Perturbed Stationary-State Wave Functions

is real, this lessens the complexity of the equations, and one may obtain  

Equation (190) is comparable to Eq. (178). The wave functions, however, are now stipulated by Eqs. (187) and (188), and the interaction potential is supplanted by terms from the small relative kinetic energy. One may treat Eq. (190) as we have Eq. (178), ignoring most of the nondiagonal elements in the series. It is important to understand, nevertheless, that even if such a procedure is adopted, the perturbed stationary-state wave functions still implicitly include some contribution from nondiagonal terms in the sense of Eq. (178). Thus, the method here is of greater accuracy than those based on Eq. (178), but note that Eq. (187) must be solved to apply this method. 

Numerical Solution of the Time-Dependent SchrSdinger Equation 

An approach of considerable promise using electronic computers has been put forth by Mazur and Rubin. They propose a technique wherein the equation is solved numerically for a wave packet representing the initial reactants. The spread of this wave packet into a product region of space determines the reaction probability. Because their primary purpose is to demonstrate the method, Mazur and Rubin make rather vigorous simplifications in their model, which is related to the colinear case of the reaction in Eq. (163). 

For a linear system of three atoms undergoing a reaction, 

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A review of the semidassical method is given by Cottrell and McCoubrey [9] and by Rapp and Kassal [13]. In this method, the translational motion is treated classically, while the molecule BC is assumed to have quantized vibrational levels. By converting the force V (x) on the oscillator due to the incident atom to V (t) by utilization of the classical trajectory x(t), one may apply time dependent perturbation theory. The wave function for the perturbed system is written as a sum of the stationary-state wave functions Y (y)exp( —icojf), with coefficients ck given by... 

Equation 6.33 is completely general. For the two-spin system, it results in the transitions we identified in Fig. 6.2, while the double quantum transition between and 4, and the zero quantum transition between 02 and 03 are forbidden. Note that this statement is true for this treatment, which employs stationary state wave functions and time-dependent perturbations, but as we shall see in Chapter 11, it is easy with suitable pulse sequences to elicit information on zero quantum and quantum double processes. For our present purposes in the remainder of this chapter we accept the validity of Eq. 6.33. 

The distribution of electric charge in a molecule is intimately related to its structure and reactivity. Knowledge of the distribution gives us a feeling for the physical and chemical properties of the molecule and provides a valuable assessment of the accuracy of approximate molecular wavefunctions. The charge distribution in the nth stationary state is determined by the many-electron wave function 0 of the free molecule. If the molecule interacts with an external electric perturbation E, the wave function determines the distortion and,... 

The perturbed stationary state method leads to the same conclusion. Here, the wave function of the colliding system (non-stationary state) is expressed in terms of the stationary state electronic wave functions Xx( ) of the Aj molecule (X denotes Eigen-value). Again, only two states, x (-R) and X (-S) with energy E R) and E R), respectively, are important (two-state approximation), and... 

There is another way of looking at this coupled ion system, namely, in terms of stationary states. From this point of view, one considers that the excitation belongs to both ions simultaneously. To determine the wave functions of the two-ion system, one resorts to degenerate perturbation theory. The coupling H can be shown to remove the degeneracy, and two new states that are mixtures of X20 and X11 are formed. For each the excita-... 

Let us consider two stationary states n and m of an unperturbed system represented by the wave function V and such that Em > Let us assume that at / = 0, the system is in the state n. At this time, the system comes under the perturbing influence of the radiation of a range of frequencies in the neighbourhood of vm of a definite field strength E. 

In order to evaluate 5s/ISv from Eq. (282), we further need the functional derivatives dqfjjdvs and ScpflSv. The stationary OPM eigenfunctions (< /r), = 1,..., oo) form a complete orthonormal t, and so do the time-evolved states unperturbed states, remembering that at t = ti the orbitals are held fixed with respect to variations in the total potential. We therefore start from t = ti, subject the system to an additional small perturbation (5i)s(r, t) and let it evolve backward in time. The corresponding perturbed wave functions [Pg.135]

When a time-independent external perturbation is applied on a molecule the molecular charges re-orient the nuclei relax into a new configuration and the electronic motion is altered, thereby representing the equilibrated ground state of the molecule in the presence of the field. The corresponding equilibrated wave function is stationary, as opposed to the case when a time-dependent perturbation is applied, and we can thus determine the molecular energy as the time-independent expectation value of the Hamiltonian... 

The QTS is a part of a reaction mechanism. These species may be found related to stationary arrangements of the external Coulomb sources. Solutions to eq.(8) coming as saddle points of at least index one (one imaginary frequency) are natural candidates to play the role of QTS. If the saddle point wave function has closed electronic shell structure, its electronic parity is positive. In this case one would expect a situation similar to the symmetry-forbidden electronic absorption bands. The intensity is borrowed from the excited states having the correct parity via couplings at second-order perturbation theory [21]. 

There have been developed two essentially different wave-mechanical perturbation theories. The first of these, due to Schrodinger, provides an approximate method of calculating energy values and wave functions for the stationary states of a system under the influence of a constant (time-independent) perturbation. We have discussed this theory in Chapter VI. The second perturbation theory, which we shall-treat in the following paragraphs, deals with the time behavior of a system under the influence of a perturbation it permits us to discuss such questions as the probability of transition of the system from one unperturbed stationary state to another as the result of the perturbation. (In Section 40 we shall apply the theory to the problem of the emission and absorption of radiation.) The theory was developed by Dirac.1 It is often called the theory of the variation of constants the reason for this name will be evident from the following discussion. 

Consider a particular electronic quantum state, which is stationary in absence of external perturbation U. We postulate that this stationary state exists and is characterized by an universal electronic state Whenever it is the case, this state function I k) determines a particular poK geometry endowed with the symmetries of the system. In a particular reference frame, one has a wave function, say 3>k(x). This one transforms according to one of the irreducible representations of the symmetry group of the stationary (equilibrium) nuclear frame poK- This is postulate PI. 

A basic problem in quantum chemistry is the determination of the wave functions for stationary states of electrons in the presence of fixed nuclei (sometimes including the effects of external fields and perturbations). 

The time-dependent perturbation changes the system s state function from exp -iE t/hyif to the superposition (9.122). Measurement of the energy then changes to one of the energy eigenfunctions exp -iE i,t/h)il/ i, (reduction of the wave function, Section 7.9). Tie net result is a transition from stationary state n to stationary state m, the probability of such a transition being... 

For the perturbed particle in a box of Problem 9.2, find the first-order correction to the wave function of the stationary state with quantum number n. 

The two structure formulae, (48 3), (48 4), are at variance with chemical usage. The perturbation calculus of Heitler and London for this case shows, however, that the actual stationary state of the group CNH corresponds to a mixture of the three pure valence states, (48 2), (48 3), (48 4), in this sense that the quantum-mechanical wave function of the binding electrons is a linear superposition of the wave functions associated with each of the above pictures. The justification of the formula (48 2) lies in the fact that in the wave function associated with a stable molecule the part which it contributes predominates over the parts arising from the other idiotic structure formulae. 

Actually, the problem is to describe the perturbed wave-function i/r from the information contained within the unperturbed state t), having both time-evolution. To begin, the evolution of the stationary states (under the Hamiltonian Ho) is considered as represented on the ortho-normalized... 

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