Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 11.
When we consider infinitesimal transformations of some dynamical variable, there is a correspondence between classical and quantum mechanics which we can see as follows. First, we’ll summarize the results from classical mechanics. We can define a canonical transformation generated by a variable as
Here, is an infintesimal amount and and are the infinitesimal amounts by which the coordinates and momenta vary. It follows from these definitions that, for any dynamical variable , its variation is given by a Poisson bracket
For the special cases of coordinates and momenta, this is
If the generator is the momentum , then
Thus, in classical mechanics, is the generator of translations in direction .
What do these things correspond to in quantum mechanics? [I find Shankar’s treatment in section 11.2 to be almost tautological, since it merely repeats the derivation given earlier. I’ll try to be a bit more general.]
Suppose we have some infinitesimal transformation given by a unitary operator . We can then define the changes in and by
Since describes an infinitesimal transformation, we can expand it to first order in :
where is some Hermitian operator known as the generator of the transformation. (We’ve seen a proof that the translation operator (a special case of ) is unitary and that its generator is Hermitian earlier, and the current case follows the same reasoning.) Using this form we have from 10 and 11, to order :
If , then
Comparing this with 6 and 7 we see that (in one dimension, where the classical coordinate is given by and momentum by ) there is a correspondence between the classical Poisson bracket and quantum commutator:
The momentum operator in quantum mechanics is thus the generator of translations, just as generates translations in classical mechanics.
More generally, we can define the variation in some arbitrary dynamical operator in a similar way, using 12 to expand the RHS:
The correspondence with classical mechanics is then
The general rule is that a quantum commutator is times the corresponding classical Poisson bracket.
If and , then by Ehrenfest’s theorem, and the average value of is conserved.
One other feature is worth noting. A canonical transformation preserves the Poisson brackets 8 in the new coordinate system. In quantum mechanics, it is the commutators that get preserved. For example, using the fact that is unitary so that :