Schrodinger equation interaction representation

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schrodinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes. 

The use of c instead of a is said to constitute the interaction representation it is designed to remove H from the Schrodinger equation for c. Passage from a to c is equivalent to a change from ua to e immug, and in the special case where ua is an eigenfunction of H, the latter may be written e liln)B< tua. Hence, all matrices with components PIm(it) in the basis are transformed to P, m by the rule... 

We now regard Eq. (8-233) as analogous to Schrodinger s equation, and proceed to carry out the transformation to the interaction representation described in Chapter 7, Section 7.7. We define the transformed density matrix R and the transformed potential U by... 

In an exact representation of the interaction between a solute and a solvent, i.e., solvation, the solvent molecules must be explicitly taken into account. That is, the solvent is described on a microscopic level, where the individual solvent molecules are considered explicitly. The interaction potential between solvent molecules and between solvent molecules and the solute can, in principle, be found by solving the electronic Schrodinger equation for a system consisting of all the involved molecules. Typically, in practice, a more empirical approach is followed where the interaction potential is described by parameterized energy functions. These potential energy functions (often referred to as force fields) are typically parameterized as pairwise atom-atom interactions. 

The operators P and obey the usual equal time anticommutation relations. The time-dependence of the field operators appearing here is due to the Heisenberg representation in the L-space. In view of the foregoing development which parallels the traditional Schrodinger quantum theory we may recast the above Green function in terms of the interaction representation in L-space. This leads to the appearance of the S-matrix defined only for real times. We will now indicate the connection of the above to the closed-time path formulation of Schwinger [27] and Keldysh [28] in H-space. Equation (82) can be explicitly... 

In 1997, Pakiari and Mohammadi used the FSGO basis set for a perturbation variation Rayleigh Ritz (PV = RR) calculation. We used a matrix representation Schrodinger equation for the configuration interaction calculation. 

We shall start from methods similar to that previously described, characterized by the use of the apparent surface charge (ASC) description of the electrostatic interaction term Ve/, passing then to consider other continuum methods, which use a different description of Ve/.To complete the exposition we shall introduce, where appropriate, methods not based on the solution of a Schrodinger equation, and hence not belonging to the category of continuum effective Hamiltonian methods. We shall pass then to a selection of methods based on mixed continuum-discrete representation of the solvent, to end up with the indication of some approaches based on a full discrete representation of the solvent. 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

The Schrodinger equation of the Floquet Hamiltonian in JT, where 9 is a dynamical variable, is equivalent, in an interaction representation, to the semiclassical Schrodinger equation in where 0 is considered as a parameter corresponding to the fixed initial phase. The dynamics of the two models are identical if the initial photon state in the Floquet model is a coherent state. 

Interaction Representation The Schrodinger equation of the Floquet Hamiltonian in... 

Note that an equation similar to (10.19a) that relates the interaction representation of any other operator to the Schrodinger representation... 

Using this reduction operation we may obtain interesting relationships by taking traces over bath states of the equations of motion (10.15) and (10.21). Consider for example the Liouville equation in the Schrodinger representation, Eq. (10.15). (Note below, an operator A in the interaction representation is denoted Ai while in the Schrodinger representation it carries no label. Labels S and 5 denote system and bath.)... 

Equations (10.126) and (10.127) represent the quantum master equation in the interaction representation. We now transform it to the Schrodinger picture using... 

If the Hamilton operator depends on time in a harmonic fashion, the time dependence can be eliminated by transformation into a rotating reference frame in analogy to the transformation of the Bloch equations. A representation in the rotating frame is also called interaction representation in quantum mechanics. If the time dependence is more general, the Schrodinger equation is solved for small enough time increments, during which H is approximately constant. For each of the n time increments At a solution of the form (2.2.46) applies. The complete evolution operator is the time ordered product of the incremental evolution operators. This operator is written in short hand as... 

Even this procedure still contains an element of arbitrariness and reveals an unsolved problem of quantum chemistry. The representation of the exchange interaction by the local electron probability density (multiplied by a suitable numerical factor, of order 0.5) seems a plausible way to approximate a complex interaction that actually depends on the locations of pairs of electrons. However as yet, there has been no derivation from the many-electron Schrodinger equation of any systematic series of successive approximations to its solution, of which the local density approximation would be the first approximation. There is a belief and a hope that such a derivation will be achieved, but it remains a tantalizing challenge now. 

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