Nuclear Magnetic Resonance

A spin-1/2 nucleus is placed in a large magnetic field B₀ in the z-direction. An oscillating field B₁ << B₀ of radio frequency is applied in the xy-plane, so the total magnetic field is

(1)

The Hamiltonian is H = μB . σ, where μ is the magnetic moment. Use the notation hΩ_|| = μB₀, hΩ_┴ = μB₁.

a) If the nucleus is initially pointing in the +z-direction at t = 0, what is the probability that it points in the -z-direction at later times?

b) Discuss why most NMR experiments adjust B₀ so that Ω_|| ~ ω/2.

SOLUTION

a) Let and denote the probability of spin up and spin down as a function of time. The time-dependent Hamiltonian is

(1)

(2)

(3)

The equations for the individual components are

(4)

(5)

where the overdots denote time derivatives. We solve the first equation for β and substitute this into the second equation:

(6)

(7)

(8)

We assume that

(9)

We determine the eigenvalue frequency λ by inserting the above form for α(t) into (8), which gives a quadratic equation for that has two roots:

(10)

(11)

(12)

(13)

(14)

We have introduced the constants a₁, a₂, b₁, b₂. They are not all independent. Inserting these forms into the original differential equations, we obtain two relations which can be used to give

(15)

(16)

This completes the most general solution. Now we apply the initial conditions that the spin was pointing along the z-axis at t = 0. This gives α(0) = 1, which makes a₁ + a₂ = 1, and β(0) = 0, which gives b₁ + b₂ = 0. These two conditions are sufficient to find

(17)

(18)

(19)

(20)

The probability of spin up is |α(t)|² and that of spin down is |β(t)|²:

(21)

(22)

b) In the usual NMR experiment, one chooses the field B₀ so that 2Ω_|| ≈ ω, in which case  and and  The spin oscillates slowly between the up and down states.