Stationary-state formalism

Here, we call the MEP for situations involving isolated or series of unstable states that are treatable within stationary-state formalisms, the "time-independent MEP" (TIMEP). 

On the other hand, there are many dynamic phenomena whose quantitative description cannot be achieved via a stationary-state formalism, whose hallmark, as already indicated, is the form of Eq. (1) or (2) for the eigenfunction. In other words, now, the complete solution of the TDSE for all t cannot be written as a product of two terms, one of which is the phase that contains time and the other is a time-independent eigenfunction in coordinate space. In these cases, in most real situations one faces a genuine time-dependent many-electron problem (TDMEP), whose solution must be based on the quantitative knowledge of time-dependent, nonstationary (unstable) states, l> q, t). 

The well-undersfood facf of fhe breakdown of "lowest-order perturbation theory" (LOFT) in the area of atom (molecule)—EMF interactions ushered theoretical research into a new age, whose substantial progress depends, apart from esfablishing fhe fundamentals of phenomenology, on the transparency and efficiency of fheoretical approaches to handle the multifarious MEP wifhin compufafional schemes that go beyond the LOPT. These fall into two types of frameworks One which employs, where appropriate, stationary-state formalism (e.g., see Refs. [1, 30-32]), and another in which the aim is the computation of P(f) by solving the TDSE. 

With time-dependent computer simulation and visualization we can give the novices to QM a direct mind s eye view of many elementary processes. The simulations can include interactive modes where the students can apply forces and radiation to control and manipulate atoms and molecules. They can be posed challenges like trapping atoms in laser beams. These simulations are the inside story of real experiments that have been done, but without the complexity of macroscopic devices. The simulations should preferably be based on rigorous solutions of the time dependent Schrddinger equation, but they could also use proven approximate methods to broaden the range of phenomena to be made accessible to the students. Stationary states and the dynamical transitions between them can be presented as special cases of the full dynamics. All these experiences will create a sense of familiarity with the QM realm. The experiences will nurture accurate intuition that can then be made systematic by the formal axioms and concepts of QM. 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

The first rigorous derivation of such a relativistic Hamiltonian for a two-fermion system that makes use of Feynman [13,14] formalism of QED was due to Bethe and Salpeter [30,31]. Recently, Broyles has extended it to many-eleetron atoms and molecules [32]. A detailed account of Broyle s derivation ean be found elsewhere [32,33] and will not be repeated here. Following Broyles, the stationary state many-fermion Hamiltonian based on QED ean be written as... 

We have seen examples of a hysteresis loop being unfolded, opening out to leave a monotonic stationary-state locus. The conditions for this change to occur can be written formally as... 

For the no reaction state ass = a0, the relaxation time given by eqn (8.10) is simply equal to the residence time. In terms of the eigenvalue, we have A = - l/tres, which is negative. The stationary state is always stable, irrespective of a0 and kl. Chemistry makes no contribution (formally we have l/tch,ss = 0, so the chemical time goes to infinity) the perturbation of a does not introduce any B to the system, so no reaction is initiated. The recovery of the stationary state is achieved only by the inflow and outflow. 

The differential curves are peak shaped in all cases even under the stationary state, with a peak potential equal to the formal potential if the current is plotted versus 1Ildex (given by equations (7.3) or (7.7)), and the peak current is given by... 

The effective hamiltonian is energy dependent, complex and nonlocal. To get a formal solution for P T >, the Q-projected hamiltonian (QHQ = Hqq) must be diagonalized. This amounts, in chemistry, to finding out stationary states for the supermolecule that might be relevant for describing different mechanistic pathways. 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

V-clcctron state T, correlation energy can be defined for any stationary state by Ec = E — / o, where Eo = ( //1) and E = ( // 4 ). Conventional normalization ) = ( ) = 1 is assumed. A formally exact functional Fc[4>] exists for stationary states, for which a mapping — F is established by the Schrodinger equation [292], Because both and p are defined by the occupied orbital functions occupation numbers nt, /i 4>, E[p and E[ (p, ] are equivalent functionals. Since E0 is an explicit orbital functional, any approximation to Ec as an orbital functional defines a TOFT theory. Because a formally exact functional Ec exists for stationary states, linear response of such a state can also be described by a formally exact TOFT theory. In nonperturbative time-dependent theory, total energy is defined only as a mean value E(t), which lies outside the range of definition of the exact orbital functional Ec [ ] for stationary states. Although this may preclude a formally exact TOFT theory, the formalism remains valid for any model based on an approximate functional Ec. 

