A peculiar property of the wavefunctions for confining potentials

Jorge Munzenmayer, Derek Frydel

This article examines the suggestion made in Ref. [EPL, 115 (2016) 60001] that a solution to a particle in an infinite spherical well model, if it is square-integrable, is a physically valid solution, even if at the precise location of the singularity there is no underlying physical cause, therefore, the divergence would have to be a nonlocal phenomenon caused by confining walls at a distance. In this work we examine this claim more carefully. By identifying the correct differential equation for a divergent square-integrable solution and rewriting it in the form of the Schroedinger equation, we infer that the divergent wavefunction would be caused by the potential V(r)=-r delta(r), which is a kind of attractive delta potential. Because of its peculiar form and the fact that it leads to a divergent potential energy = - infinity, the potential V(r) and the divergent wavefunction associated with it are not physically meaningful.

R.J. Lombard, J. Mareš

M. Andrews

Tovie Leon Rome

In studying the diamagnetism of free electrons in a uniform magnetic field it was found that reducing the field to zero in the wavefunction did not yield the experimentally indicated free particle plane wave wavefunction. However, solving the Schroedinger Equation resulting from setting the field equal to zero in the original equation did yield a plane wave wavefunction. This paradox was not found to be peculiar to the case of a charged particle in a uniform magnetic field but was found to occur in a number of other systems. In order to gain an understanding of this unexpected behavior, the following systems were analyzed: the one-dimensional square well potential; a charged, spinless particle in a Coulomb field and in a uniform electric field; a one-dimensional harmonic oscillator; and a charged, spinless particle in a uniform magnetic field. From these studies the following were obtained: conditions for determining the result of reducing the potential in a wavefunction; the condition under which the potential of a system may be switched off while maintaining the energy of the system constant; the relationship between the result of physically switching off a potential, the result of reducing it in the wavefunction, and the solution of the Schroedinger Equation obtained by decreasing the potential to zero in the original wave equation; and a general property of any wavefunction with respect to reducing any parameter within this wavefunction.

Alberto Alonso y Coria

Pier Paolo Marchisio, J. P. Coe, Irene D’Amico

We study the ground state entanglement, energy and fidelities of a two-electron system bounded by a core-shell potential, where the core width is varied continuously until it eventually vanishes. This simple system displays a rich and complex behavior: as the core width is varied, this system is characterized by two peculiar transitions where, for different reasons, it displays characteristics similar to a few-particle quantum phase transition. The first occurrence corresponds to something akin to a second order quantum phase transition, while the second transition is marked by a discontinuity, with respect to the driving parameter, in the first derivatives of quantities like energy and entanglement. The study of this system allows to shed light on the sudden variation of entanglement and energy observed in S. Abdullah et al. [Phys. Rev. B 80, 235302 (2009)]. We also compare the core-shell system with a system where a core well is absent: this shows that, even when extremely narrow, the core well has a relevant `pinning' effect. Interestingly, depending on the potential symmetry, the pinning of the wavefunction may either halve or double the system entanglement (with respect to the no-core-well system) when the ground state is already bounded to the outer (shell) well. In the process we discuss the system fidelity and show the usefulness of considering the particle density fidelity as opposed to the more commonly used -- but much more difficult to access -- wavefunction fidelity. In particular we demonstrate that -- for ground-states with nodeless spatial wavefunctions -- the particle density fidelity is zero if and only if the wavefunction fidelity is zero.