PHYS 402 Homework---Due Friday April 8
This homework assignment concerns a spin ½ system. The Hamiltonian for this system is of the form
xz tfH
σσ
ˆ
)(
ˆ
ˆ2+= Ω. Where f(t) is some time dependent function. This system can be realized in the lab by putting
the spin in a constant magnet field in the z direction and a time dependent one in the x direction. We will work in the
basis in of eigenstates of z
σ
ˆ. At t=0 the system in the spin up state, i.e ↑.
1. Consider the case where
α
θ
θ
)()()( tTttf
−
=
. That is the perturbation is of constant strength
α
for 0<t<T and
zero elsewhere.
a. Use first order perturbation theory to compute the state function for 0<t<T.
b. Compute the probability of finding the particle in the down state ( ↓) as a function of time.
c. From the form of this answer find an expression for the regime in which one expects perturbation theory
to be valid. Express this in terms of T,
α
, and Ω.
2. The preceding problem can be solved exactly: it is a precession problem similar to those we have considered
before.
a. Find the exact expression for the state as a function of time.
b. Expand the exact solution as a Taylor series in α and show it yields the perturbative result.
3. Consider the case where
α
θ
θ
)()()( tTttf
−
=
t/T. That is the perturbation is of strength
α
t/T for 0<t<T and zero
elsewhere.
a. Use first order perturbation theory to compute the state function for 0<t<T.
b. Compute the probability of finding the particle in the down state ( ↓) as a function of time.
c. From the form of this answer find an expression for the regime in which one expects perturbation theory
to be valid. Express this in terms of T,
α
, and Ω.
4. Consider the case where
α
θ
θ
)()()( tTttf
−
=
sin(ωt). That is the perturbation is of strength sin(ωt). for 0<t<T
and zero elsewhere.
a. Use first order perturbation theory to compute the state function for 0<t<T.
b. Compute the probability of finding the particle in the down state ( ↓) as a function of time.
c. From the form of this answer find an expression for the regime in which one expects perturbation theory
to be valid. Express this in terms of T,
α
,
ω
and Ω.
5. Consider the case in problem 1) .
a. Compute the state of the system to second order in perturbation theory.
b. Compute the probability of finding the particle in the down state ( ↓) as a function of time.
c. Verify that the exact solution expanded to second order in a gives this result.
