfunctions { 2 sin ( n π x ) } n = 1 ∞ of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to [ a , b ] , where a and b are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in R d turns out to be appropriate also for problems that are softly confined by U -shaped potentials, including those with strong singularities at r = 0 . Specific examples are discussed in detail, along with some bound N -boson systems.

DOI
10.1155/2013/258203

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It is shown that the spanning set for L 2 ( [ 0,1 ] ) provided by the eigenfunctions { 2 sin ( n π x ) } n = 1 ∞ of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to [ a , b ] , where a and b are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in R d turns out to be appropriate also for problems that are softly confined by U -shaped potentials, including those with strong singularities at r = 0 . Specific examples are discussed in detail, along with some bound N -boson systems.
It is shown that the spanning set for provided by the eigenfunctions of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to , where and are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in turns out to be appropriate also for problems that are softly confined by U -shaped potentials, including those with strong singularities at . Specific examples are discussed in detail, along with some bound -boson systems.
The particle-in-the-box, with or without an additional potential, is proposed as an excellent laboratory to teach and explore the details of the linear variational method using a non-orthogonal basis. The x n ( a − x ) n and x n ( a /2 − x )( a − x ) n polynomials are shown to form a complete basis for the even and odd states, respectively, of the particle confined to the x ∊ [0, a ] interval. A short and simple Octave code is presented as the natural extension to the hand calculations when the basis set grows in size.
Exact eigenfunctions for a particle in a box are obtained using Gaussian wavepacket dynamics. The eigenfunctions are obtained by propagating, without approximation, an infinite set of Gaussian wavepackets that collectively satisfy the boundary conditions of the problem, being coherent states appropriate to this problem. The method of images is applied to enforce these boundary conditions. This technique may be applied to the quantum billiard problem whenever the particle is confined to any open or closed region that tessellates space, regardless of the dimension of the region. Also, it is shown that the use of frozen Gaussians along with the De Leon-Heller spectral quantisation method gives the exact solution for the one-dimensional problem as well as for the above multi-dimensional problems, provided the components of the momentum of the wavepackets are chosen at random.
Many-particle systems pose commonly known computational challenges in quantum theory. The obstacles arise from the difficulty in finding sets of eigenvalues and eigenvectors of the underlying Hamiltonian while enforcing fermion or boson statistics, not to mention the prohibitive increase in the computational cost with the system’s size. The first obvious step in this direction is to elaborate the theory for Fermi or Bose gases without inter-particle interactions. The traditional approach to the work is with the ideal gases confined in a cubic container with impenetrable walls (in arbitrary dimensions). This approach allows one to find the particle’s spectra and compute all thermodynamic quantities of the confined gas. In the present work, we consider the gas confined in a spherical container (in other words, an infinitely deep spherical potential well in D dimensions), solving the corresponding Schrödinger equation using zero boundary conditions. We address the case of a finite number of particles N, either bosons or fermions, in the spherical potential box, as well as the thermodynamic limit, corresponding to N → ∞. Owing to Weyl’s relations, in the latter limit, the results do not depend on the shape of the box and thus approach the commonly known ones valid in the infinite space. Owing to the underlying SO(D) symmetry, we are dealing with particles carrying a well-defined angular momentum that, together with sorted energy eigenvalues, imparts a shell structure to the system.
Trajectories of a Bohmian particle confined in time-dependent cylindrical and spherical traps are computed for both contracting and expanding boxes. Quantum effective force is considered in arbitrary directions. It is seen that in contrast to the problem of a particle in an infinite rectangular box with one wall in motion, if particle initially is in an energy eigenstate of a tiny box the force is zero in all directions. Trajectories of a two-body system confined in the spherical trap are also computed for different statistics. Computations show that there are situations for which distance between bosons is greater than the fermions. However, results of average separation of the particles confirm our expectation about the statistics.