How to handle the bound state within a Dirac-Potential well?
- Boundedness: your bound state shall not blow up at
; - For a Dirac-like potential function e.g.
, your bound state must have a negative energy to be trapped in the potential well; - Continuity: your bound state must have a continuous wave function (e.g.
); - Symmetricity: your wave function has the same symmetricity as the potential function, which helps you reduce the total number of unknown constants (e.g. The w.f. should be odd or even w.r.t.
for our given potential. This gives rise to the or form of our w.f. in the region ); - Schrödinger equation must hold true under integral. For simplicity we do the integral in infinitesimally small intervals like
.
