Hermitian properties

One easily shows by differentiation with respect to t and using the hermitian property of H that this scalar product is independent of time, if and < ( )> are both solutions of (9-40). The probability... 

Applying the hermitian property of H X) to the third integral on the right-hand side of equation (3.69) and then applying (3.66) to the second and third terms, we obtain... 

As before, we apply the hermitian property of introduce the abbreviation and use the orthogonality relation (9.26) to obtain... 

Applying the hermitian property of we see that the left-hand side vanishes. Substitution of the expansion (9.54) for 0 2 using y as the dummy expansion index gives... 

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

Finally, making use of the anti-Hermitian properties of the derivative operator,... 

The proof is based on the fact that the diagonal elements of the commutator [A,H] vanish in the basis of energy eigenfunctions. Because of the Hermitian properties of A... 

Also, since S + is a raising operator, we must have / — ca, where c is some constant. We can evaluate c by use of the normalization of the spin functions and the Hermitian property of Sx and Sy one finds (Problem 1.8) c = h. Choosing the phase of c as zero, we have... 

The Hermitian property (1.13) expressed in bracket notation and in matrix-element notation is... 

Both of these formulas can be shown to give monotonic convergence for p. More importantly, Shavitt showed how use of the hermitian property of H could be used to write HC as... 

If these properties satisfy the same conditions as Ak and Bk in the previous sections, then from the Hermitian property of Lin function space it immediately follows that... 

Here, the Hermitian property of E — T has been assumed in shifting its position, because of which the scattering amplitude has been "proved" to be zero irrespective of the potential V and of the energy E. 

We shall return to the derivation of the individual terms of the effective Hamiltonian listed in (7.43) for some particular cases later in this chapter. To conclude this section, we now consider some of the general properties of the operator Xeff (0) and its eigenfunctions. Each of the terms listed in (7.43) is composed of products of the three operators Pq, Qo and X7. Although each of these operators is individually Hermitian (i.e. the operator is self-adjoint, pj, = P0), a product of any of them is not necessarily Hermitian. In fact, a term is only Hermitian when it is palindromic, that is, when it reads the same forwards as backwards. Inspection of equation (7.43) reveals that 3Ceff(0) is Hermitian up to and including terms in X2 but that there are non-Hermitian terms in the /.3 and higher-order contributions. The nature ofthese non-Hermitian properties can... 

Since we shall usually be more interested in the eigenvalues than the eigenfunctions ofXett (0), the possible non-Hermitian properties need not concern us too deeply. However, there may be certain situations, for example in the use of computer programmes to extract the eigenvalues of Xeff (0), where we must take account of the non-orthogonal eigenfunctions. In these situations, it is easiest to use a symmetrised form of the non-Hermitian terms in Xeff (0) in the third order, this would be... 

The latter two integrals can be represented as a square matrix, for which each matrix element corresponds to a particular combination for the values of /r and v. It is noted that because of the Hermitian properties of H, and S/j,v = Equation 5.2... 

Noting the Hermitian properties of the Hamiltonian and setting our zero... 

This result follows from evaluating Eqn. (37) at the equilibrium and then integrating with i/ o d using the Hermitian properties of the operators. Equation (37) is used to swap second-order differentiation of the operators in the same way, this time integrating with... 

But by the hermitian property (4.6), the left-hand side of Eq (4.9) equals zero. Thus,... 

The hermitian property was defined in Eq (4.6). A linear operator is one which satisfies the identity... 

Alternatively, this result can be derived directly by making use of the Hermitian property of H and Schrodinger s equation for a stationary state (eqn (5.16)),... 

As noted in the previous chapter, eqn (6.4) is a consequence of the Hermitian property of H, a property not enjoyed in general, by a subsystem. 

Because of the Hermitian property of Ho, bracketing Equations (4.7) on the left by (t/r01, all the first terms in the RS equations are zero, and we are left with ... 

In formulating this new methodology one begins by taking the difference of the Schrodinger equations for nuclear motion in the ground and excited electronic states. Tlien, using the Hermitian property of the vibrational Hamiltonian, it is easy to show that [104] ... 

Applying the hermitian property of we see that the left-hand side... 

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