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.