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Physical Review A

Spin precession and energy conservation

Onuttom Narayan

Published Sep 5, 2000• 0 citations
Publisher

Abstract

In this paper, we examine the precession of a spin in a magnetic field, treating the particle and the applied magnetic field as a single closed system whose energy is conserved. Through two simple examples, it is shown that the standard expression for spin precession emerges only indirectly, and as a first-order approximate result. For the first example, deviations from the first-order result are easy to obtain, and are familiar from Stern-Gerlach spin separation. For the second example, deviations from the first-order result are not practically possible to obtain. Beyond these two examples, it is also shown within a semiclassical approach that the standard expression for spin precession in a magnetic field is always valid to first order, irrespective of how the Hamiltonian depends on other dynamical variables of the system.

DOI
10.1103/physreva.62.042101

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In this paper, we examine the precession of a spin in a magnetic field, treating the particle and the applied magnetic field as a single closed system whose energy is conserved. Through two simple examples, it is shown that the standard expression for spin precession emerges only indirectly, and as a first-order approximate result. For the first example, deviations from the first-order result are easy to obtain, and are familiar from Stern-Gerlach spin separation. For the second example, deviations from the first-order result are not practically possible to obtain. Beyond these two examples, it is also shown within a semiclassical approach that the standard expression for spin precession in a magnetic field is always valid to first order, irrespective of how the Hamiltonian depends on other dynamical variables of the system.
First, the experimental foundations for the existence of a new dynamical variable, the spin of particles, are presented. The Pauli equation is then derived in detail, two applications are given and the energy levels of a particle with spin in a constant magnetic field are studied. This is the analysis of Landau levels, which can be performed by using the known results on the spectrum of harmonic oscillators. In this chapter we also study in detail the addition of orbital angular momentum and spin, as well as the spin–orbit interaction with the resulting Thomas precession. Stern-Gerlach experiment and electron spin The hypothesis that the electron has an intrinsic magnetic moment and an angular momentum, in short a spin , was first suggested in Uhlenbeck and Goudsmit (1926). They noticed, even before the discovery of quantum mechanics, that a complete description of spectra was not possible unless a magnetic moment and a mechanical moment were ascribed to the electron, and hence the concept of an electron as a point charge was insufficient. First, let us therefore try to understand how a magnetic moment can be associated to an atomic system. For this purpose, consider for simplicity the Bohr model of an hydrogen atom, where the electron moves along a circular orbit and rotates with orbital angular momentum L .
The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck's constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution.
The classical Stern-Gerlach experiment is analyzed with an emphasis on the spin dynamics. The central question asked is whether there occurs a relaxation of the spin angular momentum during the time the particle passes through the Stern-Gerlach magnet. We examine in particular the transverse relaxation, involving angular momentum exchange between the spin of the particles and the spins of the magnet. A method is presented describing relaxation effects at an individual particle level. This leads to a stochastic equation of motion for the spins. This is coupled to a classical equation of motion for the particle translation. The experimental situation is then modeled through simulations of individual trajectories using two sets of parameter choices and three different sets of initial conditions. The two main conclusions are: (A) if the coupling between the magnet and the spin is solely described by the Zeeman interaction with the average magnetic field the simulations show a clear disagreement with the experimental observation of Stern and Gerlach. (B) If one, on the other hand, also allows for a T(2) relaxation time shorter than the passage time one can obtain a practically quantitative agreement with the experimental observations. These conclusions are at variance with the standard textbook explanation of the Stern-Gerlach experiment.
The Stern-Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. But that theory is based on mathematical errors in the way it (mis)treats spinors and group theory.We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry.Contrary to what Pauli has been reading into the Stern-Gerlach experiment,the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured.Surprizingly this leads to only two energy states, which should be qualified as precession-up and precession-down rather thanspin-up and spin down.Indeed, despite the presence of the many different possible angles θ\thetaθ betweenthe spin axis s{\mathbf{s}}s and the magnetic field B{\mathbf{B}}B, the fermions can only have two possible energies m0c2±μBm_{0}c^{2}\pm\mu Bm0​c2±μB. The values ±μB\pm\mu B±μB do thus not correspond to the continuum of values −μ⋅B-{\boldsymbol{\mu\cdot}}{\mathbf{B}}−μ⋅B Einstein and Ehrenfest had conjectured. The energy term V=−μ⋅BV= -{\boldsymbol{\mu\cdot}}{\mathbf{B}}V=−μ⋅B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies ±μB\pm\mu B±μB, and as such not valid for individual fermions. The two fermion states with energy ±μB\pm\mu B±μB are not potential-energy states. We also explain themathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clearand understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.
Quantization of spin and orbital angular momenta in standard quantum mechanics is rooted in the fundamental rotational symmetry of space in conjunction with the results of experiments on atomic and subatomic particles conducted with the Stern-Gerlach apparatus. We explain some of the most important features of spin and orbital angular momenta as inescapable consequences of elementary Stern-Gerlach experiments viewed in light of the rotational symmetry of space. The rotation operator is first derived for spin-½ particles, then extended to spin-1 particles by treating the case of a hydrogen atom in its ground state, where the spin of the proton combines with that of the electron to create a spin-1 triplet state. Along the way, we examine the evolution of a particle’s spin state under an applied magnetic field, where the interaction energy arises from the existence of a magnetic dipole moment for each particle in proportion to its spin angular momentum. Whereas a constant magnetic field B0 causes the particle’s magnetic moment to gyrate around the direction of B0, it will be seen that the presence of a small oscillating magnetic field perpendicular to B0 brings about the so-called Rabi oscillations, which involve periodic flippings of the particle’s spin orientation. In the case of a hydrogen atom in its ground state, the interaction between the magnetic dipole moments of the proton and the electron splits the ground state between a lower energy singlet having a net angular momentum of zero (spin-0 state), and a higher energy triplet having a total angular momentum of ħ (spin-1 state). This hyperfine splitting is the source of the well-known 21-cm line observed in the spectrum of the hydrogen atom, which arises from transitions between the atom’s singlet and triplet states. In the final section, we put forward an argument (originally made by Richard Feynman) that ties the rotational symmetry of space to the conservation of angular momentum in quantum mechanics. The argument is, in fact, quite general, enabling one to deduce other conservation laws from various inherent symmetries of quantum mechanical systems.