An interesting limiting case of the finite square well discussed in Section 6.8 is the case where the well depth approaches infinity but the width of the well goes to zero such that V0a remains constant. Such a well may be represented by the potential energy satisfying
ℏ22mV(x)=−bλδ(x)
where δ(x) is the Dirac delta function. Note that λ/b is a constant having the units of inverse length and we have taken the top of the well to be at V=0.
(a) Show by integrating the time-independent Schr\"odinger equation that the derivative of the energy eigenfunction is not continuous at the origin, but satisfies
(dxdψ)0+−(dxdψ)0−=−bλψ(0)
(b) Determine the energy eigenvalue(s) for this well. Sketch the energy eigenfunction(s).
Suggestion: Solve the Schr\"odinger equation for E<0 in the regions x<0 and x>0, where V=0, and join the solutions together, making sure that the boundary conditions on continuity of the wave function and discontinuity of the derivative of the wave function at x=0 are satisfied.