Hamiltonian equation classical techniques

In cases where the classical energy, and hence the quantum Hamiltonian, do not contain terms that are explicitly time dependent (e.g., interactions with time varying external electric or magnetic fields would add to the above classical energy expression time dependent terms discussed later in this text), the separations of variables techniques can be used to reduce the Schrodinger equation to a time-independent equation. 

By a routine mathematical technique, the equation can be split into a space dependent part and a time dependent part. These two parts of the Schroedinger equation are set equal to the same constant (the energy E) and solved separately. For our purposes, we are interested in the space dependent, time independent part, which describes a system that is not in a state of change. In Eq. (3.4), H is an operator called the Hamiltonian operator by analogy to the classical Hamiltonian function, which is the sum of potential and kinetic energies, and is equal to the total energy for a conservative system... 

Molecular Dynamics is the term used to refer to a technique based on the solution of the classical equation of motion for a classical many-body system described by a many-body Hamiltonian H. 

The constant value, E, is termed the eigenvalue and this value is, in fact, the energy of the system in quantum mechanics. T is usually termed the wavejunction. The operator H Hamiltonian) in Equation (1), like the energy in classical mechanics, is the sum of kinetic and potential parts. Equation (1) is usually so complicated that no analytical solutions are possible for any but the simplest systems. However, numerical techniques, to be briefly discussed is this section, enable Equation (1) to be converted to an algebraic matrix eigenvalue equation for the energy, and such equations can be effectively handled by powerful computers today. 

The first detailed application of wave mechanics to a stable existent molecule was Heitler and London s 1927 paper in Zeitschrift fur Physik, which served as the cornerstone for many succeeding treatments. The paper employs a perturbation technique to solve the time-independent Schrodinger equation for an electronic wavefunction constructed to represent the H2 molecule. The Hamiltonian for the system is a classical function ... 

The solution of the dynamical problem for the gas and surface atoms requires in principle solution of the quantum mechanical equations of motion for the system. Since this problem has been solved only for 3-4 atomic systems we need to incorporate some approximations. One obvious suggestion is to treat the dynamics of the heavy solid atoms by classical rather than quantum dynamical equations. As far as the lattice is concerned we may furthermore take advantage of the periodicity of the atom positions. At the surface this periodicity is, however, broken in one direction and special techniques for handling this situation are needed. Lattice dynamics deals with the solution of the equations of motion for the atoms in the crystal. As a simple example we consider first a one-dimensional crystal of atoms with identical masses. If we include only the nearest neighbor interaction, the hamiltonian is given by ... 

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