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American Journal of Physics

The exact solution of magnetic resonance for arbitrary spin using an operator-based formalism

Sanya Bharti, James K. Freericks

Published Jan 22, 2026• 0 citations
Publisher

Abstract

The time evolution of a spin precessing in a large static field with a perpendicular rotating field is a classic problem in magnetic resonance that appears in nearly all quantum textbooks. It can be solved exactly by going to the rotating frame. Often, however, the final transition probabilities are determined only for the spin 1/2 case. In this work, we show how an operator-based approach naturally produces a time-ordered product via a Trotter product formula and then by employing simple operator identities such as the Hadamard lemma, the Baker–Campbell–Hausdorff formula, and the exponential disentangling identity, the problem can be directly solved for arbitrary spin.

DOI
10.1119/5.0248238

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The time evolution of a spin precessing in a large static field with a perpendicular rotating field is a classic problem in magnetic resonance that appears in nearly all quantum textbooks. It can be solved exactly by going to the rotating frame. Often, however, the final transition probabilities are determined only for the spin 1/2 case. In this work, we show how an operator-based approach naturally produces a time-ordered product via a Trotter product formula and then by employing simple operator identities such as the Hadamard lemma, the Baker–Campbell–Hausdorff formula, and the exponential disentangling identity, the problem can be directly solved for arbitrary spin.
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.
In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic fields. One of these fields is static (time-independent) and directed along a specific axis, while the other is timedependent, has a different amplitude, and rotates in the plane perpendicular to the first magnetic field. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin flip,” i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic field along the z-axis and a circularly polarized time-varying (ac) magnetic field of different amplitude, with a specific frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a different approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.
In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic fields. One of these fields is static (time-independent) and directed along a specific axis, while the other is timedependent, has a different amplitude, and rotates in the plane perpendicular to the first magnetic field. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin flip”, i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic field along the z-axis and a circularly polarized time-varying (ac) magnetic field of different amplitude, with a specific frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli Equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a different approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.
For tutorial purposes, the simple problem of a classical magnetic moment interacting with various magnetic field configuration is reformulated in terms of coordinate-free, time-dependent rotation operators. In this approach, the corresponding equation of motion assumes the form of an operator equation, identical with the Schrödinger equation of a spin-1 interacting with the same field configurations. It is solved utilizing deliberately the concepts of a Hermitian Hamiltonian and a unitary time evolution operator to familiarize students early in their curriculum with these important ingredients of the quantum-mechanical formalism. It turns out that besides furnishing useful exercises for the manipulation of three-dimensional rotation operators, being an end in itself, the approach also provides simple but nontrivial examples for a number of theorems for which the students will have frequent use in their quantum-mechanics course.
The solution of the time-dependent wave equation describing the interaction of two coupled spins with a rotating magnetic field in the presence of a steady field, as in the usual magnetic resonance situation, has been reexamined. By extending the method previously developed by Salwen, an exact expression is derived for all components of the wave function describing a three-state system under magnetic-resonance conditions. This solution describes in analytical form all the single and multiple resonances of which the system is capable for fields of any intensity. It is shown that when zzz is the axis of quantization, the projection of the wave function on an eigenstate of Fx{F}_{x}Fx​ oscillates at twice the resonance frequency at exact double quantum resonance. An approximate form is derived for the transition probability near double quantum resonance, showing that our solution is consistent with Salwen's in the limit of well-separated single and double quantum resonances. The theory is applied to the uppermost three substates of the upper hyperfine multiplet of an alkali metal of arbitrary nuclear spin, a case of interest to the analysis of magnetic resonance in optically pumped vapors.