Eigenstates degenerate

The energy level of an an /-fold degenerate eigenstate can be labelled according to an /-fold degenerate irreducible representation of the synmietry group, as we now show. 

We note here that the qiiantnm levels denoted by the capital indices I and F may contain numerous energy eigenstates, i.e. are highly degenerate, and refer to chapter A3.4 for a more detailed discussion of these equations. The integration variable in equation (A3.13.9) is a = 7 j / Ic T. 

Equations (16)-(20) show that the real adiabatic eigenstates are everywhere smooth and continuously differentiable functions of Q, except at degenerate points, such that E (Q) — E, [Q) = 0, where, of com se, the x ) are undefined. There is, however, no requirement that the x ) should be teal, even for a real Hamiltonian, because the solutions of Eq. fl4) contain an arbitrary Q dependent phase term, gay. Second, as we shall now see, the choice that x ) is real raises a different type of problem. Consider, for example, the model Hamiltonian in Eq. (8), with / = 0 ... 

Each of these states is also an eigenstate of J2 because J2 and and, therefore, 2 and (e/+)n, commute. Furthermore, each of these states belongs to the single eigenvalue b2 of 2. The various (X + i Y)n>fi are degenerate eigenstates of J 2. 

Some or all of the eigenvalues may be degenerate, but each eigenfunction has a unique index i. Suppose further that the system is in state aj), one of the eigenstates of A. If we measure the physical observable A, we obtain the result aj. What happens if we simultaneously measure the physical observable B To answer this question we need to calculate the expectation value (B) for this system... 

Geometric phase effect (GPE) conical intersections, 4-8 adiabatic eigenstates, 8-11 topographical energy, 568-569 curl equations, 11-17 degenerate states chemistry, x-xiii electronic states ... 

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. 

The essential principle of coherent control in the continuum is to create a linear superposition of degenerate continuum eigenstates out of which the desired process (e.g., dissociation) occurs. If one can alter the coefficients a of the superposition at will, then the probabilities of processes, which derive from squares of amplitudes, will display an interference term whose magnitude depends upon the a,. Thus, varying the coefficients a, allows control over the product properties via quantum interference. This strategy forms the basis for coherent control scenarios in which multiple optical excitation routes are used to dissociate a molecule. It is important to emphasize that interference effects relevant for control over product distributions arise only from energetically degenerate states [7], a feature that is central to the discussion below. 

It is much more than mere pedanticism to emphasize that the zero-order energy levels of the two subsystems described above cannot be considered to be eigenstates of the Hamiltonian of the total system. The zero-order levels of the two subsystems are degenerate or quasidegenerate, and therefore extensive configuration mixing is induced by the (small)... 

We wish to introduce here a very reliable calculation procedure to determine the eigenstates of a general Renner-Teller problem. It is convenient to start writing the matrix representation of the Hamiltonian of a Renner-Teller system on two degenerate p-like electronic states fx, fy and with the interaction part in the form given in Section 2, neglecting at the moment the quartic or higher order terms ... 

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