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Cambridge University Press eBooks

Two-Body Rotation and Angular Momentum

Uri Peskin

Published May 11, 2023• 0 citations
Publisher

Abstract

The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.

DOI
10.1017/9781108877787.010

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The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.
Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.
This chapter presents a detailed examination of a number of relatively simple problems pertaining to the scattering and bound states of two structureless particles interacting through a central field. It describes the spectrum of the angular momentum operators by a method that does not refer at all to the structure of the system. The spherical harmonics are the eigenfunctions appropriate to a single particle that can be described by the three classical degrees of freedom, or for a pair of such particles in their center-of-mass frame. The formulation of collision theory that is based on the one-body Schrodinger equation with a Hermitian interaction. The chapter introduces and discusses the observables pertaining to the two-body problem and explores the eigenvalues and eigenfunctions of these observables. Since the degeneracies of the energy spectrum are a consequence of the symmetries of the Hamiltonian, they can be determined experimentally by destroying this symmetry.
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.
In this work, the quantum-mechanical problem of the motion of two material points of different masses on a three-dimensional sphere with a non-fixed position of the center of mass of the system is formulated on the basis of the previously solved classical problem. It is shown that the established Schrödinger equation includes two different reduced masses, depending on the distance between the points. For the case of the interaction potential of points, depending only on the distance between them, this equation allows the separation of variables into a radial, depending on the relative distance and both the reduced masses and the spherical part. The equation for the spherical part depends only on one of the above reduced mass and allows one to formulate and solve the problem of a rigid rotator - the distance between the points is fixed. The solution and spectrum of the problem of a rigid rotator are found. It is shown that the spectrum of the system has an upper limit that does not depend on the distance between points, in contrast to the spectrum in a flat space.
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.