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Revista Mexicana de Física

Total angular momentum of water molecule and its interaction with a constant magnetic field

Cristian Heber Zepeda Fernández, Carlos Alfredo López Téllez, Yuridia Flores Orea, E. Moreno Barbosa

Published Sep 1, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.31349/revmexfis.70.050401

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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 .
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.
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
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.
The mechanical angular momentum and magnetic moment of the electron and proton spin have been calculated semiclassically with the aid of the uncertainty principle for energy and time.The spin effects of both kinds of the elementary particles can be expressed in terms of similar formulae.The quantization of the spin motion has been done on the basis of the old quantum theory.It gives a quantum number n = 1/2 as the index of the spin state acceptable for both the electron and proton particle.In effect of the spin existence the electron motion in the hydrogen atom can be represented as a drift motion accomplished in a combined electric and magnetic field.More than 18,000 spin oscillations accompany one drift circulation performed along the lowest orbit of the Bohr atom.The semiclassical theory developed in the paper has been applied to calculate the doublet separation of the experimentally well-examined D line entering the spectrum of the sodium atom.This separation is found to be much similar to that obtained according to the relativistic old quantum theory.