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American Journal of Physics

Quantum solutions for the delta ring and delta shell

Luis F. Castillo-Sánchez, Julio C. Gutiérrez-Vega

Published Jun 20, 2025• 2 citations

Abstract

The quantum description of a free particle subject to a delta circular ring and a delta spherical shell is studied. We determine the bound eigenstates and energy spectra for different angular momentum states. The normalization constants of the bound states and the transmission and reflection coefficients of the scattering states are calculated in closed form. The ground state of the delta ring is always present regardless of the potential strength. In contrast, the delta shell requires a threshold of the delta strength for the first eigenstate to appear. For the ring and the shell, the wave functions tend to be degenerate at high energies, and the effect of angular momentum on the wave functions becomes negligible. This work generalizes the standard problem of the one-dimensional delta potential studied in textbooks.

DOI: 10.1119/5.0256321

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The quantum description of a free particle subject to a delta circular ring and a delta spherical shell is studied. We determine the bound eigenstates and energy spectra for different angular momentum states. The normalization constants of the bound states and the transmission and reflection coefficients of the scattering states are calculated in closed form. The ground state of the delta ring is always present regardless of the potential strength. In contrast, the delta shell requires a threshold of the delta strength for the first eigenstate to appear. For the ring and the shell, the wave functions tend to be degenerate at high energies, and the effect of angular momentum on the wave functions becomes negligible. This work generalizes the standard problem of the one-dimensional delta potential studied in textbooks.

The problem of a spin-free electron with mass

m

, charge

e

confined onto a ring of radius

R_0

and with an attractive Dirac delta potential with scaling factor (depth)

κ

in non-relativistic theory has closed form analytical solutions. The single bound state function is of the form of a hyperbolic cosine that however contains a parameter

d>0

which is the single positive real solution of the transcendental equation

\coth(d) = λd

for non zero real

λ=\frac{2}{πκ}

. The energy eigenvalue of the bound state

\varepsilon=-\frac{d^2}{2π^2}\approx \frac{q e m R_0}{2 \hbar^2}

. In addition a discretly infinite set of unbounded solutions exists, formally these solutions are obtained from the terms for the bound solution by substituting $d \to i d $ yielding

\cot(d) = λd

as characteristic equation with the corresponding set of solutions

d_k, k\in\mathbb{N}

, the respective state functions can be obtained via

\cosh(x)\overset{x \to i x}{\longrightarrow}\cos(x)

.

The exact solution to the one-dimensional Schr\"odinger equation which describes the motion of a particle scattering off a finite set of $\ensuremath{\delta}$-function potentials of arbitrary strengths and positions is found. The behavior of the

T

matrix both on and off the energy shell is also precisely given. Localization of states is discussed.

The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.

We derive a generalized zero-range pseudopotential applicable to all partial wave solutions to the Schrödinger equation based on a delta-shell potential in the limit that the shell radius approaches zero. This properly models all higher order multipole moments not accounted for with a monopolar delta function at the origin, as used in the familiar Fermi pseudopotential for s-wave scattering. By making the strength of the potential energy dependent, we derive self-consistent solutions for the entire energy spectrum of the realistic potential. We apply this to study two particles in an isotropic harmonic trap, interacting through a central potential, and derive analytic expressions for the energy eigenstates and eigenvalues.