Mathematical operator hermitian operators

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

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. 

For a discussion of the form of the distribution N(E), see, for instance, P. G. Nevai, Orthogonal Polynomials, Memoires American Mathematical Society, Providence, RI, 1979. In essence, in handling dN(E) = n(E) dE, one can consider n( ) as the (projected) density of states of a Hermitian operator and N(E) as the corresponding integrated density of states. 

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... 

This postulate is more a mathematical than a physical postulate. Since there is no mathematical proof (except in various special cases) of the completeness of the eigenfunctions of a linear Hermitian operator, we must assume completeness. Postulate 4 allows us to expand the wave function for any state as a superposition of the orthonormal eigenfunctions of any quantum-mechanical operator ... 

The self-adjoint and Hermitian operators differ in mathematics (the matter of domains), but we will ignore this difference in this book. 

Postulate 3. There is a linear hermitian mathematical operator in one-to-one correspondence with every mechanical variable. 

The third postulate asserts that there is a hermitian mathematical operator for each mechanical variable. To write the operator for a given variable (1) write the classical expression for the variable in terms of Cartesian coordinates and momentum components, (2) replace each momentum component by the relation... 

As defined in Section 2.5, any hermitian operator, < , signifies a mathematical operation to be done on a wavefunction, v, which will yield a constant, o, if the wavefunction is an eigenfunction of the operator. 

An operator with the property exhibited in eqn (5.5) is said to be Hermitian if it satisfies this equation for all functions P defined in the function space in which the operator is defined. The mathematical requirement for Hermiticity of H expressed in eqn (5.5) places a corresponding physical requirement on the system—that there be a zero flux in the vector current through the surface S bounding the system To illustrate this and other properties of the total system we shall assume, without loss of generality, a form for H corresponding to a single particle moving under the influence of a scalar potential F(r)... 

How to reconcile this diagonal or vector form without the presence of the imaginary unit with the hermitian nature of momentum is a matter of mathematical formalism. In the EH space fimnework, it is sufficient to consider extended momentum operators, d, as elements of a Minkowsky space of signature (1,3) [78]. Admitting this space structure, then the property 5 = 1 -is automatically fulfilled. Also, it is easy to consider the extended momentum operator as a linear transformation ... 

Here the general mathematical rules are utilized for taking the Hermitian conjugate, and the fact that the creation and annihilation operators are the Hermitian conjugates of each other [cf. Eq. (2.45)]. Using the symmetry rule of Eq. (4.42) one gets ... 

---