[NOTES/QM-13009] Delta Function Potential --- An overview

We can obtain the solutions to the energy eigenvalue problem for the Dirac $\delta$ function potential by the following three methods \begin{equation} V(x) = - g \delta(x) \end{equation}
where $g>0$ is a constant. The three methods are

1.  Dirac delta function potential as a limit of square well potential.
2.  Solution of the eigenvalue problem by direct integration of the Schr\"{o}dinger equation.
3.  Solution of the eigenvalue problem in momentum space.

The bound state exists only for $E<0$. It will be shown that there is only one bound state with energy level given by \begin{equation} E= -\frac{mg^2}{\hbar^2}. \end{equation}

For $E> 0$ the eigenvalues will be seen to be continuous and doubly degenerate.  All the three methods to solve for eigenfunctions  have  the same steps as for the bound states.