Angular momentum raising operator

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). 

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. 

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

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 + =... 

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. 

By utilizing eqn (14.28) and the well-known property of the raising and lowering operator of angular momentum,the evaluation of the matrix element of eqn (14.25) is straightforward. For example, we can obtain... 

Next, we can work out the effect of the raising and lowering operators by applying them to an angular momentum function. Using T to designate any particular eigenfunction of... 

If the raising operator is applied to an angular momentum function where the z-component is already at the maximum (i.e., where M = /), the result is zero. For instance,... 

The application of a raising or a lowering operator changes the M quantum number of the angular momentum state, thereby changing one function into another. However, that... 

Three important examples of pure spin operators are the raising and lowering operators S and Si (also known as the step-up and step-down operators or as the shift operators) and the operator for the z component of the spin angular momentum S. From the effect of these operators on the spin functions (again assuming a one-particle state)... 

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