Quantum mechanics system Hamiltonian

Up until now, little has been said about time. In classical mechanics, complete knowledge about the system at any time t suffices to predict with absolute certainty the properties of the system at any other time t. The situation is quite different in quantum mechanics, however, as it is not possible to know everything about the system at any time t. Nevertheless, the temporal behavior of a quantum-mechanical system evolves in a well defined way drat depends on the Hamiltonian operator and the wavefiinction T" according to the last postulate... 

The Hamiltonian models are broadly variable. Even for an isolated molecule, it is necessary to make models for the Hamiltonian - the Hamiltonian is the operator whose solutions give both the static energy and the dynamical behavior of quantum mechanical systems. In the simplest form of quantum mechanics, the Hamiltonian is the sum of kinetic and potential energies, and, in the Cartesian coordinates that are used, the Hamiltonian form is written as... 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

Here, for notational convenience, we have assumed that Vnm = We would like to emphasize that the mapping to the continuous Hamiltonian (88) does not involve any approximation, but merely represents the discrete Hamiltonian (1) in an extended Hilbert space. The quantum dynamics generated by both Hamilton operators is thus equivalent. The Hamiltonian (88) describes a general vibronically coupled molecular system, whereby both electronic and nuclear DoF are represented by continuous variables. Contrary to Eq. (1), the quantum-mechanical system described by Eq. (88) therefore has a well-defined classical analog. 

On different atoms these orbitals are assumed orthogonal. However atomic orbitals are not eigenfunctions of the considered quantum-mechanical system, as the Hamiltonian matrix elements between orbitals of various atoms are not equal to zero and we use, for account of diagonal elements, the data [7] for energy of s- and d- states of elements. 

The model that is outlined above is generated from a one-electron Hamiltonian and is only an approximation to the tme wavefimction for a multielectron system. As suggested earlier, other components may be added as a linear combination to the wavefimction that has just been derived. There are many techniques used to alter the original trial wavefimction. One of these is frequently used to improve wavefimctions for many types of quantum mechanical systems. Typically a small amount of an excited-state wavefimction is included with the minimal basis trial fimction. This process is called configuration interaction (Cl) because the new trial function is a combination of two molecular electron configurations. For example, in the H2+ system a new trial fimction can take the form... 

For a quantum mechanical system one must replace p by the corresponding operator, giving the following simple spin Hamiltonian for a free electron in a magnetic field ... 

For quantum-mechanical systems, the moment definitions in Eqn. (8) can be made into definitions of moment operators by replacing position coordinates with their corresponding position operators. The molecular Hamiltonian for a molecule experiencing an external electrical potential following the convention of Eqn. (7) is... 

This form is applicable to any quantum-mechanical system, given the appropriate Hamiltonian and wavefunction. Most applications to chemistry involve systems containing several particles—the electrons and nuclei in atoms and molecules. An operator equation of the form... 

We will focus attention on computational methods based on a quantum mechanical wave function treatment of the system, supplemented with a classical interaction with the external electric fields. The quantum mechanical system should be possible to describe with a time-independent Hamiltonian, whereas the external perturbation can be time-independent or time-dependent. 

In response theory one considers a quantum mechanical system described by the time-independent Hamiltonian which is perturbed by a time-dependent perturbation F(t, e)... 

The starting point for the MCSCF/MM response method is the Hamiltonian for the total system containing both the quantum mechanical and the classical system and it is given by the sum of three terms Are Hamiltonian of the quantum mechanical system in vacuum (Hqm), the Hamiltonian, represented as a force field, for the classical system (Hmm) and the interactions between the quantum mechanical and the classical system (Hqm/mm)- This is written as... 

Translated to quantum mechanics, the Hamiltonian operator provides the tool to analyze the spectra. The form of the operator in the principal axis system, called the PAM form, is... 

In the case of a quantum mechanical system, the problem in general is to compute expectation values, in particular the energy, of bosonic or fermionic ground or excited eigenstates. For systems with n electrons, the spatial coordinates are denoted by a 3n-dimensional vector R. In terms of the vectors ( specifying the coordinates of electron number i, this reads R = (r1 . .., r ). The dimensionless Hamiltonian is of the form... 

Hamiltonian (H) - An expression for the total energy of a mechanical system in terms of the momenta and positions of constituent particles. In quantum mechanics, the Hamiltonian operator appears in the eigenvalue equation Hv /= v /, where E is an energy eigenvalue and / the corresponding eigenfunction. 

It is also interesting to consider the classical/quantal correspondence in the number of energized molecules versus time N(/, E), Eq. (8.22), for a microcanonical ensemble of chaotic trajectories. Because of the above zero-point energy effect and the improper treatment of resonances by chaotic classical trajectories, the classical and quantal I l( , t) are not expected to agree. For example, if the classical motion is sufficiently chaotic so that a microcanonical ensemble is maintained during the decomposition process, the classical N(/, E) will be exponential with a rate constant equal to the classical (not quantal) RRKM value. However, the quantal decay is expected to be statistical state specific, where the random 4i s give rise to statistical fluctuations in the k and a nonexponential N(r, E). This distinction between classical and quantum mechanics for Hamiltonians, with classical f (/, E) which agree with classical RRKM theory, is expected to be evident for numerous systems. 

We now present a simple application of the above formalism. Let us consider a simple quantum mechanical system containing only two states I) and II>, which are eigenfunctions of the Hamiltonian... 

The theory we have developed so far is applicable to any quantum mechanical system. In this section we consider the important special case where the unperturbed Hamiltonian is a sum of one-particle Hamiltonians... 

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