Hermitean Operators

The summation convention for double indices, for example, k in Eq. (113), is assumed, as before. However, we no longer make distinction between covariant and contravariant sets.) We set ourselves the task to find anti-Hermitean operators Xf, such that... 

Since the orbital functions, (0 and mL>l) are orthogonal, the second term vanishes. The absolute value square of the matrix element of a Hermitean operator can be written as ... 

It can be shown for Eq. (195) that Mxy My> MZ1 and M2 are Hermitean operators, which means they have the following property ... 

In the previous section we discussed the Hermitean operator -a (t) It is just a one particle operator with a complete, discrete spectrum, and the following relations hold, for <.0 i... 

These equations reduce the problem to that of determining the eigenfunctions and eigenvalues of a three-dimensional linear Hermitean operator F which itself involves the eigenfunctions which are to be determined. In principle, therefore, the equations can be solved by an iterative procedure in which one postulates an initial set of eigenfunctions, calculates a new set from Fock s equations, and rep>eats this process until convergence is obtained. This, however, is an extremely arduous task, even for atoms, and for molecules it is absolutely necessary to imp>ose further restrictions on the molecular orbitals y>, before it becomes even remotely practicable to determine the form of these orbitals. The most important of these further restrictions is the assumption embodied... 

This choice of W[ satisfies all constraints, namely that it is an odd and anti-hermitean operator of first order in V. Note that W depends on the beforehand arbitrarily chosen coefficients and i.e., it is linear in For later... 

Some comments of caution are required here. First, as also discussed by Blount [30] and by Kirtman et al. [29], one has to be careful with the phases of the Bloch functions. Whereas these functions are continuous in r space, this may not be the case in k space. Thus, when considering the limit K oo (i.e., Ak — 0), it is important that the limit exists, i.e., is continuous and differentiable in k space. In that case, one may neglect the first term on the left-hand side and, accordingly, arrive at the approximation of equation (27). On the other hand, only in that limit the expression on the right-hand side corresponds to a Hermitean operator. 

The last integral in Eq. (3.6) cancels the second integral, as shown below. H vs a. real, Hermitean operator, so that... 

A general Hermitean operator acting in the antisymmetrized part of the two-electron space would be specified by ten matrix elements, and even after the specification of the state energy zero and the introduction of the operators Jy and six degrees of freedom... 

Here H may be any hermitean operator of interest, but in our case is identified with the vibronic Hamiltonian. Given these states, one may start the following three-term recurrence relations, also called Lanczos iteration ... 

Furthermore, for the operator Q to have real eigenvalues, it has to be a Hermitean operator with the property... 

The number calculated is a real value, called the ppectation value of Q, since Q is a Hermitean operator. If T is an eigenfunction of Q, the expectation value is equal to the eigenvalue, as is easily seen in Equation 1.38. 

Formally, any physical observable is then a linear, hermitean operator 0,... 

In section 4.1 it was stated that all physical information is contained in the quantum mechanical state Y, but the question is how this information can be extracted. Any observable of a physical system is described by an hermitean operator acting on the corresponding Hilbert space. We further postulate that any experimentally measured value of a physical observable O must be identical to one of the eigenvalues of the corresponding operator 0,... 

If U is unitary and D is an hermitean operator, then UOU is also hermitean since UOU Y = = UOU. Moreover, since... 

The uncertainty principle requires that our information, even after preparation of a system state, must remain incomplete. This is because two Hermitean operators do not necessarily commute i BA and thus the result of the simultaneous measurement depends on the order of operations. The most prominent example is the position-momentum... 

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