Schrodinger equation of motion

The exact Schrodinger equation of motion, equation (1), may be equivalently Stated in a manner which shows the neglected terms arising from the assumption of the product form for the wavefunction, equation (4). The exact eigenstate Yj(x, R) is expanded in terms of the complete orthonormal set of functions y>i(x R) obtained from the solutions of the electronic equation, equation (5), in which case the nuclear wavefunctions (R) appear as the coefficients in the expansion. This procedure yields the following infinite set of coupled equations for the x (R)6... 

Therefore T (t)) obeys the time-reversed Schrodinger equation of motion. The time-reversal operator 9 is the complex conjugation operator u. 

The sytem is observed before and after the collision in time-dependent channel states ,(t)). The channel index i stands not only for the channel quantum numbers n, j, m, v but also for the relative momentum kj. The entrance channel is denoted by i = 0. The Schrodinger equation of motion for the channel i is, in atomic units. 

To re-express this result in the form given in eqn (8.143) for the field-free case, we need the Heisenberg equation of motion for F(Q, t) for a system in the presence of a magnetic field. This is obtained in the same manner as eqn (8.141) in the field-free case, using eqn (8.211) for the Schrodinger equation of motion to give... 

The motion of a material object is governed by mechanics. Down to the micron scale, classical mechanics describes the relation between forces and motion. The movement of the mobile parts follows Newton s equations of motion, which determine the coordinates and speed of each part of the machine relative to a given origin. In contrast, on the nanoscale, only a correspondence between forces and quantum eigenstates of a nanomachine is available through the Schrodinger equation of motion. Therefore, the concept of movement is not as immediately intuitive as it is at the macroscopic scale... 

The coupled nuclear Schrodinger equations of motion for the nuclear wave-function vector (jxi) 1x2) Ixm)), which is equivalent to Eq. (2.6), has the form of... 

We have already observed that the full phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the deterministic Newton (or Schrodinger) equations of motion, while the time evolution of a small subsystem is stochastic in nature. Focusing on the latter, we would like to derive or construct appropriate equations of motion that will describe this stochastic motion. This chapter discusses some methodologies used for this purpose, focusing on classical mechanics as the underl)dng dynamical theory. In Chapter 10 we will address similar issues in quantum mechanics. 

We will study the equations of motion that result from inserting all this in the full Schrodinger equation, Eq. (1). However, we would like to remind the reader that not the derivation of these equations of motion is the main topic here but the question of the quality of the underlying approximations. 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

This relation defines a time-dependent column vector a. Because = 1, Eq. (7-50) implies afa = 1 a is a unit vector. This is true of all state vectors that correspond to normalized state functions. Substitution of (7-50) into (7-49), subsequent multiplication by u, and integration yield the Schrodinger equation (sometimes called the equation of motion ) for the component ar... 

The u and v representations are sometimes distinguished as the Schrodinger and the Heisenberg representation. For stationary operators P, then, the Heisenberg equation of motion is... 

Compare this with Eq. (7-59) and observe the order of the factors on the right Equation (7-83) is the equation of motion of the statistical matrix in the Schrodinger representation p is constant, of course, in the Heisenberg representation. 

To obtain the equation of motion for p, we consider the time-dependent Schrodinger equation... 

It is possible to parametarize the time-dependent Schrodinger equation in such a fashion that the equations of motion for the parameters appear as classical equations of motion, however, with a potential that is in principle more general than that used in ordinary Newtonian mechanics. However, it is important that the method is still exact and general even if the trajectories are propagated by using the ordinary classical mechanical equations of motion. 

In a classical limit of the Schrodinger equation, the evolution of the nuclear wave function can be rewritten as an ensemble of pseudoparticles evolving under Newton s equations of motion... 

This gives the equation of motion (in one dimension), known as Schrodinger s equation ... 

We would like to point out some steps of derivation of the nonrelativistic limit Hamiltonians by means of the Foldy-Wouthuyisen transformation (Bjorken and Drell, 1964). The method is based on the transformation of a relativistic equation of motion to the Schrodinger equation form. 

In complete analogy to the diabatic case, the equations of motion in the adiabatic representation are then obtained by inserting the ansatz (29) into the time-dependent Schrodinger equation for the adiabatic Hamiltonian (7)... 

It is impossible to read much of the literature on viscosity without coming across some reference to the equation of motion. In the area of fluid mechanics, this equation occupies a place like that of the Schrodinger equation in quantum mechanics. Like its counterpart, the equation of motion is a complicated partial differential equation, the analysis of which is a matter for fluid dynamicists. Our purpose in this section is not to solve the equation of motion for any problem, but merely to introduce the physics of the relationship. Actually, both the concentric-cylinder and the capillary viscometers that we have already discussed are analyzed by the equation of motion, so we have already worked with this result without explicitly recognizing it. The equation of motion does in a general way what we did in a concrete way in the discussions above, namely, describe the velocity of a fluid element within a flowing fluid as a function of location in the fluid. The equation of motion allows this to be considered as a function of both location and time and is thus useful in nonstationary-state problems as well. 

The Schrodinger equation for the interacting pair of linear molecules is again separated into the equations of center-of-mass motion and relative motion, exactly as this was done in Chapter 5. The equation of the center-of-mass motion is of little interest and will be ignored. The Schrodinger equation of relative motion is given by [354]... 

The radial wavefunctions tp R ,Et)/R needed for the computation of Eq. 6.55 are solutions of the Schrodinger equation of relative motion,... 

Equation (4.9) is the equation of motion of the Lagrange multiplier that restricts the solution to satisfy the Schrodinger equation it is to be solved subject to the final-state condition (4.10). Equation (4.11) is the Schrodinger equation for our system it is to be solved subject to the initial condition (4.12). The field that results from these calculations is given by... 

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