Homework # 3 - The Astro Home Page

Department of Physics
Temple University
Atomic, Nuclear and Particle Physics, 4702
Instructor: Z.-E. Meziani
Homework # 3
Wednesday September 24, 2014
Due Thursday October 02, 2014
Problem 1. (10pts)
We wish to study the phase shift δ1 (k) produced by a hard sphere on the p wave (l=1). In particular, we
want to verify that it becomes negligible compared to δ0 (k) at low energy.
a) Write the radial solution for the function Uk,1 (r) for r > r0 . Show that its general solution is of the
form:
cos kr
sin kr
uk,1 (r) = C
− cos kr + a
+ sin kr
(1)
kr
kr
where C and a are constants.
b) Show that the definition of δ1 (k) implies that: a = tan δ1 (k)
c) Determine the constant a from the condition imposed on uk,1 at r = r0 .
d) Show that, as k approaches zero, δ1 (k) behaves like (kr0 )3 , which makes it negligible compared to δ0 (k)
Problem 2. (10pts)
Consider a one dimensional harmonic oscillator of mass m, angular frequency ω0 and charge q. Let |φn >
and En = (n + 1/2)¯hω0 be the eigenstates and eigenvalues of its Hamiltonian H0 . For t < 0, the oscillator
is in the ground state |φ0 >. At t = 0, it is subjected to an electric field “pulse” of duration
tau. The corresponding perturbation can be written
W (t)
=
−qEX
for
0≤t≤τ
W (t)
=
0
for
t < 0 and t > τ
and
(2)
(3)
E is the field amplitude and X is the position observable. Let P01 be the probability of finding the oscillator
in the state |φn > after the pulse.
a) Calculate P01 by using first-order time dependent perturbation theory. How does P01 varies with τ ,
for fixed ω0 ?
b) Show that, to obtain P02 , the time-dependent perturbation theory calculation must be pursued at least
to second order.
Problem 3. (10pts)
~ˆ1 and S
~ˆ2 coupled by an interaction of the form a(t)S
~ˆ1 · S
~ˆ2 ; a(t) is a function
Consider two spins 1/2’s, S
of time which approaches zero when |t| approaches infinity, and takes a non-negligible values (of the order
of a0 ) only inside an interval, whose width is of the order of τ , about t = 0
a) At t = −∞, the system is in the state |+, − > (and eigenstate of Sˆ1z and Sˆ2z with the eigenvalues
+¯h/2 and −¯
h/2). Calculate, without approximations, the state of the system at t = +∞. Show that
the probability P(+− → −+) of finding, at t = +∞, the system in the state |+, − > depends only on
the integral
Z
+∞
a(t)dt.
−∞
1
b) Calculate P(+− → −+) by using first-order time-dependent perturbation theory. Discuss the validity
conditions for such an approximation by comparing the results obtained with those of the preceding
question.
~ 0 parallel to Oz. The
c) Now assume that the two spins are also interacting with a static magnetic field B
corresponding Zeeman Hamiltonian can be written:
H0 = −B0 (γ1 S1z + γ2 S2z )
(4)
where γ1 and γ2 are the gyromagnetic ratios of the two spins, assumed to be different. Assume that
2
2
a(t) = a0 e−t /τ . Calculate P(+− → −+) by first order time dependent perturbation theory. With
fixed a0 and τ , discuss the variations of P(+− → −+) with respect to B0
2

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