Oxford University Press eBooks

Rotational motion in three dimensions

Peter Joshua Atkins, Julio de Paula, Ronald Friedman

Published Dec 12, 2013• 0 citations

Abstract

Contents A particle on a sphere 111 The wavefunctions 112 Brief illustration 14.1: The angular nodes of the spherical harmonics 114 The energies 114 Example 14.1: Using the rotational energy levels 115 Angular momentum 115 The angular momentum operators 115...

DOI: 10.1093/hesc/9780199609819.003.0019

Related papers

Angular Momentum and Rotational Motion in Three Dimensions

Jochen Autschbach

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.

Oxford University Press eBooksDec 21, 20200 citations

Rotational motion in two dimensions

Peter Joshua Atkins, Julio de Paula, Ronald Friedman

Contents A particle on a ring 103 Brief illustration 13.1: The moment of inertia 104 The qualitative origin of quantized rotation 105 Brief illustration 13.2: The rotational energies 106 The solutions of the Schrödinger equation 106 Example 13.1: Using the particle-on-a-ring model...

Oxford University Press eBooksDec 12, 20130 citations

Rotation

Peter Atkins, Julio de Paula, James Keeler

This Topic switches attention from motion in a line to motion first on a circle and then on the surface of a sphere. By imposing certain constraints on the wavefunction it shows how the energy and angular momentum of a particle rotating in two and three dimensions are quantized. The conclusions are crucial to the description of electrons in atoms and of molecules rotating in space.

Oxford University Press eBooksDec 26, 20250 citations

Rotation

Peter Joshua Atkins, Julio de Paula, David Young Smith

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.

Oxford University Press eBooksDec 15, 20160 citations

Angular Momentum

Victor Anisimov, James J. P. Stewart

Angular momentum theory provides a mathematical foundation for the rotation of the eigenvectors of the angular momentum operator, the spherical harmonics functions. A useful property of spherical harmonics is that their rotation in 3D space corresponds to rotation in the virtual space of their azimuthal degrees of freedom. In addition to that, spherical harmonics functions that have different orbital indices do not mix. Given that spherical harmonics functions have known rotation properties, and that any arbitrary function can be expanded in a series of spherical harmonics, this property allows the rotation of an electrostatic potential when it is expanded in the series of spherical harmonics. In this chapter, the exponential form of the rotation operator, the rotation matrix for a 3D rotation of the coordinate system, and the principles of rotation of vectors and functions are described. The chapter provides a detailed self-contained introduction to the theory of angular momentum, derives the commutative properties of the angular momentum operator, and explains shift operators, while minimizing references to quantum mechanics and entirely avoiding group theory.

CRC Press eBooksDec 6, 20190 citations

Translational motion in one dimension

Peter Joshua Atkins, Julio de Paula, Ronald Friedman

Contents Free motion 73 Brief illustration 9.1: The wavefunction of a freely moving particle 74 Confined motion: the particle in a box 74 The acceptable solutions 75 Brief illustration 9.2: The energy of a particle in a box 76 The properties...

Oxford University Press eBooksDec 12, 20130 citations