Homework 1 Solutions

We now simply integrate (using Mathematica or a book of integrals or knowldege
about the Fourier transform of a Gaussian distribution)
∫
ρ(t) = p(ξ)( 1 2 I + 1 2 (cos ω(1 + ξ/B 0)tσ x − sin ω(1 + ξ/B 0 )tσ y ))dξ (19)
= 1 2 I + 1 ∫
p(ξ)(cos ω(1 + ξ/B 0 )tσ x − sin ω(1 + ξ/B 0 )tσ y )dξ (20)
2
= 1 2 I + 1 2 exp(−(ωξ 0t) 2
)(cos ωtσ x − sin ωtσ y ) (21)
2B 2 0
The 〈σ x (t)〉 = exp(− (ωξ 0t) 2
) cos ωt.
2B0
2
Note that ρ(t) rapidly approaches the totally mixed state, ρ(∞) = 1 I, with the
2
rate determined by the ratio of the width of the inhomogeneity ξ 0 and the applied
field B 0 . We will see later that NMR is possible because B 0 is much larger than ξ 0 .
2 Density Matrices and Harmonic Oscillators
Alice and Bob are trying to build a device that generates I 2 in the ground electronic
state, a translational velocity of 300 m/s, and a vibrational state of |ψ〉 = 1/2 |0〉 +
√
1/2 |1〉 + 1/2 |3〉. (For this problem, assume the molecule has a classical velocity
along ẑ, and the molecular axis is fixed to point along ˆx. We can then treat I 2 as
a 1d harmonic oscillator). Alice builds three boxes and Bob sets up a detector z=1
meter away. Bob’s detector can measure the velocity of the molecule, the number
state, and 〈X〉 and 〈P x 〉. (Hint: Write ˆX and ˆP x in terms of a † and a).
Alice makes three boxes that satisfy 〈N〉=1.25 (where ˆN = a † a) and the translational
velocity of I 2 is 300 m/s.
1. a. For |ψ〉 what is 〈X〉
b. If one of Alice’s boxes has produced |ψ〉 at the box, how does 〈X〉 change with
the distance to Bob’s detector How does 〈P x 〉 change
2.1 Solution
Remember that ˆX =
√ ¯h
2MΩ (a† + a) = x 0 (a † + a) Define |ψ ′ 〉 = a |ψ〉
First, we calculate 〈X〉
〈X〉 = 〈ψ| ˆX |ψ〉
= x 0 〈ψ| (a † + a) |ψ〉
4