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

Question

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.

Answer 1

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.

Answer 2

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:

Analytic results in the position-dependent mass Schrodinger problem
Position-dependent effective masses in semiconductor theory

Answer 3

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).