Multiple approaches to incorporating scattering states in non-degenerate perturbation theory

The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square well with a perturbation is used to illustrate the importance of including scattering states into the calculation. The second-order correction to the ground-state energy is obtained in three distinct ways. The first involves altering the problem by imposing additional boundary conditions at large distances to make the positive-energy spectrum discrete. The second makes use of the continuum scattering states directly. The third bypasses the use of scattering states by solving a differential equation for the first-order correction to the wave function.