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American Journal of Physics

A Spin Flip Problem

R. T. Robiscoe

Published Feb 1, 1971• 23 citations
Publisher

Abstract

We examine the problem of calculating the spin up and spin down amplitudes for a spin 12 particle, initially aligned with respect to a constant magnetic field, and then subjected to a rotation of the field through an arbitrary angle in a finite time. We find that the solutions to the problem are entirely determined by an adiabaticity parameter α which is the ratio of the particle Larmor frequency to the field rotation rate. As expected, a spin flip occurs readily only in the nonadiabatic case, α≪1.

DOI
10.1119/1.1986080

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We examine the problem of calculating the spin up and spin down amplitudes for a spin 12 particle, initially aligned with respect to a constant magnetic field, and then subjected to a rotation of the field through an arbitrary angle in a finite time. We find that the solutions to the problem are entirely determined by an adiabaticity parameter α which is the ratio of the particle Larmor frequency to the field rotation rate. As expected, a spin flip occurs readily only in the nonadiabatic case, α≪1.
In order to probe the transitions of a nuclear spin sss from one of its substate quantum numbers mmm to another substate m′m'm′, the experimenter applies a magnetic field B0{\bm B}_0B0​ in some particular direction, such along z^\hat{\bm z}z^, and then applies an weaker field B1(t){\bm B}_1(t)B1​(t) that is oscillatory in time with the angular frequency ωωω, and is normally perpendicular to B0{\bm B}_0B0​, such as B1(t)=B1x^cos⁡(ωt){\bm B}_1(t)=B_1\hat{\bm x}\cos(ωt)B1​(t)=B1​x^cos(ωt). In the rotating wave approximation, B1(t)=B1[x^cos⁡(ωt)+y^sin⁡(ωt)]{\bm B}_1(t)=B_1[\hat{\bm x}\cos(ωt)+\hat{\bm y}\sin(ωt)]B1​(t)=B1​[x^cos(ωt)+y^​sin(ωt)]. Although this problem is solved for spin 12\frac{1}{2}21​ in every quantum mechanics textbook, for the general spin sss 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 sss values. We found a new result: unlike the s=12s=\frac{1}{2}s=21​ 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.
We examine spinor evolutions in a general time-dependent magnetic field B. We derive a condition the B variation must satisfy in order that the effective field direction ${\mathbf{r}\mathbf{\^{}}}_{\mathrm{B}}^{\ensuremath{'}}$ for a frame rotating with B becomes stationary in that frame. This condition is both sufficient and necessary for B to bring about a spinor evolution which exactly maintains a fixed angle between the spin and B. The adiabatic evolution used by Berry is just a special case, albeit only approximate, of these fixed-angle evolutions. To the other extreme of the fixed-angle evolutions, the nontrivial parallel transportation occurs, which is exact, necessarily nonadiabatic, and yields a pure geometric phase. Hence an experiment aimed at observing the geometric phase should seek to effect not an adiabatic evolution but a parallel transportation.
The time-dependent Schr\"odinger equation is solved formally for an atomic or molecular system which is subjected simultaneously to a rotating magnetic field of constant amplitude and angular velocity and to a constant magnetic field along the axis of rotation. The method yields the transition probabilities in terms of the solutions to an eigenvalue problem. This eigenvalue problem is solved both for (a) a normal Zeeman effect and for (b) the case where a transition from a given level is isolated in frequency from other transitions from the same level. Case (a) is exactly soluble and yields a solution which is shown to be the same as that of Bloch and Rabi, but is in a form which is more convenient for integration over the velocity distribution. Case (b) must be solved by an approximate method which results in a prediction of multiple quantum transitions as observed by Kusch.