Eigenfunction, definition

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

Consider now the case where an electron with a spin is moved from orbital i to orbital a. The first S-type determinant in Figure 4.1 is of this type. Alternatively, the electron with /3 spin could be moved from orbital i to orbital a. Both of these excited determinants will have an value of 0, but neither are eigenfunctions of the operator. The difference and sum of these two determinants describe a singlet state and the 5 = 0 component of a triplet (which depends on the exact definition of the determinants). 

For a degenerate energy eigenvalue, the several corresponding eigenfunctions of H may not initially have a definite parity. However, each eigenfunction may be written as the sum of an even part V e(q) and an odd part V o(q)... 

Since any component of M commutes with M2 it can be shown that if M2 has a definite eigenvalue associated with the eigenfunction ip, this value is not changed if any component of M operates on ip. 

The paper is organized in the following way. For the sake of completeness we shall give a short review of some elementary definitions and facts on quantum graphs. We shall then show how pairs of isospectral domains in R2 can be converted to isospectral pairs of quantum graphs, and discuss their spectra and eigenfunctions. 

Hence, = I + 1 if k > 0 and = I — 1 if k < 0. Consequently, in the Dirac-Pauli representation and have definite parity, (—1) and (—1) respectively. It is customary in atomic physics to assign the orbital angular momentum label I to the state fnkm.j- Then, we have states lsi/2, 2si/2) 2ri/2, 2p3/2, , if the large component orbital angular momentum quantum numbers are, respectively, 0,0,1, ,... while the corresponding small components are eigenfunctions of to the eigenvalues 1,1,0,2,. 

The following properties of Hermitian operators follow from the definition (1.23). The eigenvalues of a Hermitian operator are real. Two eigenfunctions of a Hermitian operator that correspond to different eigenvalues... 

The expansion in eigenfunctions leads to expressions of the various quantities pertaining to the stochastic process - as in equations (7.13) through (7.16). It also simplifies some of the derivations, in particular the proof of the approach to equilibrium. In fact, according to (7.13) it is sufficient to prove that all X other than X = 0 are positive, i.e., that W is negative semi-definite. In the same notation as in V.5 one has for any vector pn = x pl in the Hilbert space... 

Exercise. Consider the right-hand side of (3.5) as a linear operator W acting on the space of functions P(X) defined for 0 < X < oo and obeying (3.6). Verify that it has the symmetry property (V.7.5) and is negative semi-definite, the only eigenfunction with zero eigenvalue being (3.7). 

The wavefunction which fits the equation and leads to discrete values of V is the eigenfunction. The search for such eigenfunctions and eigenvalues can be a most demanding mathematical excercise, and need not be considered here. Let us note however that the solutions of the Schrodinger equation lead to the definitions of the orbital quantum numbers n, l and m. The quantum numbers of rotational and vibrational levels are also derived from the Schrodinger equation. 

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