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European Journal of Physics

Effect of infinite discontinuities on the motion of a particle in one dimension

U. Oseguera

Published Jan 1, 1990• 9 citations
Publisher

Abstract

The author solves the Schrodinger equation for an attractive delta function potential flanked by infinite walls, and then considers the limit as the intensity of the delta potential becomes infinite. The states apparently become degenerate in the process, the degeneracy being of the same type as the one that appears in the one-dimensional Coulomb potential. The author finds a way to decide whether the degeneracy is real.

DOI
10.1088/0143-0807/11/1/006

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The author solves the Schrodinger equation for an attractive delta function potential flanked by infinite walls, and then considers the limit as the intensity of the delta potential becomes infinite. The states apparently become degenerate in the process, the degeneracy being of the same type as the one that appears in the one-dimensional Coulomb potential. The author finds a way to decide whether the degeneracy is real.
The quantum mechanical problem of a particle moving in one dimension (1D) under the influence of a singular potential is reexamined. An attractive delta function potential of infinite strength and the 1D Coulomb potential is considered in particular. In both cases the singularity acts as an impenetrable barrier, thus dividing space into two independent regions. Analysis of the scattering states gives results in agreement with this conclusion. The implications that the bound states of the 1D Coulomb potential cannot be classified according to parity, even though the potential is symmetric, is also discussed herein.
The Schrödinger equation for a particle in the V-shaped potential decorated by a repulsive or attractive Dirac delta function interaction at the center is solved, demonstrating the crucial influence of point interaction on the even-parity states of the original system without decoration. As strength of the attraction increases, the ground state energy falls down without limit; and in limit of infinitely large attraction, the ground state approaches a singular state. Our analysis and conclusion can be readily generalized to any one-dimensional system a particle interacts with symmetrical potential plus the Dirac delta function interaction at the center.
Abstract The stationary states of a particle under the influence of a delta potential confined by impenetrable walls are investigated using the method of expansion in orthogonal functions. The eigenfunctions of the time-independent Schrödinger equation are expressed in closed form by using a pair of closed-form expressions for series available in the literature. The analysis encompasses both attractive and repulsive potentials with arbitrary couplings. Confinement significantly impacts the quantum states and introduces a scenario of double degeneracy including the ground state. Analysis extends to discuss the transition to unconfinement. This research holds particular significance for educators and students engaged in mathematical methods applied to physics and quantum mechanics within undergraduate courses, offering valuable insights into the complex relationships among profiles of potentials, boundary conditions, and the resulting quantum phenomena.
A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.
A simple exact solution of a one-dimensional Schr\"odinger equation which describes the motion of a particle in a continuous potential V(x)V(x)V(x) and scattering from a finite set of $\ensuremath{\delta}$-function potentials of arbitrary positions and strengths is found.