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

Richard L. Hall, Alexandra Lemus Rodríguez

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
Spatial confinement of quantum systems has been used to model atoms under high pressure, quantum dots, and Bose–Einstein condensates. Spherical boxes are commonly employed, yet they are energy-minimizing, and other box geometries will be more relevant for some applications. We study generic single-particle quantum systems comprising a central physical potential confined within a three-dimensional infinite potential well corresponding to an impenetrable box having prescribed geometric properties; we refer to this as geometric confinement. There is a marked distortion of the wavefunction and energy spectrum, relative to the standard unconfined system, when a confining box is made small enough. We employ asymptotic analysis in combination with numerical simulations to better understand this behaviour, determining lengthscales on which each of the physical potential and confining box respectively dominate the structure of quantum states, with Padé approximants employed to connect asymptotic scalings between these two regimes. This combination of methods permits us to determine, in a fairly general manner, how and to what extent box size and shape modify a given single-particle quantum system held under geometric confinement. We provide a variety of examples of single-particle central potentials and confining geometries to illustrate our methods and motivate the theoretical findings.
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.
We describe a multi-scale resolution approach to analyzing problems in Quantum Mechanics using Daubechies wavelet basis. The expansion of the wavefunction of the quantum system in this basis allows a natural interpretation of each basis function as a quantum fluctuation of a specific resolution at a particular location. The Hamiltonian matrix constructed in this basis describes couplings between different length scales and thus allows for intuitive volume and resolution truncation. In quantum mechanical problems with a natural length scale, one can get approximate solution of the problem through simple matrix diagonalization. We illustrate this approach using the example of the standard quantum mechanical simple harmonic oscillator.