Nuclear Magnetic Resonance for Arbitrary Spin Values in the Rotating Wave Approximation

TL;DR

This paper provides an exact solution for the time-dependent substate probabilities of arbitrary nuclear spin values under the rotating wave approximation, extending beyond the well-known spin-1/2 case.

Contribution

It introduces an elementary method to solve the general spin s case exactly, revealing complex time dependencies of substate probabilities not previously published.

Findings

Exact time dependence of substate probabilities for arbitrary spin s.

Discovery of more complex oscillation patterns for higher spins.

Figures illustrating probability dynamics for various initial conditions.

Abstract

In order to probe the transitions of a nuclear spin $s$ from one of its substate quantum numbers $m$ to another substate $m'$, the experimenter applies a magnetic field ${\bm B}_0$ in some particular direction, such along $\hat{\bm z}$, and then applies an weaker field ${\bm B}_1(t)$ that is oscillatory in time with the angular frequency $\omega$, and is normally perpendicular to ${\bm B}_0$, such as ${\bm B}_1(t)=B_1\hat{\bm x}\cos(\omega t)$. In the rotating wave approximation, ${\bm B}_1(t)=B_1[\hat{\bm x}\cos(\omega t)+\hat{\bm y}\sin(\omega t)]$. Although this problem is solved for spin $\frac{1}{2}$ in every quantum mechanics textbook, for the general spin $s$ case, its general solution has been published only for the overall probability of a transition between the states, but the time dependence of the probability of finding the nucleus in each of the substates has not previously been published. Here we present an elementary method to solve this problem exactly, and present figures for the time dependencies of the various substates states for a variety of initial substate probabilities for a variety of $s$ values. We found a new result: unlike the $s=\frac{1}{2}$ case, for which if the initial probability of finding the particle in one of the substates was 1, and the time dependence of the probabilities of each of the substates oscillates between 0 and 1, for higher spin values, the time dependencies of the probabilities finding the particle in each of its substates, which periodic, is considerably more complicated.