Operator canonical momentum

Canonical momentum operator, 248 Car and Parrinello (CP) method, 388 CASMP2, CASPT2 methods, 132 CASVB method, 202 Cavity, in reaction field models, 393 CBS-4, CBS-q, CBS-Q and CBS-APNO methods, 167... 

The canonical momentum operator is indicated by p and a X V(r) is a spin-orbit contribution. The symbol 8 in eq. (14) represents the delta function and toa/S is the transition frequency between occupied a) and empty states /3). 

The connection of the canonical momentum operator p with the effect of external vector potentials A on a moving electron with charge (/g = — e is often simply written as... 

The form invariance of the Schrodinger equation will then lead to gauge invariant expectation values of the Hamiltonian. However, this will not be the case for an arbitrary operator. In particular, it turns out that expectation values of the canonical momentum operator, given in Eq. (2.45), are not gauge invariant, whereas expectation values of the mechanical or kinematical momentum operator, given in Eq. (2.97), are gauge invariant [see Exercise 2.15]... 

The components of the position operator, therefore, commute with one another furthermore they are canonically conjugate to the momentum operators... 

The B cyclic theorem is a Lorentz invariant construct in the vacuum and is a relation between angular momentum generators [42], As such, it can be used as the starting point for a new type of quantization of electromagnetic radiation, based on quantization of angular momentum operators. This method shares none of the drawbacks of canonical quantization [46], and gives photon creation and annihilation operators self-consistently. It is seen from the B cyclic theorem ... 

Indeed, in this context, position and momentum operators are canonically conjugated operators, through the relations (62). When eq.(61) is used as... 

A new momentum operator Pj a must therefore be introduced, defined in such a way to be canonically conjugated to A.y-/V through the commutation relations... 

With that in mind, we have to suggest a second important correction which concerns the role of the momentum operators, haDjmomentum operator PjjCt is the canonically conjugated partner of Xj>a, we understand that the use of haDj Ct in the representation Phi,h2(g 1532), although legitimate, is improper and subtle. Moreover, since hi and h2 are connected by the relation (68), it should be useful to introduce the new parameter /ieff in the formulation. Both these requirements can be fulfilled if we determine a new unitary irreducible and infinite-dimensional representation of the group D"... 

Having expressed the Hamiltonian in terms of the canonical momenta, we can readily quantize the particles dynamics. To do so we replace each particle s canonical momentum by the momentum operator in the coordinate representation,... 

The quantum Hamiltonian of the classical kicked rotor, defined by the classical Hamiltonian function (5.1.2), is easily obtained by canonical quantization. On replacing the classical angular momentum L by the quantum angular momentum operator L according to... 

The electronic magnetic multipoles (25)-(27) are unperturbed, or permanent, moment operators. In the presence of a vector potential A(r, t) (we simplify the notation, omitting the index), the canonical momentum is replaced by the mechanical momentum... 

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,... 

The canonical real d-orbital wave functions are linear combinations of the complex eigen functions (solutions) of the hydrogenic Schrodinger equation and the orbital angular momentum operator. Thus, instead of a set of five degenerate orbitals that may be indexed by values of orbital... 

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

This is the quantum mechanical analog of the classical expression Eq. (11), with the canonical momentum replaced by the expectation value of the momentum operator (P)t. It is then clear that the choice... 

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