The eigenvalue spectrum

The net effect of these restrictions is that in some cases, E may have any value then we speak of a continuous spectrum of eigenvalues of E. In other cases, E may be restricted to certain particular values then we have a discrete spectrum of eigenvalues. In these latter cases we say that E is quantized. Ordinarily, for each boundary condition we impose, we introduce a quantization of some observable. 

SO that in general we deal with a set of eigenfunctions, 1/ 2 If energy 

The existence of a degeneracy poses a problem in describing the state. Suppose the eigenstate with energy E is three-fold degenerate, with eigenfunctions ij/2, and This means that 

Since the Hamiltonian operator is linear, this implies that if we construct a linear combination, 

The linear combination cj) is also an eigenfunction of the Hamiltonian operator with E as an eigenvalue therefore is an appropriate description of the eigenstate. We may construct two additional independent linear combinations of the same type  

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In quantum mechanics, the eigenvalues of an operator represent the only numerical values that can be observed if the physical property corresponding to that operator is measured. Operators for which the eigenvalue spectrum (i.e., the list of eigenvalues) is discrete thus possess discrete spectra when probed experimentally. 

The reason for pursuing the reverse program is simply to condense the observed properties into some manageable format consistent with quantum theory. In favourable cases, the model Hamiltonian and wave functions can be used to reliably predict related properties which were not observed. For spectroscopic experiments, the properties that are available are the energies of many different wave functions. One is not so interested in the wave functions themselves, but in the eigenvalue spectrum of the fitted model Hamiltonian. On the other hand, diffraction experiments offer information about the density of a particular property in some coordinate space for one single wave function. In this case, the interest is not so much in the model Hamiltonian, but in the fitted wave function itself. 

In order to understand the subsequent discussion it is helpful first to elucidate the eigenvalue spectrum of the triatomic compound [see also Nesbitt (1988a,b)]. The body-fixed theory for total angular momenta / 0 outlined in Section 11.1, especially the set of coupled equations... 

With the abovementioned structure of the eigenvalue spectrum, the term with l = 1 in Eq. (4.131), which is proportional to exp(—e Gt/xo), at ct > 1 is far more long-living than any other one. The dominating role of the decrement A had been proved by Brown, who derived [47] the following asymptotic expression for it ... 

Let us recall that the LRC potential, which is fitted on a panel of atomic IP, leads to good IP for most (if not any) atomic IPs. This is unfortunately not true anymore for the molecular systems for which a shift between experimental IPs and the eigenvalues spectrum remains [76]. This shift is definitely smaller than those of the LDA or other GGAs, but it is significantly larger than for atoms. Indeed, it corresponds to an overcorrection with respect to the LDA, and this could be related to a proportionnally smaller importance of low density domains in molecules than in atoms. This overcorrection was also noticed by Casida with the VLB potential [77]. 

Fig. 1. Regions in the (k, q) plane where the -eigenvalue spectrum corresponds to unitary irreducible representations of so(3) (horizontal shading) and so(2, 1) (vertical shading). See text for details. 

Inserting the eigenvalue spectrum (3) leads to an expression for the partition function, which can be numerically evaluated and expressed a power series ... 

The form of the eigenvalue spectrum for a given dimer depends on the relative spreading of the 3d and 45 atomic orbitals, which varies markedly across the series. The consequences for the eigenvalue behaviour are shown in Figures 16 and 17 for ground-state configurations of Ti2 and Cu2, respectively. 

Des Cloizeaux has shown that one can explicitly construct a similarity transformation of 9if which leaves the eigenvalue spectrum invariant, but enforces hermiticity and therefore orthogonal eigenvectors for distinct eigenvalues,... 

Figure 10. k as a function of the intensity of the multiplicative noise, k is defined as (0) with (jc(t))/(ac(0)). The two arrows on the left denote the point where the discrete branch of the eigenvalue spectrum disappears (see Schenzle and Brand ). The two arrows on the right denote the phase transition threshold. [Taken from S. Faetti et al., Z. Ffiys., B47, 353 (1982).]... 

The use of the function theorem can be seen in conjunction with the representation theorem. We choose the spectral representation of the observable a, that is the representation in which the basis states are the eigenstates (corresponding to the eigenvalue spectrum) of a. 

A referee has drawn out attention to previous calculations of the eigenvalue spectrum of icosahedral C q [ 14]. 

The calculation of the eigenvalue spectrum starts with the generation of the nanocrystal in real space by specifying the coordinates of the atoms. The nano-... 

The density of states concept was introduced in Section 1.3. For any operator A characterized by the eigenvalue spectrum nJ we can define a density function Pa (a such that the number of eigenvalues that satisfy a [Pg.82]

Fig. 3b shows the case where the eigenvalue spectrum of the B and M matrices remains the same but with slightly stronger coupling between the orbital and CSF variations. In this case the lowest eigenvalue of the... 

The diagonal eigenvalues in this matrix make up the eigenvalue spectrum. The decomposition... 

The eigenvalue spectrum of the continuum electron gas in one-dimension is calculated. New collective states are found and identified as soli-tons or bound states of these solitons. These new states carry current and can participate in transport. They are thermally activated at low temperatures, with a gap which is calculted for all values of the coupling strength. This exact calculation confirms a previous result based on integrating the renormalization group equations up to the coupling strength where a solution could be found. 

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