Classical Hamiltonian involving product of factors whose quantum analogues don't commute

Dirac remarked in his quantum mechanics book:

One can usually assume that the Hamiltonian is the same function of the canonical coordinates and momenta in the quantum theory as in the classical theory. There would be a difficulty in this, of course, if the classical Hamiltonian involved a product of factors whose quantum analogues do not commute, as one would not know in which order to put these factors in the quantum Hamiltonian, but this does not happen for most of the elementary dynamical systems whose study is important for atomic physics.

I think some examples include: a free particle, a harmonic oscillator, or a charged particle in a magnetic field.

But what are some examples (and consequences) of elementary dynamical systems whose quantum Hamiltonian differs from its classical Hamiltonian? In particular, they should be relatively simple and in the realm of Mechanics /Atomic Physics, in the spirit of Dirac's remark.

• You are misreading Dirac's quote, I think. The feature he claims is rare is the existence of terms like $x p$ (i.e. products of non-commuting operators) in the Hamiltonian. In that case the quantum $\to$ classical limit is well defined (i.e. the classical version of the quantum hamiltonian coincides with the original classical hamiltonian) but there may be several quantum hamiltonians to choose from (i.e. the classical $\to$ quantum step is not well defined). Sep 30 '14 at 12:07
• Strictly speaking, these are all just models and the connection between the classical and the quantum mechanical version is being made ad-hoc. The one and only Hamiltonian/Lagrangian that nature may have implemented is the one of the TOE. We don't know what it looks like. Everything else is either a mean field theory that follows from that one, or it's a man made abstraction. Free particles do not exist any more than perfect harmonic oscillators do. You define what you want to call a "quantum harmonic oscillator" and then derive a classical model from that, but that mapping is ambiguous. Sep 30 '14 at 14:33
• @EmilioPisanty, I did not misread it at all. I'm asking for examples of systems where this "rare existence" arises. Sep 30 '14 at 17:07
• In that case, I suggest you edit your question, and particularly the title, to better reflect that. Sep 30 '14 at 18:34
• Here is a simple quantum mechanical example where operator ordering of the Hamiltonian is non-trivial. Oct 1 '14 at 19:48

I don't know how elementary you consider a simple position dependent mass, but due to ordering ambiguity in the kinetic term $\hat{p}^2/2m(\hat{r})$ such a system will have a quantum Hamiltonian different from the classical one. For example:

The problem here is how to quantize systems whose classical hamiltonian involves factors of the form (for example) $p^nx^m$, because these cannot be unambigously represented in a formalism where $p$ and $x$ do not commute.

As such there are many alternatives (all of which are classically equivalent) but only one is quantum-mechanicaly relevant.

In most cases the factors have to be "de-composed symmetricaly" (for example):

$p^{1/4}x^{1/4}p^{1/4}x^{1/4}$ etc..

Another aspect of where classical and quantum hamiltonians differ is when spin is involved which is a purely quantum-mechanical concept. For example the hamiltonian for the hydrogen atom can be similar to the classical hamiltonian but only a crude approximation since electron spin should be taken into account and this makes the quantum analog differ from the classical one.

• This doesn't answer my question at all. I know what the problem is; and the last paragraph cheats in a way (it's not in the spirit of Dirac's remark). Sep 30 '14 at 17:08
• @ChrisGerig, om its your choice, for my defense i will simply say that the first part relates to tge Dirac quote on your question and the second part relates (at least that is the motivation) to the last part of your question hinting at simple systems "But what are some examples (and consequences) of elementary dynamical systems whose quantum Hamiltonian differs from its classical Hamiltonian?". For specific elementary systems which have factors of the form $p^mx^n$ there are some examples in curved spaces (there are others also) Sep 30 '14 at 18:23

Consider the kinetic energy operator

$$\hat{K}=\frac{1}{2m}\hat{p}^{2}.$$

Then $[r^{-1}, \hat{K}]$ is ambiguous depending on the coordinate system.

Moreover, in Cartesian coordinates, it is quite peculiar compared to its Poisson bracket...

(At least, if I did my math correctly, which is possible considering how sloppy/quick it was done, as I am pressured at the moment).

• ...What is your elementary system / Hamiltonian? It can't be just $\hat{K}$ because that is a free particle, and subsequently this is irrelevant to the question! Oct 1 '14 at 19:44
• @ChrisGerig I think he means it in the context of the Hydrogen atom Hamiltonian, although I don't see the direct connection to the original question, as $\hat K$ and $1/\hat r$ do not mix at first glance. I would be glad if Alex could elaborate. Oct 1 '14 at 19:51