Eigenfunction nonstationary

This nonstationary state may be represented as a superposition of the eigenfunctions of the total Hamiltonian, i.e.,... 

As mentioned in Section III.B.l, ata> 1 the most long-living nonstationary solution of Eq. (4.138) is the eigenfunction with 1=1, whose eigenvalue is exponentially small see Brown s estimation (4.132). We use this circumstance for approximate evaluation of tp, in the ct 1 limit by neglecting the right-hand side of Eq. (4.138) for l = 1. On so doing, the equation obtained for the... 

Note that eigenfunctions with A,. n = 0 are stationary eigenfunctions and those with 0 are nonstationary. Furthermore, the stationary pi- n with n = 0 are uniform distributions on the I(p,q) = I torus, whereas the non-... 

Our emphasis has been on the fundamental role of the eigendistributions of Lc and La, and it is within this eigenfunction basis that classical-quantum correspondence assumes its most simple form. At present the correspondence is clearly understood for stationary and nonstationary eigendistributions in regular systems and for stationary eigendistributions in the chaotic case. Further work is necessary to clarify the picture for nonstationary chaotic eigendistributions. 

The average value of the dipole moment will be calculated by means of Dirac s perturbation theory for nonstationary. states, up to third order the zero order refers to the free molecules in the absence of the field. Let the wave function of the system of the two interacting molecules in- the external field be specified by y, an eigenfunction of the total Hamiltonian H. This wave function y> may be expanded in a complete set of the energy eigenfunctions unperturbed system the index n labels the various unperturbed eigenstates characterized by the energy En. We may then write... 

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). 

Taking into account the orthogonality of these eigenfunctions the nonstationary Schrodinger equation... 

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