*Due October 1, 11:00 am.*

1. (a) For the finite square well potential *V* = 0 for
*x* <
0, *V* = -*V*_{0}
for 0 < *x*
< *L*, *V* = 0 for *L* < *x*, prove that
the transmission amplitude *S*(*k*) for an electron coming in from
the left with momentum *k* is

_{}

where
*k*_{1} is the momentum inside the well. Compare this with the
transmission through a square *barrier* in http://www.phys.virginia.edu/classes/751.mf1i.fall02/OneDimSchr.htm and comment on the similarities and
differences.

(b) Sketch the probability of transmission carefully as a
function of energy, or plot it with *Maple* or *Mathematica*. For
what energies is there perfect transmission?
Can you give any physical explanation?

(c) Now regard the incoming momentum *k* as a *complex*
variable. Note that *S*(*k*) become *infinite* when

_{}

and
use the formula _{} to show that this is
equivalent to the formulas for bound states in the finite square well. Give a physical interpretation of why an
infinite value of *S*(*k*) would correspond to a bound state.

**Shankar problems: 5.2.1, 5.2.2.**