I will make a few observations (if any of these is incorrect please let me know) and then ask my question :-
i) For a Quantum Mechanical Harmonic Oscillator (QMHO) we have, at least, two kinds of representations : WaveFunction representation (in Hilbert Space, $\mathcal{H}$) and Occupation Number representation (in Fock Space, $\mathcal{F}$).
ii) The energy levels of a QMHO are equally separated.
iii) In the $\mathcal{F}$ for QMHO, each particle created out of quantization is identical.
iv) We have creation ($a^{\dagger}$) and annihilation ($a$) operators which allow us to create and destroy particles. These operators are only available when we work in $\mathcal{F}$ (correct?)
Now my question :-
We never see (atleast in typical textbooks) any Fock Space representation for other QM potentials like 1D Square Well Potential etc. Is this because there is no such representation? If yes, then is it connected to the QMHO having equally spaced energy levels whereas other potentials typically lack this feature? If no, how can we construct the Fock Space for other potentials? The latter basically means having a ladder operator structure with identical particles which can be created or destroyed by the ladder operators.
Without even going into mathematical formalism can I say that since the energy levels are not equally spaced in a potential like 1D Square Well Potential the particles that we create out of those cannot be identical because if they are identical then adding one particle should amount to same increase in energy at each addition which isn't true for this potential. Or we could have more than one particle (basically a collection of $m$ QMHO, each uniquely defined with the energy their specific particle, $\omega_m$) such that the interplay among them gives us the desired result (sort of like how we do Fourier Analysis on general functions to decompose them into individual frequencies, which is quite nice as an analogy here IMHO).