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.