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An Electron in a Precessing Magnetic Field: a Tutorial

Anindya Biswas, Bhabani Prasad Mandal

Published Feb 1, 2020• 0 citations
OpenAlex

Abstract

In this pedagogical article, we elucidate on direct derivation of wave function of an electron in a precessing magnetic field of constant magnitude.

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The spin polarization of an electron moving nonrelativistically through a potential barrier V(z)V(z)V(z), with also a constant uniform magnetic field (0,0,B0,0,B0,0,B) applied, is calculated using the WKB approximation for the zzz dependence of the wave function. It is found that, in a classically allowed region, the polarization precesses at fixed angle $\ensuremath{\theta}$ about the magnetic field, with phase angle the same as a classical particle with a magnetic dipole moment. The particle penetrates through the barrier with no change in the phase of the precession but with a discontinuity in $\ensuremath{\theta}$.
The purpose of this tutorial is to derive the precessional characteristics of the magnetic moments of hydrogen nuclei in the presence of a constant external magnetic field using the Dirac bra-ket formulation of quantum mechanics (QM). This behaviour has many applications, most notably in nuclear magnetic resonance (NMR) and magnetic resonance (MR) imaging. Many NMR and MR imaging textbooks claim that the QM expectation value of the magnetic moment of a proton in a magnetic field reduces to the classical picture of a precessing magnetic dipole. This paper validates this conclusion by reducing the cumbersome QM integrals using Dirac notation and matrix algebra, and then comparing this result with the classical picture. It illustrates the connections between the quantum and classical pictures and demonstrates quantitatively how they differ and how they are similar. This tutorial targets students and researchers interested in the fundamental physics behind the MR phenomenon and assumes that the reader has a basic understanding of QM. This may be helpful for undergraduate QM students learning about spin angular momentum and Dirac notation.