The quantum harmonic oscillator with λδ' (x) potential

Abstract

In this work, the problem of the quantum harmonic oscillator with delta derivative
0potential λδ' (x), where λ is a coupling constant, is solved using the Green’s func-
tions technique. A transcendental equation that governs the energy eigenvalues of
the problems for each coupling constant is obtained. The eigenfunctions and their
first derivative are proved to be discontinuous at the origin. The values of the dis-
continuities jumps are found to agree with the requirements of having a self-adjoint
extension Hamiltonian. In the large coupling limit, the even energy eigenvalues and
eigenfunctions for the quantum harmonic oscillator are annihilated, and only the
odd parts survive. The dependence of the energy eigenvalues and eigenfunctions on
the sign of λ was made clear. A mapping between the sign of λ and the positions
of the particle was used to explain the discontinuity of the solution. In the large
λ regime, an educated guess for the wave functions was proposed. The proposed
solutions led to the correct energy eigenvalues and obeyed the required conditions
to have a self-adjoint extension Hamiltonian.