I am confused as to what hermiticity of an operator means when given a basis set.
My course notes say that hermitian operators in Hilbert space stay unchanged under it's complex conjugate:
$$<n|A|m>^* = <m|A|n>.$$
And that mathematically hermiticity imply that the eigenvalues are always real, and there exists an orthonormal set of eigenvectors.
The 'completeness relation' says that:
$$\hat A\Sigma_n|n><n|=\Sigma_n\lambda_n|n><n|, $$
With $\hat A$ the operator, $|n><n|$ the orthonormal set and $\lambda$ the eigenvalues.
My question is:
Given an orthonormal basis $|\psi_n>$ in Hilbert space of an operator so that:
$$\hat A|\psi_n>=a_i|\psi_n>,$$
or a linear combination (eg. $\hat A|\psi_n>=a_i|\psi_n>+\space b_i|\psi_n>+…$),
Does this also mean that operator $\hat A$ is always hermitian, or is it only hermitian if the coëfficients $a_i$ are real? Are in this case $a_i$ eigenvalues or are they coëfficients determining the chance a certain observation $|a_i|^2/N^2<\psi_n|\psi_n>$, ($N$ is normalisation), takes place? If these are not the eigenvalues, how are they determined from this basis set?
Best Answer
Hermitian operator
Your definition is ok and equivalent to
$\langle a| A b \rangle = \langle A a| b \rangle$
Eigenspace of a Hermitian operator
It's possible to prove that a Hermitian operator has eigenvectors that form an orthogonal vector base, associated with real eigenvalues
$A | n \rangle = \lambda_n | n \rangle$.
Than you can normalize eigenvectors to get a orthonormal basis.
Generic wave function
Given a generic wave function $| \psi \rangle$ you can write it using the basis $\{ |n\rangle\}_n$, as a superposition of states
$| \psi \rangle = a_n |n\rangle $
so that you can write the action of the operator $A$ as
$ A | \psi \rangle = a_n A |n\rangle = a_n \lambda_n |n\rangle $.
Normalization condition and interpretation as a probability
With the normalization condition for the wave function $\langle \psi|\psi \rangle = 1$ (holding also for eigenvectors), you readily get
$1 =\langle \psi|\psi \rangle = \langle a_n n | a_m m \rangle = \sum_n a_n^2$.
If a system is ina state descirbe by wavefunction $|\psi \rangle$, the probability of measuring state $| m \rangle$ is
$|\langle \psi | m \rangle|^2 = |\langle a_n n | m \rangle|^2 = a_m^2$.
Eigenvalues and result of measurement process
On the other hand, eigenvalues of an operator are connected to the values you get from measurement process for the physical quantity associated with that oeprator. See Born's rule, https://en.m.wikipedia.org/wiki/Born_rule