Eigenstates nondegenerate

Let the eigenvalue w be fixed and assume that fit is nondegenerate and unit-normalized. The restriction to nondegenerate eigenstates will be relaxed in Section V, but for now we consider only pure-state density matrices. The A -electron density matrix for the pure state fit is... 

A general asymmetric top molecule has 2 J + 1 nondegenerate eigenstates for any given total angular momentum quantum number J J+1 states belong to the (—l)"7 parity manifold and J states have parity... 

Each nondegenerate eigenstate has a distinct wavefunction which is uniquely characterized by the totality of expansion coefficients G jp) in Equation (11.10). 

Thus, there is a very simple and analytic pulse shaping recipe for achieving ) complete population transfer between two arbitrary superposition states [VF) am P ), composed of nondegenerate energy eigenstates. 

In general the relaxation to equilibrium of E(t) is nonexponential, since the rate matrix in the master equation has an infinite number of (in principle) nondegenerate eigenvalues if there are an infinite number of states n). There are, however, two instances where the relaxation is approximately exponential. In the first instance one assumes that the initial nonequilibrium state has appreciable population only in the first two oscillator eigenstates, and further that k,. 0 k,, m and k0. t k0 m for m > 2. If one neglects terms involving these small rate constants, the master equation reduces to a pair of coupled rate equations for a two-level system ... 

Nondegenerate perturbation theory enables algebraically transparent analysis of the interactions between any number of decaying quasi-eigenstates pro-... 

For nondegenerate modes, the zero-order states n,)° can be defined analytically as harmonic oscillator eigenstates or numerically as solutions of the eigenequation ... 

Average values of the relative participation ratio for the eigenstates of orientationally disordered aggregates for nondegenerate (full line) and doubly degenerate (dotted line) cases. 

Hence the new state (a y is an eigenstate of N with eigenvalue (y + 1). Since the eigenstates of a one-dimensional harmonic oscillator are nondegenerate, this implies that... 

The application of perturbation theory to a nondegenerate state (for ko = 0) will involve no linear term in k (by symmetry) and the first significant term, which gives the curvature of the band, is given by second order and conventionally expressed as an effective (inverse) mass tensor. If the unperturbed state is denoted by 0> and all other k = 0 eigenstates by n>, the result for this is... 

For the moment these wavefunctions are to be understood to represent nondegenerate eigenstates. 

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