PHY 4605
March 23, 2012
Exam #2
Popilius pro cetera asperitate animi uirga, quam in manu gerebat,
circumscripsit regem ac ’priusquam hoc circulo excedas’ inquit
’redde responsum, senatui quod referam.’ obstupefactus tam uiolento
imperio parumper cum haesitasset, ’faciam’ inquit ’quod censet
senatus.’ tum demum Popilius dextram regi tamquam socio atque
amico porrexit.
Livy XLV.12
(1) Consider a one-dimensional particle of mass m in the potential,
2 2
V (x) = − h̄ma sech2 (ax) .
a) What is the form of the ground state energy from dimensional analysis? (15
points)
b) What is the ground state energy in the WKB approximation? (35 points) The
following indefinite integral might be useful:
√
Z
q
√
z−u
−1
dz z−u = − tan−1 (1−z)−(z−u)
√
√
√ √
− 2 u tan
.
z
1−z
2 1−z z−u
u 1−z
(2) A particle starts out (at time t = 0) in the N th state of an infinite square well. Now
the “floor” of the well rises so that the potential inside is uniform but time dependent:
V0 (t).
a) What are the various cm (t) to first order in perturbation theory? (25 points)
b) What are the various cm (t) exactly? (25 points)