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Oxford University Press eBooks

Rotational motion

Peter Atkins, Julio de Paula, James Keeler

Published Dec 5, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.1093/hesc/9780198847816.003.0038

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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.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.
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.
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 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.
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 .
This chapter discusses how solving the Schrödinger equation leads to the conclusion that orbital angular momentum is quantized and how the same conclusion can also be reached from the angular momentum operators without solving the Schrödinger equation explicitly. It analyses the commutation properties of the angular momentum operators, noting that the same conclusions apply to observables that are described by operators with the same commutation properties. Angular momentum has many more mundane applications, including being central to the discussion of the structures of atoms and the rotation of molecules, as well as to virtually all forms of spectroscopy. The chapter mentions classical mechanics that puts no constraints on the magnitude of the angular momentum, which is consistent with the kinetic energy of rotation being continuously variable. The property of hermiticity can be demonstrated explicitly in the position representation.