Hermitian operators definition

Definition 15 A -body operator is a Hermitian operator that can be represented as a polynomial of degree 2 A in the annihilation and creation operators, and is of even degree in these operators. In addition, a A -body operator must be orthogonal to all k — l)-body operators, all k — 2)-body operators,. .., and all scalar operators, with respect to the trace scalar product. 

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

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... 

Exercise. In the space of stochastic variables a scalar product may be defined by (17). Prove that with this definition the projection onto the average is a Hermitian operator. 

In the formalism of quantum mechanics, observables are associated to hermitian operators that act on the Hilbert space of square integrable functions representing the state of the quantum system. In the following, for the sake of definiteness, we shall consider hermitian operators B which can be written as hermitain combinations of position and momentum operators,... 

It follows also from Eq. (1-3) that there exist electron states having discrete or definite values for energy (or, states with discrete values for any other observable). This can be proved by construction. Since any measured quantity must be real, Eq. (1-3) suggests that the operator 0 is Hermitian. We know from mathematics that it is possible to construct eigenstates of any Hermitian operator. However, for the Hamiltonian operator, which is a Hermitian operator, eigenstates are obtained as solutions of a differential equation, the time-independent Schroedinger equation. 

An operator that satisfies the condition in Eq. (20.33) is said to be Hermitian. Conversely, the eigenvalues of Hermitian operators are all real quantities. In quantum mechanics we deal only with Hermitian operators. The definition of an Hermitian operator is somewhat more general than Eq. (20.33) would imply. The operator H is Hermitian if... 

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

Although aromaticity concept has been talked about by all chemists for almost two centuries there exists no unambiguous quantitative definition of aromaticity mainly because it is not an experimentally measurable quantity. It is also not possible to theoretically calculate it as the expectation value of a linear hermitian operator in quantum mechanics, mainly due to the same reason stated above. 

The reason for specifying that M must be hermitian is that the eigenvalues of a hermitian operator must be real numbers. We shall discuss this and other aspects of hermiticity (including its definition) later in this chapter. 

We now apply the definition of a hermitian operator, Eq. (16.3-28) to the left-hand side... 

This can also be stated in words that an operator A usually operates to the right on a function but it can also operate to the left on the complex adjoint i j = Hermitian operator. This also means that in the matrix mechanics form of quantum mechanics a given matrix-element of a Hermitian matrix is related to another element on the other side of the (upper left to lower right) diagonal of the matrix by the relationship Amn = A. The following theorem shows why this definition is useful. 

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