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Oxford University Press eBooks

Angular momentum

Peter Joshua Atkins, Ronald Friedman

Published Nov 11, 2010• 0 citations
Publisher

Abstract

This chapter considers rotational motion as the second class of motion of an object around a fixed point and angular momentum, which is one of the most important topics in quantum mechanics. It discusses the rotational motion and angular momentum in terms of solutions of the Schrödinger equation and see how its properties emerge from the operators for angular momentum and their commutation relations. It also examines the quantum mechanical description of a particle travelling on a circular ring, which applies to the motion of a bead on a circle of wire and any body rotating in a plane. The chapter describes the moment of inertia the property, which determines the characteristics of the rotational motion of a body. It cites a particle of mass m travelling on a circle of radius r in the xy-plane and its potential energy that is constant and taken to be zero.

DOI
10.1093/hesc/9780199541423.003.0004

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This chapter considers rotational motion as the second class of motion of an object around a fixed point and angular momentum, which is one of the most important topics in quantum mechanics. It discusses the rotational motion and angular momentum in terms of solutions of the Schrödinger equation and see how its properties emerge from the operators for angular momentum and their commutation relations. It also examines the quantum mechanical description of a particle travelling on a circular ring, which applies to the motion of a bead on a circle of wire and any body rotating in a plane. The chapter describes the moment of inertia the property, which determines the characteristics of the rotational motion of a body. It cites a particle of mass m travelling on a circle of radius r in the xy-plane and its potential energy that is constant and taken to be zero.
This chapter focuses on the motion first on a circle and then on the surface of a sphere. It discusses the imposition of certain constraints on the wavefunction which shows how the energy and angular momentum of a rotating particle are quantized. It also examines the energy and the angular momentum of a rotating object which are quantized. The chapter reviews quantum mechanics and shows what they reveal about rotational motion. The chapter also considers a particle of mass m moving in a circular path of radius r in the xy-plane. It highlights translational motion which focuses on linear momentum, p and analogous angular momentum, J if rotational motion is considered.
This chapter focuses on the concept of rotational motion. It explains that energy and angular momentum for a particle rotating in two- or three-dimensions are quantized and satisfied cyclic boundary conditions, which imply the quantization of angular momentum's magnitude and z-component. The electronic structure of atoms showcases rotational motion because electrons orbit around nuclei and spin on their axis. The chapter details the solutions for the Schrödinger equation and the vector model of angular momentum. It also discusses the concept of space quantization revolving around the quantum mechanical result of a rotating body that may not take up an arbitrary orientation concerning some specified axis.
This chapter discusses the classic circular motion as a basis for the circular motion of a particle. The derivation of the bidimensional Laplace operator in polar coordinate system is presented next in order to solve the Schrödinger equation of the particle in a ring (whose wave function is a part of the solution of the hydrogen atom). Subsequently, there appears the derivation of the wave equation for the particle in a ring which leads to an energy equation and its comparison with the classic equation of the angular momentum leads to the corresponding quantum angular momentum equation. Afterwards, there appears the derivation of the angular momentum in spherical polar coordinate which is part of the tridimensional Laplace operator in spherical polar coordinate which is obtained in the next section. Then, the tridimensional Laplace operator in spherical polar coordinate is used in Schrödinger equation of the particle in a sphere (a product of angular equation and azimuthal equation) where the azimuthal equation was already solved (from the particle in a ring given in a previous section) and then this section is devoted to derive the angular or polar equation and the corresponding quantum angular momentum equation. (195 words).
When a classical particle moves in a central field the energy and the three components of angular momentum are all constants of the motion. The motion is an orbit whose orientation in space is fixed. To discuss the corresponding situation in quantum mechanics we must examine the commutation properties of the associated operators: If they all commute, then the quantities which they represent may all simultaneously be constants of the motion; if not, we must find as many mutually commuting operators as we can and obtain a maximal set of constants of the motion.
The motion in the angular dimensions described by the solutions in Chapter 14 are related to angular momentum. We explore that relationship in this chapter. This gives insight into the angular TISE, showing that the equation amounts to the classical equation relating angular momentum, moment of inertia, and rotational kinetic energy: KE = L 2 /2I. This also shows that angular momentum is both conserved and quantized in a radial potential. These quantized angular momenta levels are well known to students of chemistry as the orbitals, for which the orbital angular momentum quantum number equals 0, 1, 2, and 3 for the s, p, d, and f orbital, respectively. This quantum number represents the amount of orbital angular momentum of the electron, but the calculation is a little bit more complicated, and is related to the separation constants introduced in Chapter 14 .
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.