Angular momentum identities

The formulas in this section (and those in the main text that derive therefrom) apply only for integer values of the indices, so no distinction is made between, e.g., ( — 1) and ( — 1) . These formulas are equivalent to those in the books by Edmonds [17] and Brink and Satchler [73]. 

Writing in terms of the Wigner 3-j symbols, two spherical harmonics F (f) and couple as follows to form a resultant angular momentum 

The 3-j symbols evaluate to real numbers, so equation (Al) continues to be satisfied if each Y is replaced by the corresponding Y.  

The product of two spherical harmonics of the same argument can be written as a linear combination of harmonics according to the equation 

Complex conjugate signs can be added or removed using the formula 

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Appendix A. Angular Momentum Identities Appendix B. Differential Properties of the References... 

When the three principal moment of inertia values are identical, the molecule is termed a spherical top. In this case, the total rotational energy can be expressed in terms of the total angular momentum operator J2... 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

This list of postulates is not complete in that two quantum concepts are not covered, spin and identical particles. In Section 1.7 we mentioned in passing that an electron has an intrinsic angular momentum called spin. Other particles also possess spin. The quantum-mechanical treatment of spin is postponed until Chapter 7. Moreover, the state function for a system of two or more identical and therefore indistinguishable particles requires special consideration and is discussed in Chapter 8. 

The many-electron wave function (40 of any system is a function of the spatial coordinates of all the electrons and of their spins. The two possible values of the spin angular momentum of an electron—spin up and spin down—are described respectively by two spin functions denoted as a(co) and P(co), where co is a spin degree of freedom or spin coordinate . All electrons are identical and therefore indistinguishable from one another. It follows that the interchange of the positions and the spins (spin coordinates) of any two electrons in a system must leave the observable properties of the system unchanged. In particular, the electron density must remain unchanged. In other words, 4 2 must not be altered... 

A spinning electron also has a spin quantum number that is expressed as 1/2 in units of ti. However, that quantum number does not arise from the solution of a differential equation in Schrodinger s solution of the hydrogen atom problem. It arises because, like other fundamental particles, the electron has an intrinsic spin that is half integer in units of ti, the quantum of angular momentum. As a result, four quantum numbers are required to completely specify the state of the electron in an atom. The Pauli Exclusion Principle states that no two electrons in the same atom can have identical sets of four quantum numbers. We will illustrate this principle later. 

This operator can now be shown to be identical with the operator for an infinitesimal rotation of the vector field multiplied by i, i.e. J = — M. The components of the angular momentum operator satisfy the commutation relations... 

Formula (58) shows that the angular momentum operator for the photon consists of two terms. The first term is identical with the usual quantum-mechanical operator L for the orbital angular momentum in the momentum... 

If j = 0 there is only one vector spherical harmonic which is identical with the longitudinal harmonic Y 1 = nVoo- From this observation it follows that there are no transverse spherical harmonics for j = 0. It also means that the state with angular momentum zero represents a spherically symmetrical state, but a spherically symmetrical vector field can only be longitudinal. Thus, a photon cannot exist in a state of angular momemtum zero. 

Optimization of a series of correlation functions for the total correlated energy of the atom using a large HF base set - this establishes the identity and quantity of angular momentum functions that will be included in each correlation consistent basis set. 

In the following, we pay special attention to the connections among the spherical, Stark and Zeeman basis. Since in momentum space the orbitals are simply related to hyperspherical harmonics, these connections are given by orthogonal matrix elements similar (when not identical) to the elements of angular momentum algebra. 

The identity operator within the space of functions with a hxed value of J and the parity (denoted by p), and that are associated assymptotically with a quantum number K of the body-fixed z component of the total angular momentum, is... 

The last equation is formally identical to the radial Schrodinger equation with a non-integer value of the angular momentum quantum number. Its spectrum is bounded from below and the discrete eigenvalues are given by... 

Since the angular momentum operator and the spatial part of a spin-orbit operator have the same symmetry, from a computational point of view the process of calculation of CjS0,1 or Cj 2 is almost identical to that of obtaining Aj or Bj, respectively. 

In die Dunning family of cc-pVnZ basis sets, diffuse functions on all atoms are indicated by prefixing with aug . Moreover, one set of diffuse functions is added for each angular momentum already present. Thus, aug-cc-pVTZ has diffuse f, d, p, and s functions on heavy atoms and diffuse d, p, and s functions on H and He. An identical prescription for diffuse functions has been used by Jensen (2002) in connection with the pc-n basis sets. 

The Debye theory [220] in which a sphere of volume V and radius a rotates in a liquid of coefficient of viscosity t has already been mentioned. There is angular momentum transfer across the sphere—liquid interface that is, the liquid sticks to the sphere so that the velocity of the sphere and liquid are identical at the sphere s surface. Solution of the rotational diffusion equation... 

Since s 4, it is obvious that every value of / will have two values of j equal to I + I and / - 4. The only exception is / — 0. for which j — 4 these values are identical since it is the absolute magnitude of/ that determines the angular momentum. 

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