[AntonioLao]

There is a class of linear operators in quantum mechanics called Hermitian operators and probably named in honor of the French mathematician Charles Hermite (1822-1901) who proved that e is transcendental and that the general quintic equation can be solved using elliptic functions. These are function defined on the complex plane for which f(z)=f(z+a)=f(z+b) where a/b is not real. It follows that f(z+ma+nb)=f(z) for all integers m and n. Moreover, f(z) is periodic in two distinct directions on the complex plane.

The adjoint or conjugate of linear operator A is symbolized by A+. For two given kets |fñ and |yñ the following is always true áy|A+|fñ=áf|A|yñ*. Furthermore, a linear operator is Hermitian if it is identical to its adjoint: A=A+. Therefore, in order to obtain the Hermitian of any given expression, it is sufficient and necessary to apply the following steps: (1) replace the constants by their complex conjugates, (2) replace the kets by the associated bras, vice versa, (3) replace the linear operators by their adjoints, and (4) reverse the order of the factors, for example: láf|AB|yñ®l*áy|B+A+|fñ. The Hermitian adjoint of a product of operators A, B, C, D, E, F, G, H is given by reversing their order and then forming the adjoint of each operator: (ABCDEFGH)+=H+G+F+E+D+C+B+A+. There are two distinctive properties of Hermitian operators. First, is that their eigenvalues are real numbers. Second, is that their eigenvectors are orthogonal. Reality of Hermitian operators makes them fundamentally important in the precise quantum measurement of energy and linear momentum in spectral analysis.