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.
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.
A standard description of the angular momentum in atomic or molecular physics is based on quantum mechanics. However, especially at large angular-momentum limit in molecules, sometimes a classical approach greatly simplifies problems that are extremely complicated from a quantum-mechanical viewpoint. The aim of this paper is to examine the relationship between the classical and quantum descriptions of the angular-momentum distribution in an ensemble of particles. At first glance, quantum and classical approaches appear to be absolutely different and have very little in common. In this paper this relationship is analyzed. It is shown that similarities can be found between the precession of angular momentum in the external field and the influence of this field upon the wave-function phase of a microparticle. The importance of wave function phases is stressed even for stationary conditions, and some examples of coherent superposition of quantum states are presented. The general method of how to pass from the quantum approach to a classical one is formulated. Results are visualized in the form of 3D distribution functions. This approach allows a graphical interpretation and thus results in a better understanding, sometimes at first glance, of counterintuitive quantum results. As an example the phenomenon of alignment- orientation transition is considered. Rather than presenting only formal derivations, the emphasis is on the instructional aspects, and on illustrating the general approach.
The object of this paper is to call to the attention of those teaching quantum mechanics several of the problems arising from the behavior of nuclei possessing a magnetic moment in an external magnetic field. These problems are simple enough to serve as student exercises. They are presented both from the classical and quantum-mechanical points of view, and this is particularly instructive in connection with the problems considered because both techniques of calculation lead to the same answers, but the results in spite of formal similarity are to be interpreted quite differently. This provides the teacher with an excellent opportunity to discuss the conceptual differences of the old and new mechanics.
Abstract In this chapter we discuss the intrinsic angular momentum called spin. Starting from the magnetic dipole moment related to orbital angular momentum, we focus on the hydrogen atom and consider the effect of an external magnetic field, resulting in Larmor precession and Zeeman splitting. We then turn to the discovery of spin in the Stern-Gerlach experiment. After providing a mathematical description of spin-1/2 systems in terms of two-component spinors and Pauli matrices, we illustrate projective measurements for the example of spin. The relationship between the Lie groups SU(2) and SO(3) is discussed in the context of the transformation of spinors under rotations. We analyse how the presence of spin alters the description of the hydrogen atom. Discussing the addition of angular momenta and the Pauli exclusion principle leads us to the structure of electronic orbitals. Finally, we discuss the occupation of electronic energy levels in heavier nuclei
This chapter begins showing the basic concepts of the magnetic moment and its relation with the angular momentum of an electron with and without external magnetic field applied. The former leading to the Larmor frequency. After that, it is presented the Landé’s core model in which the total angular momentum is a sum of angular momentum of valence and core electrons (plus nuclei for the core). Subsequently, normal and anomalousZeeman effect, their relations with frequency of precession and magnetic moment, and Landé’s g-factor are presented. The Heisenberg’s core model is also presented. Afterwards, Pauli concept of the electron twofoldness and Pauli’s g-factor are given followed by Pauli’s exclusion principle from his contribution to Aufbau principle. Information about the experiments that led to the discovery of the spin are given in the sequence. Finally, Pauli and Dirac’s works on spin yielding the Pauli’s spin matrix and Dirac’s spin matrix are presented along with Dirac’s four spinor equation which gave the theoretical support for the existence of the spin (168 words).
The water molecule has many biological functions and is one of the most abundant molecules in the human body. Then, in order to carry out a study of the molecule in physical and chemical phenomena, the model used depends on the phenomenon. For some cases, it is necessary to consider the electronic distribution, while in other cases, it is necessary to consider the protons of the hydrogen atoms, for example, to explain the physics in magnetic resonance imaging. In this work, the water molecule model considered is conformed by three particles: The two nuclei of the hydrogen atom and the oxygen atom negatively double charged and unstructured. The spatial wave function and the interaction of the angular momentum of protons with a constant magnetic field has been studied in a previous work. The present work is a completion, in order to have the complete wave function of this model, considering the spin of the protons, where the energy is degenerated (B = 0). Finally, the interaction of the spin of the nuclei of hydrogen atoms with a magnetic field is studied, representing the case of magnetic resonance imaging, where it is obtained a break in the degeneration of the energy levels, which are in the order of radiofrequency.
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 .