Generalized momentum operator

This means that the generalized momentum operator is... 

Generalized momentum operator, 248 Genetic Algorithms (GA) for global—... 

Generalized momentum operators as defined by Eq. (2.77) can be used in wave mechanical as well as in matrix mechanical formulations. It ensures that the operators are Hermitian, and that momenta, 7r, conjugated to generalized coordinates, qh fulfil commutation relations similar to the canonical relations of Cartesian coordinates and momenta,... 

Expressions for the calculation of magnetic properties at the relativistic level of theory can be derived by replacing the momentum operator p in the Dirac equation (1) with the generalized momentum operator n... 

The presence of a magnetic field can be included in the so-called minimal coupling by addition of a vector potential A to the momentum operator p, forming a generalized momentum operator jt, which for an electron (charge of -1) is given by eq. (8.21). 

With the generalized momentum operator n replacing p, the time-independent Dirac equation may be separated analogously to the procedure in Section 8.2.1 to give the equivalent of eq. (8.13). 

The introduction of the generalized momentum operator in the one-electron kinetic energy part of the Dirac equation leads to three new interaction terms, as shown in eq. (8.29) and (8.30). It should be noted that the last two terms will also show up in a non-relativistic treatment when the magnetic vector potential is included, and only the s B term should be considered a relativistic effect. 

When both nuclear spins and an external magnetic field are present, there is an additional mixed Aext Aa term arising from the expansion of the generalized momentum operator. 

The expansion of the generalized momentum operator only involves the magnetic interactions in the electronic part of the wave function. Since the corresponding nuclear part has been separated out by the Bom-Oppenheimer approximations, we need to add a few terms corresponding to the (classical) interaction of the nuclear magnetic moments with an external magnetic field and between nuclei. 

Next, let us consider the kinetic energy in the first term on the left-hand side of Eq. (6.101). The square of the generalized momentum operator. A, is derived as... 

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