Eigenfunctions orthogonality

The last identity follows from the orthogonality property of eigenfunctions and the assumption of nomralization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important corollary to the fiftli postulate is established ... 

The ability to assign a group of resonance states, as required for mode-specific decomposition, implies that the complete Hamiltonian for these states is well approxmiated by a zero-order Hamiltonian with eigenfunctions [M]. The ( ). are product fiinctions of a zero-order orthogonal basis for the reactant molecule and the quantity m. represents the quantum numbers defining ( ).. The wavefimctions / for the compound state resonances are given by... 

SEMIEMPIRICAL CALCULATIONS ON LARGER MOLECULES The spin eigenfunctions are orthogonal... 

Since j/[ and xpl are eigenfunctions of the same hamiltonian, namely that in Equation (1.65), they are orthogonal, which means that, when v f v". 

The severe restriction of orthogonality towards the exact eigenfunctions of lower energy has recently been removed, see ref. 33. 

Let us now consider the possibilities for deriving an eigenfunction for a particular excited state. The straightforward application of the variation principle (Eq. II.7) is complicated by the additional requirement that the wave function Wk for the state k must be orthogonal to the exact eigenfunctions W0, Wv for all the lower states although these are not usually known. One must therefore try to proceed by way of the secular equation (Eq. III.21). A well-known theorem15 25 says that, if a truncated... 

This implies that the absolute values of the overlap integrals clk are vanishing as A[Jk—Ek]l for Jk->Ekt ensuring automatic orthogonality between the function 0k and all the lower exact eigenfunctions F0, Wv. . Wk x when Jk = Ek. Substituting this... 

Substitution of this eigenfunction in an expression of the type of Equation 21 permits the evaluation of the perturbation energy W1, in the course of which use is made of the properties of orthogonality arid normalization of the spin eigenfunctions namely,... 

This eigenfunction is equivalent to and orthogonal to pi, and has its maximum value of 2 at 6 = 19°28

angle between the lines drawn from the center to two corners of a regular tetrahedron. The third and fourth best bond eigenfunctions... 

In light of our supposition, and are eigenfunctions corresponding to distinct eigenvalues, that is, A, the orthogonality of y and... 

If Ip I and ip2 ( fe eigenfunctions of a hermitian operator A with different eigenvalues a and ai, then ipi and p2 are orthogonal. To prove this theorem, we begin with the integral... 

From any pair 0i, 02 which initially are not orthogonal, we can construct by selecting appropriate values for ci and C2 a new pair which are orthogonal. By selecting different sets of values for ci, C2, we may obtain infinitely many new pairs of eigenfunctions which are mutually orthogonal. 

Thus, the quantity S is an upper bound to the energy Ek corresponding to the state tpk. For situations in which 0 can be made orthogonal to each exact eigenfunction xpo, xp, ipk-i, the coefficients ao, a, a i vanish according to equation (9.4) and the inequality (9.7) applies. 

Each of the eigenfunctions is orthogonal to all the other unperturbed eigenfunctions for k n, but is not necessarily orthogonal to the other eigenfunctions for E Any linear combination of the members of the set... 

Applying the hermitian property of and noting that is orthogonal to all eigenfunctions belonging to the eigenvalue we have... 

They are assumed to be known. Furthermore, it should be recalled that the eigenfunctions VrJ form a complete orthogonal set. The equation of interest is... 

An tJjer fundamental property concerns the orthogonality of eigenfunctions. Given that... 

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