Revisiting the bound states of a confined delta potential

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.