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Oxford University Press eBooks

Rotation

Peter Joshua Atkins, Julio de Paula, David Young Smith

Published Dec 15, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1093/hesc/9780198727873.003.0048

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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.
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 considers the quantum mechanics of translation and vibration. Both types of motion can be solved in certain cases, and both are important not only in their own right but also because they form a basis for the description of more complicated types of motion. It mentions translational motion, which has the advantage of introducing in a simple way many of the striking features of quantum mechanics. The chapter describes certain features of wavefunctions that are common to all the problems. The combination of features of wavefunctions with the solution of the Schrödinger equation results in one of the most characteristic features of quantum mechanics, the quantization of energy.