Momentum canonical

In order to obtain the Hamiltonian for the system of an atom and an electromagnetic wave, the classical Hamilton function H for a free electron in an electromagnetic field will be considered first. Here the mechanical momentum p of the electron is replaced by the canonical momentum, which includes the vector potential A of the electromagnetic field, and the scalar potential O of the field is added, giving [Sch55]... 

The variational formalism makes it possible to postulate a relativistic Lagrangian that is Lorentz invariant and reduces to Newtonian mechanics in the classical limit. Introducing a parameter m, the proper mass of a particle, or mass as measured in its own instantaneous rest frame, the Lagrangian for a free particle can be postulated to have the invariant form A = mulxiilx = — mc2. The canonical momentum is pf, = iiiuj, and the Lagrangian equation of motion is... 

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,... 

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 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... 

In the presence of the external fields the vector potential appears in the Hamiltonian at Eq. (1) and the space translation symmetry is, therefore, lost. However, there exists a generalization, i.e. the phase space translation group [3] which provides a symmetry associated with the CM motion of the system in the presence of the external fields. The new conserved quantity which is the corresponding generalization of the total canonical momentum of the field-free case is the so-called pseudomomentum K [3,4]... 

Here k = (a>/c)k defines the propagation direction of the laser beam. Employing the relation between the canonical momentum, the principal function S and the kinetic momentum of the electron, i.e. n = VS = p — ([Pg.12]

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... 

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). 

We now abandon conventional quantum chemical wisdom, and embrace the relativistic theory of electromagetic interactions wholeheartedly. In a singleparticle theory, the interaction between an electron and a vector potential, A(r), is included in the Dirac hamiltonian by modifying the canonical momentum, so that... 

For the computation of the second derivatives given in Eq. (6.8) for the shielding tensor, it is necessary to specify the molecular (electronic) Hamiltonian H in the presence of a magnetic field. The latter is obtained by replacing the canonical momentum p in the kinetic energy part of H by the kinetic momentum n... 

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