Hamiltonian wave-mechanical

Eigen function In wave mechanics, the Schrodinger equation may be written using the Hamiltonian operator H as... 

Before considering the quantum-mechanical vibrational wave functions and energies, we must find the classical-mechanical Hamiltonian for vibration. (Quantum mechanics is peculiar in that it depends on classical mechanics to formulate the Hamiltonian, and yet classical mechanics is only a limiting case of the more general theory quantum mechanics.)... 

For both methods, we describe the interactions between the quantum subsystem and the classical subsystem as interactions between charges and/or induced charges/dipoles and a van der Waals term [2-18]. The coupling between the quantum subsystem and the classical subsystem is introduced into the quantum mechanical Hamiltonian by finding effective interaction operators for the interactions between the two subsystems. This provides an effective Schrodinger equation for determining the MCSCF electronic wave function of the molecular system exposed to a classical environment, a structured environment, such as an aerosol particle. 

The correspondence between Hamiltonian mechanics and wave mechanics is rigorous, not requiring the neglect of powers of h, if the symbols are understood in a certain way [137]. 

In wave mechanics, the electron density is given by the square of the wave function integrated over - 1 electron coordinates and the wave function is determined by solving the Schrddinger equation. For a system of Mnudd nuclei and iVeiec electrons, the electronic Hamiltonian operator contains the terms given in eq. (B.8). 

Hamiltonian operator An operator that describes the kinetic and potential energy of a system treated by wave mechanics. 

This Hamiltonian, which was introduced by Schmickier [12], is equivalent to earher formulations by Levich and Dogo-nadze in terms of wave mechanics [5] it is also related to the spin-boson model for homogeneous electron exchange [13] and to the Anderson—Newns model for specific adsorption [14]. 

When the quantum mechanical Hamiltonian for vibration is constructed from Eq. (22.4-16) there are 3 — 5 or 3n — 6 terms, each one of which is a harmonic oscillator Hamiltonian operator. The variables can be separated, and the vibrational Schrodinger equation is solved by a vibrational wave function that is a product of 3 - 5 or 3n — 6 factors ... 

The Uncertainty Principle, 21. Wave Mechanics, 23. Functions and Operators, 25. The General Formulation of Quantum Mechanics, 27. Expansion Theorems, 31. Eigenfunctions of Commuting Operators, 34. The Hamiltonian Operator, 37. Angular Momenta, 39. 

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 reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

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