Angular momentum lowering operator

As stated above, the CG coefficients can be worked out for any particular case using the raising and lowering operator techniques demonstrated above. Alternatively, as also stated above, the CG coefficients are tabulated (see, for example, Zare s book on angular momentum the reference to which is given earlier in this Appendix) for several values of j, j, and J. 

The derivative of the core operator h is a one-electron operator similar to the nucleus-electron attraction required for the energy itself (eq. (3.55)). The two-electron part yields zero, and the V n term is independent of the electronic wave function. The remaining terms in eqs. (10.89), (10.90) and (10.95) all involve derivatives of the basis functions. When these are Gaussian functions (as is usually the case) the derivative can be written in terms of two other Gaussian functions, having one lower and one higher angular momentum. 

The raising and lowering operators for spin angular momentum as defined by equations (5.18) are... 

In the following, we shall denote the angular momentum operator of a single particle by a lower case letter and use a capital letter for the angular momentum operator of the total system. 

PROBLEM 3.5.6. One can define linear ladder operators for angular momentum (orbital or spin) the raising operator + = Lx + iLy and the lowering operator =LX — iLy. (a) Verify that brute-force expansion yields + =... 

Operation on this state with./ =J + J2+, using equation (5.20) leads to zero. This means that m already possesses its maximum value so this second state has angular momentum j = j + y 2 - 1. This can be confirmed by operating on the state with J. All lower values of m for this value of / can then be obtained by successive application of the lowering operator./ = J +. A. ... 

In Section 4.1, the identification of complefe radial and angular momentum raising and lowering operators for fhe familiar spherical harmonics is presented in our own version, as a point of reference for some extensions in Section 4.2 for fhe spheroconal harmonics. The resulfs in Section 4.1 have been known in the literature [44, 46], but here the interest is in their adaptation and extension to eigenfunctions of P and Lf, and and H. ... 

Complete radial and angular momentum raising and lowering operators for a free particle in three dimensions... 

The aim of fhis work of identifying fhe complete radial and angular momentum raising and lowering operators is implemented by Eqs. (120-122). The proof of fheir actions on any of fhe eigentstates of Eq. (119) is illusfrafed by Eqs. (123-125) for the angular part including the factorization of the radial parts, whose actions are exhibited by Eqs. (126 and 127). Their combined effects of Eqs. (128 and 129) lead to the two alternative representations of the Hamiltonian in Eq. (130). 

We consider, in the following, fhe successive application of this operator on the lower angular momentum eigenstates, starting with that for i = 0, Zo(kr). The application of the angular momentum operators give zero and the net result is as follows ... 

Of course, the connection between Sections 4.1 and 4.2.2 is provided by Euler s formula = cos mtp isin mtp. Thus, the simultaneous raising and lowering actions of the linear momentum operators on the order of the angular momentum and radial eigenfunctions is established for cartesian, spherical, and spheroconal representations. 

In fact all the factors containing a lowering operator go to zero and one obtains zero because of Eq. (A.34). This means that it is not necessary to consider the precessional effects in projecting out to lowest order the role of angular momentum. 

At this point it is useful to define creation (at) and annihilation (a) operators, which are analogous to angular momentum raising and lowering operators. These at, a operators profoundly simplify the algebra needed to set up the polyad Heff matrices, to apply some of the dynamics diagnostics discussed in Sections 9.1.4 and 9.1.7, and to transform between basis sets (e.g., between normal and local modes). They also provide a link between the quantum mechanical Heff model, which is expressed in terms of at, a operators and adjustable molecular constants (evaluated by least squares fits of spectra), and a reduced-dimension classical mechanical HeS model. 

Atoms have complete spherical symmetry, and the angular momentum states can be considered as different symmetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower symmetry. Symmetry operations for the molecule are transformations such as rotations about an axis, reflection in a plane, or inversion through a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule form a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A1A on molecular symmetry. 

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