Hermitian operators properties

Two or more properties F,G, J whose eorresponding Hermitian operators F, G, J eommute... 

Let us, therefore, assume that the amplitude >ft x) describing a relativistic spin particle is an -component object. We are then looking for a hermitian operator H, the hamiltonian or energy operator, which is. linear in p and has the property that H2 = c2p2 + m2c4 = — 2c2V2 + m2c4. We also require H to be the infinitesimal operator for time translations, i.e., that... 

Heisenberg-type descriptions for two observers, 667, 668 Heitler, W., 723 Helicity operator, 529 Hermitian operator, 393 Hermitian operator Q describing electric charge properties of particles, 513... 

In some of the derivations presented in this section, operators need not be hermitian. However, we are only interested in the properties of hermitian operators because quantum mechanics requires them. Therefore, we have implicitly assumed that all the operators are hermitian and we have not bothered to comment on the parts where hermiticity is not required. 

Because of these properties of Hermitian functions it is accepted as a basic postulate of wave mechanics that operators which represent physical quantities or observables must be Hermitian. The normalized eigenfunctions of a Hermitian operator constitute an orthonormal set, i.e. 

In the derivation above, we used the property of hermitian operators with a definite T-parity... 

Let a system be in the state T at some instant of time. Let orthonormal eigenfunctions of the Hermitian operator G that corresponds to the physical property G ... 

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... 

Consider some molecular property defined by the matrix A with the elements Ay = (( /-(l/Ur. R)l P,)) of a Hermitian operator A. One then easily derives that this matrix A obeys the following equation of motion [37],... 

Equation (26) expresses a smoothness of matrix elements of Hermitian operator A in a strictly diabatic basis. It is the key formula for deriving the expression of the ADT angle in terms of a given molecular property [37]. 

It is possible to make evident some properties of the resolvent of a Hermitian operator, by using the identity... 

The only observable values for a property 2 are one of the eigenvalues to,-of the associated linear Hermitian operator m as defined in the eigenvalue equation... 

Based on the molecular quantum similarity measmes. Molecular Quantum Self-Similarity Measures (MQS-SM) were proposed as molecular descriptors where each molecule is compared with itself and all the others, and appropriate Hermitian operators 2 are associated to each molecular property [Ponec et al, 1999]. 

It can be proved that the eigenvalues of hermitian operators are real. Accordingly, observables are represented by hermitian operators, as that guarantees that the outcome of observations is real. Another property of those operators is that their non-degenerate eigenfunctions are orthogonal, that is... 

For example, suppose that after a full Cl calculation with a complete N-electron set, we have obtained the exact wavefunction, F. The formal rela-fions regarding energy are given by Eqs. (3-7). Now consider a one-electron Hermitian operator, representing a property, say 0(0, where i are electron... 

Postulate 2 Mean-Value Postulate. To every ensemble of measurements performed on identically prepared reglicas of a system there corresponds a real linear functionalAm(P) of the Hermitian operators P o f the system such that if P corresponds to an observable property P, m(P) is the arithmetic mean P of the results of the ensemble of P-measurements. 

Associated with each system property A is a linear, Hermitian operator A. 

Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system properties and Hermitian operators, and the mathematical result that only operators which commute have a common set of eigenfunctions, a rather remarkable property of nature can be demonstrated. Suppose that one desires to determine the values of the two quantities A and B, and that the corresponding quantum-mechanical operators do not commute. In addition, the properties are to be measured simultaneously so that both refiect the same quantum-mechanical state of the system. If the wavefimction is neither an eigenfiinction of jlnor Ji, then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the wavefimction / in terms of the eigenfunctions of the relevant operators... 

As we have already seen, the Liouvillian L is an Hermitian operator and the propagator eiLt is unitary. Likewise since Q is Hermitian, QLQ is Hermitian (QLQ)+ = Q+L+Q+ = (QLQ). It follows that e QLQt is a unitary operator. These properties allow us to prove the following theorems. 

Quantum mechanics postulates that to every observable property of a system there corresponds a linear Hermitian operator. Mechanical operators having a classical analogue can be constructed when the operator forms for coordinates and momenta are known. 

Mathematiceilly, these solutions are the eigenvalues (tj) and eigenfunctions Hermitian operator (h) and, as such, they have several important properties. One of these properties is that they are complete, any function of ordinary three-space (the coordinates of a single electron) with sufficiently similar boundary conditions can be expanded as a linear combination of these functions. That is, any function /(f) can be written exactly as... 

Starting from the fact that classical quantum study of microscopic systems is essentially associated with the Algorithm 1 of Appendix B. Then, all observable property values of a known system, co, can be formally computed as expectation values, (t ), of the associated hermitian operator, Q, acting over the known state DF, p r [4,5]. In the same way as in theoretical statistics it can be written ... 

Definition of Hermitian Operators. Let A be the linear operator representing the physical property A. The average value of A is [Eq. (3.88)]... 

Therefore, a Hermitian operator A possesses the property that... 

In the previous section, we proved the orthogonality of the eigenfunctions of a Hermitian operator. We now discuss another important property of these functions this property allows us to expand an arbitrary well-behaved function in terms of these eigenfimctions. 

We postulated in Section 7.3 that the eigenfunctions of any Hermitian operator that represents a physically observable property form a complete set. Since the g/s form a complete set, we can expand the state function as... 

Postulate 2. To every physically observable property there corresponds a linear Hermitian operator. To find this operator, write down the classical-mechanical expression for the observable in terms of Cartesian coordinates and corresponding linear-momentum components, and then replace each coordinate x by the operator X and each momentum component pj, by the operator —ihd/9x. 

We saw in Section 7.2 that the restriction to Hermitian operators arises from the requirement that average values of physical quantities be real numbers. The requirement of linearity is closely coimected to the superposition of states discussed in Section 7.6. In our derivation of (7.69) for the average value of a property B for a state that was expanded as a superposition of the eigenfunctions of B, the linearity of B played a key role. 

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