The mathematical techniques most commonly used in chemical kinetics since their formulation by Bodenstein in the 1920s have been the quasi-stationary state approximation (QSSA) and related approximations, such as the long chain approximation. Formally, the QSSA consists of considering that the algebraic rate of formation of any very reactive intermediate, such as a free radical, is equal to zero. For example, the characteristic equations of an isothermal, constant volume, batch reactor are written (see Sect. 3.2) as... 

We assume that the molecule is in a stationary state initially, the wave function of which is describable by HF. In the density matrix formalism [9, 10] (which is equivalent to the usual operator form), the Fock F(0) and density matrices D(Cl> satisfy the time-independent equation... 

In our introduction to the physics of NMR in Chapter 2, we noted that there are several levels of theory that can be used to explain the phenomena. Thus far we have relied on (1) a quantum mechanical treatment that is restricted to transitions between stationary states, hence cannot deal with the coherent time evolution of a spin system, and (2) a picture of moving magnetization vectors that is rooted in quantum mechanics but cannot deal with many of the subder aspects of quantum behavior. Now we take up the more powerful formalisms of the density matrix and product operators (as described very briefly in Section 2.2), which can readily account for coherent time-dependent aspects of NMR without sacrificing the quantum features. 

It is evident that the formal stationary solution Zi = 0 is not applicable to describe the stationary state of the system in respect of the second internal variable Y (eiR cannot be equal to zero). Therefore, we find the stationary value Y2 at the nonzero Z2 by inserting equation (3.36) into the equation for the rate of changing the concentration of Y ... 

The transition between the two stationary states and is the ammonia-maser transition with transition frequency v = (E — E )/h = 23,870,110,000 s . The very existence of the ammonia-maser transition suggests that the states I. and I, of ammonia do indeed exist in reality and not just in the quantum-mechanical formalism. 

Equation (19) describes a non-Markovian process in the CV space. In fact, the forces acting on the CVs depend explicitly on their history. Due to this non-Markovian nature, it is not clear if, and in which sense, the system can reach a stationary state under the action of this dynamics. In [32] we introduced a formalism that allows to map this history-dependent evolution into a Markovian process in the original variable and in an auxiliary field that keeps track of the visited configurations. Defining... 

The ITT approach formally exactly solves the Smoluchowski equation, following the transients dynamics into the stationary state. In this way the kinetic competition between Brownian motion and shearing, which arises from the stationary flux, is taken into account in the stationary distribution function. To compute it explicitly, but approximatively, using ideas based on MCT, MCT-ITT approximates the obtained averages by following the transient structural changes encoded in the transient density correlator. 

Formally, the H-theorem valid for general Fokker-Planck equations states that the solution of (3) becomes unique at long times [47], Yet, because colloidal particles have a non-penetrable core and exhibit excluded volume interactions, corresponding to regions where the potential is infinite, and the proof of the H-theorem requires fluctuations to overcome all barriers, the formal H-theorem may not hold for nondilute colloidal dispersions. Nevertheless, we assume that the system relaxes into a unique stationary state at long times, so that f (f 1 holds. This assumption... 

The equation60 for the amplitude A is formally ft-independent and thus does not change. The classical-limit amplitude function is obtained by solving Eq. (60), with Sj substituted for S(q, t). Equations (60)—(63) define the classical limit of the TDSCF, but rather than use trajectories, these equations lead directly to Ak(qk, t), the classical probability density, which is the analog of the quantum mechanical fc( k, t) 2. For a stationary state in a time-independent potential Ak(qk, t) = Ak(qk) = const x [p (q )]- where pk(qk) is the momentum of mode k at the fixed energy of the system. 

It can be concluded, if the DF role must be preserved, that the statistical formalism of expectation values, represented by equation (1), has to be used in classical quantum mechanics for stationary states, in every circumstance. Furthermore, the following conditions must hold ... 

In standard QM, the reversibility in time is a manifestation of a Hermifian (self-adjoint) system with stationary states and is reflected in the unitarity of the S-matrix. Unitarity entails the inclusion of the contribution of fime-reversed states. In other words, for a stationary state, invariance under time-reversal implies that if is a stationary wavefunction, then so is A major tool for deriving results in the framework of a Hermitian formalism, explicitly or implicitly, is the resolution of the identity operator, I, on the real axis, which is a Hermitian projection operator. 

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