Single Dirac Potential Well

Now that we have built up the architecture for dealing with different kinds of wells, let’s set up the problem of bound states of a Dirac Potential Well.

Consider a potential that is given by:

\[\displaystyle V(x) = \begin{cases} 0 & \text{if $x < 0$} \\ -\beta \delta(x) & \text{if $x=0$} \\ 0 & \text{if $x > 0$} \end{cases}\]
  1. Question: Sketch the potential well (note the negative sign on the $\delta$ function) and write down the eigenvalue equation for locations away from $x=0$.
    • Discussion: We are seeking bound state solutions. What can you say about the sign of any expected eigenenergies? What does that indicate for the sign of $q$ in a definition of $q=\sqrt{\dfrac{-2mE}{\hbar^2}}$?
  2. Question: With that choice of $q$, what do the general solutions look like at away from $x=0$?
    • Discussion: Can you argue anything about the value of some of the undetermined coefficients?
  3. Question: What boundary conditions are appropriate here? Can you write them? No need to solve anything yet.
    • Discussion: The potential goes to infinity at $x=0$, what does that imply about your boundary conditions?