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On the angular momentum of a system of quantum particles

O Chavoya-Aceves

Published Jan 1, 2014• 0 citations
Publisher

Abstract

The properties of angular momentum and its connection to magnetic moment are explored, based on a reconsideration of the Stern-Gerlach experiment and gauge invariance. The separation of angular momentum of a quantum system of particles into orbital angular momentum plus intrinsic angular momentum is reconsidered, within the limits of the Schrödinger theory. A proof is given that, for systems of more than two particles, unless all of them have the same mass, the possibility of having eigenvalues of the form $(n+1/2)\hbar$ is not excluded for the operators of angular momentum of the centre of mass and the internal angular momentum. \noindent\textbf{PACS: 03.65.Ta} angular momentum, magnetic moment, spin crisis

DOI
10.13140/2.1.4709.5688

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The properties of angular momentum and its connection to magnetic moment are explored, based on a reconsideration of the Stern-Gerlach experiment and gauge invariance. The separation of angular momentum of a quantum system of particles into orbital angular momentum plus intrinsic angular momentum is reconsidered, within the limits of the Schrödinger theory. A proof is given that, for systems of more than two particles, unless all of them have the same mass, the possibility of having eigenvalues of the form (n+1/2)ℏ(n+1/2)\hbar(n+1/2)ℏ is not excluded for the operators of angular momentum of the centre of mass and the internal angular momentum. \noindent\textbf{PACS: 03.65.Ta} angular momentum, magnetic moment, spin crisis
Angular momentum plays a similarly important role in atomic systems, such as an electron subject to the central force provided by the Coulomb field of a nucleus. This chapter introduces the properties of angular momentum, both orbital and spin, in the quantum-mechanical scheme of things. The angular momentum of atomic systems is inseparable from the magnetic properties of atoms. The various permitted values of the quantized angular momentum play a central role in determining the number and variety of possible quantum states in atomic systems. The Stern-Gerlach experiment was deliberately designed to test whether or not the angular momentum of individual neutral atoms is quantized in a magnetic field. As in the development of the Schrodinger equation, classical physics gives some hints of the correct forms of these angular momentum operators. The chapter discusses rough estimate using the Neils Bohr theory and attributing the magnetic moment to the orbital motion of the electron.
This chapter analyses the orbital angular momentum to three dimensions and introduces the spin angular momentum of the electron. The description of a localized particle in orbit requires a superposition of eigenfunctions analogous to the packet state that describes motion in one dimension. If the distance between molecules is fixed, all the energy changes are rotational kinetic energy changes, and the quantization of total angular momentum gives rise to discrete rotational energy levels. The analysis of orbital angular momentum is a powerful tool for describing molecular rotation. The Stern-Gerlach experiment and many other lines of evidence show that the measured magnetic moment associated with an electron spin is almost exactly one Bohr magneton. When orbital angular momentum and spin angular momentum both exist in the same atom, the magnetic moments that result from orbital and spin angular momenta interact to cause a splitting in the corresponding energy level.
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 .
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.
Angular momentum is a very important and revealing property of many physical systems. In classical mechanics the principle of conservation of angular momentum is a powerful aid to the solution of such problems as the orbits of planets and satellites and the behaviour of gyroscopes and tops. The role of angular momentum in quantum mechanics is probably even more important and this will be the subject of the present chapter and the next. We shall find that the operators representing the components of angular momentum do not commute with each other, although they all commute with the operator representing the total angular momentum. It follows that no pair of these components can be measured compatibly and we shall therefore look for a set of eigenfunctions that are common to the operators representing the total angular momentum and one of its components. We shall find that the angular momentum eigenvalues always form a discrete set and that, in the case of a central field, the operators commute with the Hamiltonian, implying that the total angular momentum and one component can be measured compatibly with the energy in this case, so that their values, once measured, remain constant in time. Consideration of angular momentum will also enable us to make predictions about the behaviour of atoms in magnetic fields, and we shall find that these are in agreement with experiment only if we assume that the electron (in common with many other fundamental particles) has an intrinsic “spin” angular momentum in addition to the “orbital” angular momentum associated with its motion. The interaction between spin and orbital angular momenta leads to detailed features of the atomic spectra which are known as “fine structure.” We shall also use the measurement of angular momentum to illustrate the quantum theory of measurement discussed in the previous chapter.
It is proved that, according to Classical Mechanics and Electrodynamics, the trajectory of the center of mass of a neutral system of electrical charges can be deflected by an inhomogeneous magnetic field, even if its internal angular momentum is zero. This challenges the common view about the function of the Stern-Gerlach apparatus, as resolving the eigen-states of an intrinsic angular momentum. Doubts are cast also on the supposed failure of Schrodinger's theory to explain the properties of atoms in presence of magnetic fields without introducing spin variables.