Eigenvalues introduced

A unitary transformation is then introduced whieh diagonalizes the FG matrix, yielding eigenvalues s, and eigenvectors q,. The kinetic energy operator is still diagonal in these eoordinates. 

In solving the eigenvalue problem for the energy operator //op, we have previously always introduced a complete basic set Wlt in which the eigenfunction has been expanded ... 

The eigenfunctions and eigenvalues of Eq. (10-405) can be obtained as in the nonrelativistic case upon introducing the variables... 

Before giving further motivations, it will be sensible to introduce the eigenfunction and the eigenvalue Aj, with the number k of the... 

For an arbitrary state at a fixed time t, the ket ) may be expanded in terms of the complete set of eigenkets of In order to make the following discussion clearer, we now introduce a slightly more complicated notation. Each eigenvalue A, will now be distinct, so that A, f Xj for i 7 j. We let g, be... 

We now solve the Schrodinger eigenvalue equation for the harmonic oscillator by the so-called factoring method using ladder operators. We introduce the two ladder operators d and a by the definitions... 

We have already introduced the use of ladder operators in Chapter 4 to find the eigenvalues for the harmonic oscillator. We employ the same technique here to obtain the eigenvalues of and Jz. The requisite ladder operators and J-are defined by the relations... 

Although the functions Rniir) according to equation (6.20) form an orthogonal set with w r) = r, the orthogonal relationships do not apply to the set of functions Sxiip) with w p) = p. Since the variable p introduced in equation (6.22) depends not only on r, but also on the eigenvalue E, or equivalently on X, the situation is more complex. To determine the proper orthogonal relationships for Sxiip), we express equation (6.24) in the form... 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

The eigenvalue problem was introduced in Section 7.3, where its importance in quantum mechanics was stressed. It arises also in many classical applications involving coupled oscillators. The matrix treatment of the vibrations of polyatomic molecules provides the quantitative basis for the interpretation of their infrared and Raman spectra. This problem will be addressed tridre specifically in Chapter 9. 

We introduce the eigenstates and eigenvalues for the creation and annihilation operators (coherent states86) ... 

The required spin-orbit matrix elements are given in Table 8, whence the eigenvalues of the matrix of the first and second order contributions are easily found to be 0, and 2 — ((3/A i) + (2/A 2)), where AEl and A E2 represent respectively the energies of the 3II(o7r) and 3Il(7r6) states above the ground level. Introducing the appropriate metal mixing coefficient, c, for the a, ir, and 5 levels the required g value then becomes... 

The extra h was introduced to make the exponent dimensionless. The wavefunctions Xi 3X6 eigenfunctions of Hi with eigenvalues Et, the Xf eigenfunctions of Hf. Hence ... 

---