Square well: negative kinetic energy.

    Here is one last problem. This one, I think, is better (more relevant to what we would wish to understand about quantum) than 3.2e.
     About the title, kinetic energy cannot be negative. However, contributions to the KE from some regions of space can be negative. (This came up in class I think when we discussed the H atom problem a bit at the beginning and some people were getting a negative KE.) These negative contributions to the KE turn out to be important to understanding band states and solid state physics, so I thought I would add one problem related to that.
     This problem elucidates where negative KE contributions occur. The bottom line is that they occur where the curvature of the wave-function is upward. For example: that is no where for the infinite square well (the curvature is always downward); outside the well for the finite square well; and in the region where \(E_n \gt \frac{1}{2} k x^2\) for the 1D HO. Basically where the electron wave-function extends into a so called "forbidden region".
      3.X ) Consider a 1D finite square well that is 1 eV deep and L= 0.613 nm wide. For this well I believe that for the ground state: \(k= .595 \pi/L\), a = .27 nm, B/A=.594.
a) From normalization calculate B and A in units of nm^-1/2.
b) What fraction of the normalization integral comes from outside the well?
c) Calculate the expectation value of the KE. (You can assume that outside left is the same as outside right and do just two integrals, one for inside, one for outside.)
d) What is the KE contribution from the region of space outside the well? What is the total KE?
e) Show that the PE is the same as the fraction of the normalization integral inside the well times -1 eV.
f) What is the total energy?
g) Discuss the kinetic energy integrand and mathematically why it is positive inside the well and negative outside.