# Fock Space representation for 1D Square Well Potential

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).

Your question is closely related to this question. I believe that what you need is

Fiset, M.A. and Hussin, V., 2014. Equivalent sets of coherent states of the 1D infinite square well and properties. arXiv preprint arXiv:1410.0305.

This is based on a method by Gazeau and Klauder where they actually start with any distribution of eigenvalues and do the gymnastics to get creation and destruction operators but I can’t find that work now (I did see Gazeau give a conference talk on this but when?)

Closely related is the factorization method

Infeld, L. and Hull, T.E., 1951. The factorization method. Reviews of modern Physics, 23(1), p.21.

which is at the heart of supersymmetric quantum mechanics.

• In view of your last comment it is quite, quite interesting to note that I am currently working on SUSY QM. I hope you don't mind giving me some time so that I can check the references you provided and then accept this as an answer. – self.grassmanian Jun 3 '20 at 5:43
• The paper Cooper Fred, Avinash Khare, and Uday Sukhatme. "Supersymmetry and quantum mechanics." Physics Reports 251.5-6 (1995): 267-385. then becomes all the more relevant. – ZeroTheHero Jun 3 '20 at 13:09