2.1 Infinite square well with delta function perturbation
Consider the infinite square well problem with walls at 0 and $L$. Imagine that we place a delta
function perturbation at the center of the well of the form:
$H' = \alpha \delta(x - L/2)$
(1)
(a) Use first-order non-degenerate perturbation theory to compute the first order corrections to
the energies to all energy levels.
(b) In part (a), you should have found that some states do not shift in energy to first order. Draw
the probability density, $|\psi_n^2(x)|$, for the $n = 1$, $n = 2$, $n = 3$, $n = 4$ and $n = 5$ energy levels without
the perturbation. From these sketches, explain why, physically, some states did not exhibit an
energy shift to first order.
(c) Similarly, can you explain why the delta function potential does induce an energy shift for the
other states?
(d) Where would you put the delta function to produce the maximum first order energy correction
for the first three states that exhibit zero energy shift?
(e) How does the first-order correction vary as you move the delta function across the well starting
at $x = 0$ and moving to $x = L$? Sketch the behavior of the first order correction versus the position
of the delta function for the $n = 2$ state.
