Hermitian Operators

There is a special class of operators which are called Hermitian operators. They are of particular importance in quantum mechanics because they have the property that all of their eigenvalues are real (not complex). This is convenient for the measurement outcome of any experiment must be a real number. There are non-Hermitian operators, but they do not correspond to observable properties. All observable properties are represented by Hermitian operators (but not all Hermitian operators correspond to an observable property).

Hermiticity (the name comes from an old French mathematician by the name of Hermite) is defined as

The eigenvalues of Hermitian operators are real.

You wanna proof!!??!!

The eigenstates corresponding to different eigenvalues of hermitian operators are orthogonal.

You wannaother!!??!!

If two observables are to have simultaneously precisely defined values, the corresponding operators must commute.

And this one proved too???

This is true of all operators, but also to the hermitian operators of special importance to us here.