Expectation value of a ladder operator

Question

I am going back over old Q.M simple harmonic motion material and, as I can't see an answer on the web, I would like to confirm the validity of an assumption.

Using the ladder operators:

$$ {\displaystyle {\begin{aligned}a&={\sqrt {m\omega \over 2\hbar }}\left({\hat {x}}+{i \over m\omega }{\hat {p}}\right)\\a^{\dagger }&={\sqrt {m\omega \over 2\hbar }}\left({\hat {x}}-{i \over m\omega }{\hat {p}}\right)\end{aligned}}} $$

My early reading was on the discrete energy levels of potential wells, and the expectation values of, for example $x$, $x^n$ , $p^2$ etc. that can be calculated using these orthoganal energy eigenstates.

I know that I can easily rearrange the above to get $x$ and $p$ in terms of $a$ and $a^{\dagger } $ and that should give me all the aspects of expectation values which I am used to using in 1 D expectation values.

I also know that you can find the expectation value if $ {\displaystyle A}$ has a complete set of eigenvectors ${\displaystyle \phi _{j}}$, with eigenvalues ${\displaystyle a_{j}} $.

My question is: does trying to find the expectation value of $\langle \Psi | a |\Psi \rangle$ or $\langle \Psi | a^{\dagger } |\Psi \rangle$ implicity assume that you get $x$ and $p$ in terms of $a$ and $a^{\dagger }$ and then use those expressions in calculation of the expectation values?

Apologies, this is basic stuff but it's been a while and the answer might help someone else. I can see related questions regarding the number operator but if there is a duplicate I will remove this.

EDIT Do these expressions make any physical sense? Thanks to ACuriousMind for this answer below.

As mathematical expressions the "expectation values" of $a$ and $a^†$ are perfectly fine, but they are physically non-sensical since the operators are not self-adjoint and therefore are not observables - you're not computing expectation values because there's no measurement you could expect those values for.

END EDIT

Answer 1

The operators $a$ and $a^\dagger$ are not hermitian. From the definition of the operators above it is not hard to see that $$ x~=~\sqrt{\frac{2\hbar}{m\omega}}(a~+~a^\dagger) $$ and $$ p~=~i\sqrt{2\hbar m\omega}(a~-~a^\dagger) $$ are however hermtian. The role of the raising and lowering operators is $$ a|n\rangle~=~\sqrt{n}|n-1\rangle,~a^\dagger|n\rangle~=~\sqrt{n+1}|n+1\rangle. $$ Clearly these operators are not hermitian, which would imply $a~=~a^\dagger$

From a matrix perspective if is clear that these operators have a representation according to off diagonal entries. The lowering operator for instance would have $a_{n+1,n}~=~\sqrt{n+1}$ and the raising operator would have $a_{n,n+1}~=~\sqrt{n}$, with all other entries zero. The eigenvalues of any operator with matrix representation $M$ are computed by $det|M~-~\lambda\mathbb I|$ $=~0$. If the diagonal entries of a matrix are zero there are no eigenvalues. Think of $M~=~e^\mu$ with $det~M~=~e^{tr\mu}$, where the exponential of the trace is zero corresponding to zero diagonal entries. Therefore there are no eigenvalues.