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:

V(x)={0if x<0βδ(x)if x=00if x>0
  1. Question: Sketch the potential well (note the negative sign on the δ 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=2mE2?
  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?