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In J. Phys. A: Math. Gen. 22 (1989) 811-822, Crehan considered the classical Hamiltonian,

\begin{align} H=\frac{p^2}{2}+\frac{q^2}{2}+\lambda(p^2+q^2)^3\,. \end{align}

Due to the presence of the third term, the process of quantizing $H$ is ambiguous as we need to worry about the order of the operators. For instance, we could write $\hat{H}$ using standard ordering, anti-standard ordering, or Weyl ordering, to name a few possibilities.

Crehan showed that the eigenfunction of $H$ for all possible quantizations is the eigenfunction of the SHO, but with an eigenvalue of

\begin{align} E_n=\frac{1}{2}\hbar(2n+1)+\lambda\hbar(2n+1)^3+\lambda(3\hbar^2\alpha-4)(2n+1)\,, \end{align}

where $\alpha$ is a parameter of the quantization.

How do we know which value of $\alpha$ gives the correct quantization/operator ordering for this problem? As different quantizations yield different operator orderings and hence different energies, we have physically distinct predictions.

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    $\begingroup$ This is an excellent question. If the results are distinct, only experiment can answer the question. Moreover, it may well be that different experimental implementation are compatible with different orderings. $\endgroup$ Jul 15, 2017 at 23:36
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    $\begingroup$ Related: physics.stackexchange.com/q/323937/50583 - you seem to be expecting that there is a unique way to determine the quantization of every classical system, which is simply not the case. The linked question asks why there isn't such a map, and therefore seems to be what you really want to know. $\endgroup$
    – ACuriousMind
    Jul 16, 2017 at 0:31
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    $\begingroup$ also related is this excellent review by Paul Chernhoff: researchgate.net/publication/… published in the (obscure) Hadronic Journal in 1981. $\endgroup$ Jul 16, 2017 at 0:43
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    $\begingroup$ imo Hamiltonians can be pulled out of the hat . It is how they model a specific potential problem that is important. What is the specific physical set up for which the above hamiltonian is a correct model? $\endgroup$
    – anna v
    Jul 18, 2017 at 16:46

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But...did you specify the problem? Which problem? Of course you have physically distinct predictions. Which ones do you want to use and where? Crehan's paper finds all 2-parameter (ħ,α) deformations of this cubic oscillator, subject to his plausible constraints, but you seem to have extra conditions based on unstated physics principles? If you do not state them, your question cannot be answerable.

Either you look at your experimental system modeled by this deformed oscillator and see which α best fits its spectrum, a situation often occurring with small systems relying on such simple models, e.g. in nonlinear optics; or, else, with Robnik (cited), you search for convenient recipes and models easiest handled by some particular class of αs. (Also see the stochastic transition to quantum non-integrable systems in de Carvalho, R. E. (1993). "Classical and quantal aspects of resonant normal forms". Nonlinearity 6(6), 973.)

In a range of problems, like quantization on nonstandard manifolds (spheres, etc..) you pick the α that best preserves classical symmetry algebras through quantization-- often you want to preserve those.

Ivan Todorov's tasteful and edifying "Quantization is a mystery" covers the waterfront.

But quantization is the quintessential one-to-many map (it contains additional information over and above the classical limit---otherwise how could people have invented QM, and why?), and you never know you have the correct quantum hamiltonian, operator, etc... until it is checked to describe an experimental situation. QM is not a coordinate-change-like functor of classical mechanics, it is an extension with new information over and above what survives the classical limit.

Frankly, I'd shudder to think you observe two different quantum systems in the lab with different αs and spectra, etc, but the same classical limit, and somehow decide that only one of them is "correct" on unstated capricious metaphysical principles...

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  • $\begingroup$ Would you have a short list of examples of physical systems for which experiment eventually decided which of several mathematically possible quantization was the correct one? $\endgroup$
    – user154997
    Jul 17, 2017 at 3:39
  • $\begingroup$ No. But I keep stumbling them in quantum optics, chaos, etc... (The PT QM types did what you suggest for their racket, but they have something to prove, unlike the graceless proofs/explications here that the earth is not flat...) The problem with compiling the list is one has to do due diligence to prove all alternative prescriptions would be unworkable. For instance, in my paper linked (Not an experimental project!) we demonstrate how Weyl ordering preserves the manifold symmetry algebra, but we did not care to exclude other popular prescriptions (Husimi, BJ, etc...) let alone all... $\endgroup$ Jul 17, 2017 at 12:40
  • $\begingroup$ ...and we would not be caught dead "championing" some prescriptions at the expense of others, like some mathematicians do for BJ, completely missing the boat. In any case, nonlinear optics types relish doing these experiments to make a point or other, often bogus, about "quantum chaos"... however they would strive to define it.... $\endgroup$ Jul 17, 2017 at 12:42
  • $\begingroup$ This suggests that perhaps it is best to not think about "quantizing" classical theories at all, but rather about building quantum models from the get-go. A classical model can be a hint as to a quantum model, but only a hint; and so we should talk of "building quantum models" and definitely not "quantizing" classical models as though there is some straightforward, procedural process. $\endgroup$ May 15, 2020 at 17:54
  • $\begingroup$ (After all, if there were, then quantum gravity would be a done deal by now :) ) $\endgroup$ May 15, 2020 at 17:56

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