Angular Momentum of a One-Particle System

In Section 3.3 we found the eigenfunctions and eigenvalues for the linear-momentum operator p. In this section we consider the same problem for the angular momentum of a particle. Angular momentum is important in the quantum mechanics of atomic structure. We begin by reviewing the classical mechanics of angular momentum. 

Classical Mechanics of One-Particle Angular Momentum. Consider a moving particle of mass m. We set up a Cartesian coordinate system that is fixed in space. Let r be the vector from the origin to the instantaneous position of the particle. We have 

The particle s angular momentum L with respect to the coordinate origin is defined in classical mechanics as 

We get the quantum-mechanical operators for the components of orbital angular momentum of a particle by replacing the coordinates and momenta in the classical equations (5.39) by their corresponding operators [Eqs. (3.21)-(3.23)]. We find 

Since the commutation relations determine which physical quantities can be simultaneously assigned definite values, we investigate these relations for angular momentum. Operating on some function/(jt, y, z) with Ly, we have 

We next evaluate the commutators of with each of its components, using commutator identities of Section 5.1. 

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In conventional quantum mechanics, a wavefunction d ribing the ground or excited states of a many-particle system must be a simultaneous eigenfunction of the set of operators that commute with the Hamiltonian. Thus, for example, for an adequate description of an atom, one must introduce the angular momentum and spin operators L, S, L, and the parity operator H, in addition to the Hamiltonian operator. 

In a many-electron system, one must combine the spin functions of the individual electrons to generate eigenfunctions of the total Sz =Li Sz(i) ( expressions for Sx = j Sx(i) and Sy = j Sy(i) also follow from the fact that the total angular momentum of a collection of particles is the sum of the angular momenta, component-by-component, of the individual angular momenta) and total S2 operators because only these operators commute with the full Hamiltonian, H, and with the permutation operators Pjj. No longer are the individual S2(i) and Sz(i) good quantum numbers these operators do not commute with Pjj. 

Mathematical functions play an important role in thermodynamics, classical mechanics, and quantum mechanics. A mathematical function is a rule that delivers a value of a dependent variable when the values of one or more independent variables are specified. We can choose the values of the independent variables, but once we have done that, the function delivers the value of the dependent variable. In both thermodynamics and classical mechanics, mathematical functions are used to represent measurable properties of a system, providing values of such properties when values of independent variables are specified. For example, if our system is a macroscopic sample of a gas at equilibrium, the value of n, the amount of the gas, the value of T, the temperature, and the value of V, the volume of the gas, can be used to specify the state of the system. Once values for these variables are specified, the pressure, P, and other macroscopic variables are dependent variables that are determined by the state of the system. We say that P is a state function. The situation is somewhat similar in classical mechanics. For example, the kinetic energy or the angular momentum of a system is a state function of the coordinates and momentum components of all particles in the system. We will find in quantum mechanics that the principal use of mathematical functions is to represent quantitites that are not physically measurable. 

Here p is the radius of the effective cross-section, (v) is the average velocity of colliding particles, and p is their reduced mass. When rotational relaxation of heavy molecules in a solution of light particles is considered, the above criterion is well satisfied. In the opposite case the situation is quite different. Even if the relaxation is induced by collisions of similar particles (as in a one-component system), the fraction of molecules which remain adiabatically isolated from the heat reservoir is fairly large. For such molecules energy relaxation is much slower than that of angular momentum, i.e. xe/xj > 1. 

In the present calculation the SIC potential is introduced for each angular momentum in a way similar to the SIC one for atoms [9]. The effects of the SIC are examined on the CPs of three materials, diamond, Si and Cu compared with high resolution CP experiments except diamond [10, 11]. In order to examine the quasi-particle nature of the electron system, the occupation number densities of Li and Na are evaluated from the GWA calculation and the CPs are computed by using them [12, 13]. 

Of special importance for the application of the quantum conditions and of the correspondence principle is the case in which the Hamiltonian function is not changed by the rotation as a whole of an atomic system about a fixed direction in space. If we introduce as co-ordinates the azimuth =qf of one of the particles of the system together with the differences of the azimuths of the other particles from , and other magnitudes depending only on the relative position of the particles of the system with respect to the fixed direction in space, will be a cyclic variable and the momentum p conjugated to it is, by 6, the angular momentum of the system... 

Under no external forces, Vcm is unchanged. In addition, since the intermolecular potential is a function neither of R nor of the system total orientation, the motion of the CM can be eliminated because it is a constant. As a result, we have transformed the motion of the two particles with masses mi and m2 into the motion of just one particle with mass /a, kinetic energy (l/2)/xv and angular momentum L around the scattering centre located at r = 0. 

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