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CRC Press eBooks

Orbital Angular Momentum and Central Potentials

Gordon Baym

Published Mar 5, 2018• 0 citations
Publisher

Abstract

The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.

DOI
10.1201/9780429499265-6

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The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.
The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.
The angular momentum plays an important role in quantum mechanics. It is no longer an ordinary vector, instead it is a vector operator, whose three components do not commute with each other. In quantum mechanics, the angular momentum becomes a vector operator. A general angular momentum is defined as a vector operator whose components are Hermitian operators satisfying the commutation relations similar to those for the orbital angular momentum. In order to obtain the eigenvalues and eigenfunctions, it is more convenient to express the orbital angular momentum operators in spherical polar coordinates. Spin angular momentum is an intrinsic property of elementary particles. In quantum computing, a qubit or quantum bit is a quantum version of the classical binary bit physically realized in a two-state quantum-mechanical system.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
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.
A particle moving along a straight-line has a linear momentum mv. A revolving particle has angular momentum due to its mass and angular velocity. In classical mechanics, orbital angular momentum is the vector product of position (r) and linear momentum (p): L = r× p . (11.1) Let us imagine the electron as a small ball, revolving about some axis. It possesses angular momentum. Precisely, the angular momentum vector is a vector pointing along the axis of rotation with magnitude proportional to the speed of rotation. A vector which changes sign under space inversion (parity operation) is called a polar vector . r and p are polar vectors as they change sign. Since L = (−r)× (−p), L does not change sign under parity operation. Such vectors which do not change sign under parity operation are called axial vectors or psuedo-vectors. Classically, the angular momentum of the electron could have any magnitude. In quantum mechanics the orbital angular momentum operator is obtained by replacing p by −i~∇. The angular position of a particle is the angle at which it is located in two-dimensional polar coordinates.
Angular momentum is one of the more abstract topics in quantum mechanics. This chapter begins with a review of classical angular momentum. Quantum-mechanical angular momentum is then introduced by obtaining operators from their classical counterparts. The commutation relations amongst those operators and the corresponding uncertainty relations then follow. Eigenstates and eigenvalues of the angular momentum operators are then introduced (but not derived). From these follow the raising and lowering operators. Physical meaning is imparted to quantum angular momentum by considering predictions for sequential measurements, and by comparing the conceptual pictures of angular momentum in classical and quantum mechanics. Finally, quantum spin and orbital angular momenta, and the deep differences between them, are discussed.