Hamiltonian operator field

In continuum field theory, the field dynamical variable and the coordinates X and t are only parameters. With the conjugate momentum f (T, t), and Hamiltonian operator... 

So, ligand-field theory is the name given to crystal-field theory that is freely parameterized. The centrally important point is that ligand-field calculations, whether numerical or merely qualitative, explicitly or implicitly employ a ligand-field Hamiltonian, very much like the crystal-field Hamiltonian, operating upon a basis set of pure d orbitals. Instead of the crystal-field Hamiltonian (Eq. 6.15),... 

Treating the radiation in the semiclassical dipole approximation, the Hamiltonian operator in the presence of an electric field e(t) may be written in atomic units as ... 

If we replace the z-component of the classical angular momentum in equation (6.87) by its quantum-mechanical operator, then the Hamiltonian operator Hb for the hydrogen-like atom in a magnetic field B becomes... 

The Hamiltonian operator for a hydrogen atom in a uniform external electric field E along the z-coordinate axis is... 

Given the Hamiltonian eqn (3.1), it is reasonable to express the eigenfunctions in terms of the electron and nuclear spin quantum numbers ms,mi). Applying to this function only the two terms in the Hamiltonian operator that involve the -direction of the field B we get ... 

Here H is the Hamiltonian operator of the system without external field, M denotes the operator corresponding to the magnetization, and the Trace is the quantum mechanical equivalent of the classical integral over phase space. 

If the nucleus (which is placed in the center) has a positive charge of Ze, where e is the numerical value of the charge of the electron, we obtain the Hamiltonian operator for an electron in this central field,... 

H is the Hamiltonian operator for the total energy, h = Planck s constant / 2tc, t is the time, and is the wave function describing the electronic state. The electric field of the light adds another contribution to the Hamiltonian. Assuming that all the molecules are isolated polarization units, the perturbation part of the Hamilitonian is the electric dipole operator, -p E. Thus,... 

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form ... 

Using italic indices for spatial 3-vectors, but retaining vector notation for the abstract 3-vector index of the matrices t and the gauge field W, the fermion Hamiltonian operator is... 

On the other hand, ab initio (meaning from the beginning in Latin) methods use a correct Hamiltonian operator, which includes kinetic energy of the electrons, attractions between electrons and nuclei, and repulsions between electrons and those between nuclei, to calculate all integrals without making use of any experimental data other than the values of the fundamental constants. An example of these methods is the self-consistent field (SCF) method first introduced by D. R. Hartree and V. Fock in the 1920s. This method was briefly described in Chapter 2, in connection with the atomic structure calculations. Before proceeding further, it should be mentioned that ab initio does not mean exact or totally correct. This is because, as we have seen in the SCF treatment, approximations are still made in ab initio methods. 

Symmetrically opposite recipes are valid for a Hamiltonian operator in momentum space.) When magnetic fields are present, then the momentum vector receives an additional term, the vector potential A. At relativistic speeds the Dirac equation shall be used. 

We are interested in what happens when a magnetic moment fJt interacts with an applied magnetic field B0—an interaction commonly called the Zeeman interaction. Classically, the energy of this system varies, as illustrated in Fig. 2.1a, with the cosine of the angle between l and B0, with the lowest energy when they are aligned. In quantum theory, the Zeeman appears in the Hamiltonian operator... 

Having considered the general expressions for first- and second-order molecular properties, we now restrict ourselves to properties associated with the application of static uniform external electric and magnetic fields. For such perturbations, the Hamiltonian operator may be written in the manner (in atomic units)... 

It is conventional that the ligand field problem for systems with Na> d electrons requires the diagonalization of an effective Hamiltonian operator composed for the electronic kinetic energy T, and both one-electron ligand field terms, and two-electron Coulomb interactions ... 

---