8.421 Homework Assignment #1
Spring 2012, Prof. Wolfgang Ketterle
Due Wednesday, February 22, 2012
1. [4 points] When driven far from resonance the power dissipated in a mechanical (classical)
damped oscillator increases linearly with the damping γ, but on resonance varies as γ−1. Why
does reducing the damping increase the power dissipated on resonance?
2. [8 points] In this problem we want to study the time evolution of a system with a Hamiltonian
h¯
H=−~·~
µ B =.
2 ω0ωRe−iωt
(1)
ωReiωt −ω0!
This Hamiltonian corresponds to a magnetic moment ~µ in a combination of static and rotating
~
fields B(t) = −(B1cos ωt, B1sin ωt, B0). Here ω0=γB0and ωR=γB1are the Larmor and Rabi
frequency associated with the static field B0and
the rotating field of magnitude B1, respectively,
1! 0
and γis the gyromagnetic ratio. The basis is = |↓i ≡ |ei,!= ,
0|↑i g
1≡ | i where
|↑i,|↓i are the states where ~µ is oriented along the ∓zaxis. The time evolution of any state
|ψ(t)i=ag(t)|gi+ae(t)|eiis determined by the two coefficients ag(t), ae(t).
a.) Find the equations of motion for the system, i.e. derive explicit expressions for a˙g(t) and
a˙e(t).
b.) Solve the equations of motion and find ag(t) and ae(t) in terms of ag(0) and ae(0).
c.) Given the initial conditions ag(0) = 1 and ae(0) = 0 show that the probability to find the
system in the state |eiagrees with the classical result.
3. [8 points] Now we want to analyze the Hamiltonian of problem 2 in the density matrix
formalism. Parameterize Has H=¯h
2[V1b
σx+V2b
σy+ω0b
σz], and the density matrix as ρ=
1[r01
b+r1σ
bx+r2σ
by+r3σ
bz], where 1
bis the unity matrix, and
2
σ
bx= 0 1 !, σ
bx= 0−i!1 0
, σ
bz=,
1 0 i0 0−1!(2)
are the Pauli spin matrices.
Employing the von Neumann equation ih¯ρ˙ = [H , ρ] show that ~r =r1xˆ + r2yˆ + r3zˆ obeys the
relation d~r
dt =~ω ×~r with ~ω =V1ˆx+V2ˆy+ω0ˆz. Can you interpret this result?
