Stationary time-independent

For a stationary (time-independent) set of wavefunctions, the time derivative on the right hand side vanishes and the equation obviously transforms back into the previous one. Now the system is perturbed by an external perturbation Hp, which is chosen to have a well-defined temporal periodicity ... 

For a stationary, time-independent equilibrium structure (in the quantum chemical sense) all these effects can in principle be captured at once by the electronic wave function... 

The energy properties refers to stationary, time-independent, states of the solute molecules. This implies a complete equilibration between the degrees of freedom, electronic and nuclear, of the solute and of the solvent. For the last two categories, on the contrary, we have to consider not only an equilibrated solute-solvent system but also the dynaunics of its response to a time dependent perturbation. 

If the system were to obey classical mechanics, then E = H, where H represents the Hamiltonian for a stationary (time-independent) state. For wave motion this equation is rewritten as... 

The physics that corresponds to the stationary, time-independent, description is that there is a large number of binary collisions. Some have started a long time ago and the products are already receding, others are yet to enter the range of the force. In such a description the rate of collisions is constant. It is tike a crossed molecular beam experiment that is running under steady conditions. Like the experiment, what we want to detemtine is the flux of products. 

The understanding, interpretation and practical tools to approach the problem of resonances states in quantum chemistry and molecular physics are basically very well studied. Generally one has either (i) concentrated on the properties of the stationary time-independent scattering solution (ii) attempted to extract the Gamow wave by analytic continuation and/or (iii) considered the time-dependent problem via a suitably prepared reference function or wave-packet. In each case the analysis prompts different explanations, numerical techniques and understanding, see e.g. Ref. [15] for a review and more details. 

Some nonequilibrium phase transitions involve the passage of a system from a stationary (time-independent) nonequilibrium state to a nonstationary nonequilibrium state with rich spatiotemporal order. For example, it has been observed that certain chemically reactive systems can pass from a quiescent homogeneous state to a state characterized by spatial and/or temporal oscillations in the concentrations of certain chemical species. In nonequilibrium thermodynamics, Prigogine has argued that such phase transitions result from the violation of the stability condition... 

In Figure 2.2(a) states m and n of an atom or molecule are stationary states, so-called because they are time-independent. This pair of states may be, for example, electronic, vibrational or rotational. We consider the three processes that may occur when such a... 

In many applications of quantum mechanics in physics and chemistry, interest is primarily in the description of the stationary, or time-independent, states of a system. Thus, it is sufficient to determine the energies and wave-functions with the use of the Schitidinger equation in the form... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

Also, the continuity Eq. (258) and the stationary condition (260) allow us to prove in a straightforward manner that, in the time-independent situation, we have ... 

The unperturbed Hamiltonian 3 is the same for all systems and is time-independent. The time-dependent perturbation G(t), different for each system, is considered as a stationary stochastic variable. We may, without loss of generality, suppose that the mean value of G(t) over the ensemble is equal to zero. We denote by a,p,y,. . . the eigenstates of supposed to be non-degenerate, and by fix, the corresponding energies. 

DR. SCHATZ All mixed valence systems have stationary states. In any such state the probability distribution is time independent. Hence, it is incorrect to say that a mixed valence state is inherently time dependent. 

The stationary theory deals with time-independent equations of heat conduction with distributed sources of heat. Its solution gives the stationary temperature distribution in the reacting mixture. The initial conditions under which such a stationary distribution becomes impossible are the critical conditions for ignition. 

Note that, when expressed in terms of the reference moving frame, the distribution in the steady state becomes stationary or time independent. The transition rates in Eq. (64) can also be expressed in the reference system of the trap ... 

To stress this difference between the local and the classical Teorell s model let us work out the overall stationary resistance of the filter R(v, A, D) for a given time-independent flow rate v. We have analogously to (6.3.30)... 

Stationary electro-convection at an electrically inhomogeneous permselective membrane.7 Once again the time-independent version of (6.4.45)-(6.4.49) with the boundary conditions (6.4.54a,b) at x — 0. 

If the thickness of the diffusion layer (d) is time-independent, a steady limiting current is obtained. However, if the electrode reaction occurs at a stationary electrode that is kept at a constant potential, the thickness of the diffusion layer increases with time by the relation S (7iDt)1/2, where t is the time after the... 

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

In spectroscopy, we have a system (atom or molecule) that starts in some stationary state of definite energy, is exposed to electromagnetic radiation for a limited time, and is then found to be in some other stationary state. Let H0 be the time-independent Hamiltonian of the system in the absence of radiation. We have... 

Show that addition of a constant C to a time-independent Hamiltonian H leaves the stationary-state wave functions unchanged and adds C to each energy eigenvalue. 

Of course, there may be more than one. Each 0 is a time-independent solution of the master equation. When normalized it represents a stationary probability distribution of the system, provided its components are nonnegative. In the next section we shall show that this provision is satisfied. But first we shall distinguish some special forms of W. 

This completes the proof of the second lemma. A corollary is that for a time-independent solution either all components are nonnegative, or all non-positive. For a stationary probability distribution one has, of course, psn 0, because C— 1. 

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