Rotation and Reverse-Rotation Methods for Magnetic Resonance Quantization

Eigenvalues of a spin Hamiltonian operator, H=ω₀S_z+ω₁[S_xcos(f)+S_ysin(f)], are not directly estimable because the operator is not generally diagonal; the field continuously oscillates in the xy plane with the washboard frequency as a function of time f=f(t)= ωt + g(t), g(t+2π/ω)=g(t).

In this work, two of rotation and reverse-rotation methods are introduced. The rotation method with f(t)=ωt leads to #1 the harmonic oscillation equations of (S_x, S_y) with the variables  (β, φ), which means the xy plane oscillations are circulatory, and #2 the eigenvalue of the Hamiltonian. But, the function f shouldn't be  linear in time, f ≠ ωt, which means the oscillation is not in a circle anymore. The reverse-rotation method shows the spin characterization for general f(t).

In many spin experiments including NMR, f(t) is tunable by manipulating B(t). Any function of f could lead to a different shape of the oscillations. Here, with several types of washboard functions f, we find the general solutions and plot the spin state ψ for arbitrary spin value from their closed form in term of an elliptic integral.