Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$
I've read that all Hilbert spaces with a countable basis are isomorphic. And I can have some $2\pi$ periodic function $u(\theta)=e^{in\theta}$ such that $|n\rangle \rightarrow u(\theta)$. The number operator thus assumes the form $\hat{n}=-i\frac{\partial}{\partial\theta}$.
It is then claimed that the conjugate operator is $\hat{\theta}=\theta$.
It is then claimed that : $$[\hat{n},\hat{\theta}]=[-i\frac{\partial}{\partial\theta},\theta]=i$$
However, I can't understand how this can be possible. For example, if $\theta$ is periodic then we can express it as a Fourier series, such that : $$\theta=\sum_{m=-\infty}^{\infty}C_me^{im\theta}$$
Now if we let this act on some $u(\theta)$, we would have :
$$\hat{\theta} u(\theta)=\sum_{m=-\infty}^{\infty}C_me^{im\theta}e^{in\theta}=\sum_{m=-\infty}^{\infty}C_me^{i(m+n)\theta}$$
As we can clearly see, this would contain terms of the form $e^{-ik\theta}$, where $k\ge0$. The problem now is that, if the number operator acts on these, we will get an eigenvalue of $-k$ i.e. a negative eigenvalue. However, we know that in the Hilbert-space of the harmonic oscillator, the spectrum of $\hat{n}$ must be a whole number.
So essentially when $\hat{\theta}$ acts on $|n\rangle\rightarrow u(\theta)$, we get specific terms like $|-k\rangle$ which shouldn't belong in the Hilbert space of the harmonic oscillator. This creates a problem, as when we perform $\hat{n}\hat{\theta}|n\rangle$, we get terms that are physically unfeasible. Based on this, can I conclude, that if two operators $\hat{A},\hat{B}$ are conjugates of each other, then for some eigenfunction $|a\rangle$, $\hat{B}|a\rangle$ must also exist in the Hilbert space of $\hat{A}$ and vice-versa?
So, can we truly claim that if $\hat{n}$ is represented as $-i\frac{\partial}{\partial\theta}$, the conjugate variable is $\theta$ ? Is the claim that $[\hat{n},\hat{\theta}]=[-i\frac{\partial}{\partial\theta},\theta]=i$wrong in this example ?
p.s. I'm still in my undergrad physics and have just started to learn this, and so, I lack a lot of the technical details and the mathematical rigor to understand this completely. Please be a little lenient, if possible.