Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as:
Every dynamical variable may be represented by a Hermitian operator whose eigenvalues represent the result of carrying out a measurement of the value of the dynamical variable…
My question is: just how much can we deduce from saying that the some of the operators involved in Quantum Mechanics are Hermitian?
When I use the word Hermitian, I am referring to the property that $A=A^+$ where $A$ is an operator and $A^+$ is the adjoint of $A$. Does an operator being Hermitian automatically imply that there will exist eigenfunctions and eigenvalues corresponding to this operator? Does being Hermitian imply that, should there exist eigenfunctions corresponding to this operator, they will form a complete and orthonormal set? That any wavefunction can be expanded in terms of these eigenfunction? Just how much is hidden in the word "Hermitian"?
Best Answer
There is quite a lot of very important information hidden in the term hermitian.
For an operator $A$ on a finite-dimensional Hilbert space $\mathcal H$, one can show that there exists an orthonormal basis for the Hilbert space consisting of eigenvectors of the operator $A$. Moreover, one can show that the eigenvalues corresponding to these eigenvectors are all real. This result is the finite-dimensional Spectral Theorem. The fact that eigenvalues of hermitian operators are real is of crucial importance in quantum mechanics since eigenvalues of observables are supposed to represent real-valued, physically measurable quantities.
When the Hilbert space is inifinite-dimensional, then the analogous result, which also is called the spectral theorem, is harder to prove and there are more technical assumptions one needs to make because of issues arising regarding domains of definition of operators and so-called unbounded operators etc. In particular, one needs the notion of a self-adjoint operator which is the infinite-dimensional extension of hermitian and reduces to hermitian in the finite-dimensional case. I'd encourage you to look at the wiki article on the spectral theorem that I linked for more info.
Also, fyi, the branch of mathematics that deals with the infinite-dimensional case is functional analysis.