A. Suppose a particle starts out (t = 0) in the ground state of the initial well:
\ll(x, 0) = l sin (~x).
Show that the expansion coefficients can be written in the form
21rr . , Cn = - e-,c, z- sin(nz) sin(z) dz,
l[ 0
[10.5]
where a = mva/ 2n2n is a dimensionless measure of the speed with which
the well expands. (Unfortunately, this integral cannot be evaluated in terms
of elementary functions.)

B. Suppose we allow the well to expand to twice its original width, so the
"external" time is given by w(Te) = 2a. The "internal" time is the period of
the time-dependent exponential factor in the (initial) ground state. Determine
Te and T;, and show that the adiabatic regime corresponds to a « 1, so that
exp(-iaz2) ~ 1 over the domain of integration. Use this to determine the
expansion coefficients, en. Construct \ll(x , t), and confirm that it is consistent
with the adiabatic theorem