Eigenenergies

The constant 9 is the initial population of level Ek and thus computable from the initial data, Eq. (16). All this turns out to be true if the following assumption on the eigenspaces and eigenenergies of H q) is fulfilled ... 

For m/M small enough, the populations of the eigenstates are nearly constant and the quantal motion is given in terms of the evolution of the eigenstates and eigenenergies Ek along qgo-... 

We remind the reader that the tunneling transition energy could also be thought of as an eigenenergy of the wall s motion, but of a lower, 1=1, order, associated with the translational motion of the shell s center of mass. 

Taking the eigenenergies of the system before (after) chemisorption to be ° (cj), then the change in the DOS is... 

Diagonalization Method for Quantum Eigenenergy Calculation or for Spectral Analysis of Time Signals. 

Highly Excited Vibrational Spectrum of Sulfur Dioxide. I. Eigenenergies and Assignments up to 15000 cm-1. 

Equivalently, the unitary operator can be viewed as transforming the reference Hamiltonian. This yields an effective Hamiltonian that has as an eigenfunction and the exact eigenenergy E of the target state as its eigenvalue. Thus... 

The new diagonal elements are the eigenenergies e (t) and e (t) of the lower and upper dressed state /) and u), respectively. In intense laser fields, the dressed states split up according to... 

The Hamiltonian of relative motion is given by Eq. 5.30. The states i), I/), of this Hamiltonian, Jf, which we will call the initial and final states of a given spectroscopic transition, are associated with eigenenergies Ef. The wavefunctions of relative motion are obtained by introducing spherical coordinates R, 8, (p and the partial wave expansion, according... 

As a consequence of the attractive region of the interaction, several bound state solutions of the radial Schrodinger equation generally exist whose wavefunctions are localized in the well region. The bound state eigenenergies are negative, discrete and will be subscripted with the vibrational and rotational quantum numbers, v and the normalization... 

Let us remark that the ability of periodic-orbit methods to calculate eigenenergies even in classically chaotic systems is related to the validity... 

Now, the eigenenergies of the Hamiltonian can be detected directly if the time dependence of the above average exhibits quantum beats. This will be the case if the spectrum is not too dense and the linewidths are smaller than the level spacings. From a Fourier transform of the autocorrelation function, we then obtain an expression of the form (2.26)-(2.27), which can be evaluated semiclassically in terms of periodic orbits and their quantum phases. 

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

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