PHYS 401 Homework---Due November 29
1. Consider scattering from a repulsive delta function potential of the
form )()( xgxV
δ
=
. One way to view this problem is as the limit of a
finite potential such as a square well. Compute R and T as a function
of energy for this problem by taking the analogous expressions for the
finite square barrier and studying them in the limit 0
→
a with
Vag 2
=
(where V) is the height of the barrier.
2. Consider the same problem as in 1. However, I want it solved by a
completely different means: namely by integrating the Schrodinger
equation from in –ε to ε and looking at the limit 0
→
ε
(as we did in
class for the bound states).
3. Consider a periodic potential )()( axVxV
+
=
. The energy eigenstates
satisfy Bloch states theorem )()( xuex ikx
=
ψ
with )()( xudxu
=
+
. Thus
the energy eigenstates cannot be normalized. Never-the-less,
expectation values of a sort can be defined.
ψψ
ψψ
−
−
=2/
2/
*
2/
2/
*ˆ
d
d
d
d
dx
Adx
A.
Where
A
ˆ is the explicit differential form of the operator.
a. Show that if u(x) is real that .kp =
b. Given the definition for the expectation value is the average
value for momentum you would expect if you made many
measurements of systems in this state?
c. Clearly 2
x does not correspond to the the average value for
2
xyou would expect if you made many measurements as that
value is infinite. What does 2
x represent physically?