Day 7 In-Class Activity

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:

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 $q=\sqrt{{\displaystyle \frac{-2mE}{{\hslash }^2}}}$?
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?