Variational approach to the Schrödinger equation with a delta-function potential

Abstract

We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form Hg = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction.

Variational approach to the Schrödinger equation with a delta-function potential

“…Although in this case we do not have exact results for comparison, we are confident about the accuracy of the results because the roots of the secular equation converge to a limit from above. The actual eigenvalues given by the Rayleigh-Ritz variational method are available elsewhere [18].…”

“…The basis set chosen is suitable for moderate values of |g| as suggested by the remarkable rate of convergence shown in figures 1 and 2 (see [18] for more explicit results). For large, positive values of g the performance of the variational method is similar because ψ(0) → 0 as g → ∞.…”

“…However, this approach, based on just one trial function with adjustable parameters, only applies to the ground state. On the other hand, the Rayleigh-Ritz method outlined in this paper yields estimates for all the eigenvalues with the advantage that we can monitor the accuracy of the results because the roots of the secular determinant converge to the actual eigenvalues from above [18].…”