A lot of books find the bound state energies of the single delta function potential centred at $x=0$ by integrating the Schrodinger equation around $x=0$, using symmetric limits and making the width zero.

By heuristic argument it should not be equal to zero as the function is even.



Please, let me know what is wrong or if there is some technicality with respect to theory of distributions playing around here.

  • $\begingroup$ You are integrating over the whole space whereas you need to integrate from $-\epsilon$ to $\epsilon$ about the discontinuity so that, with $\epsilon\to 0$, you capture the change in $\psi'$ across that discontinuity. See en.wikipedia.org/wiki/Delta_potential $\endgroup$ – ZeroTheHero Sep 27 '17 at 7:25
  • $\begingroup$ What exactly is your question? $\endgroup$ – By Symmetry Sep 27 '17 at 12:23

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