A spin-1/2 particle with gyromagnetic ratio , at rest in a static magnetic field B 0 k,

A spin-1/2 particle with gyromagnetic ratio γ, at rest in a static magnetic field B₀k̂, precesses at the Larmor frequency ω₀ = ϒB₀ (Example 4.3). Now we turn on a small transverse radiofrequency

so that the total field is

(a) Construct the 2x2 Hamiltonian matrix (Equation 4.158) for this system.

(b) If χ(t) =

is the spin state at time t, show that

where Ω Ξ ϒ B_rf is related to the strength of the rf field.
(c) Check that the general solution for a(t) and b(t), in terms of their initial values a₀ and b₀, is

where

(d) If the particle starts out with spin up (i.e. a₀ = 1, b₀ = 0), find the probability of a transition to spin down, as a function of time

(e) Sketch the resonance curve,

as a function of the driving frequency ω (for fixed ω₀ and Ω).
(f) Since ω₀ = ϒB₀, we can use the experimentally observed resonance to determine the magnetic dipole moment of the particle. In a nuclear magnetic resonance (nmr) experiment the g-factor of the proton is to be measured, using a static field of 10,000 gauss and an rf field of amplitude 0.01 gauss. What will the resonant frequency be? (See Section 7.5 for the magnetic moment of the proton.) Find the width of the resonance curve. (Give your answers in Hz.)

(Example 4.3)

Transcribed Image Text:

(rf) field, Brf [cos (wt)î - sin (wt)ĵ],