References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 2.7; Exercises 2.7.4 – 2.7.5.
Here are a couple of examples of canonical variable transformations.
Example 1 We rotate the 2-d rectangular coordinates through an angle , giving the transformations
To show this is a canonical transformation, we must evaluate the Poisson brackets. Here, and . Remember that is a constant in these derivatives.
since neither coordinate depends on any momentum. Similarly since this Poisson bracket contains derivatives of with respect to and these are all zero. The remaining Poisson bracket are of the form . There are four of these, but we’ll work out only a couple. The other two have similar forms.
Example 2 The transformation from 2-d rectangular to polar coordinates is given by
For the Poisson brackets we have
because, again, the coordinates don’t depend on the momenta.
In this case, however, the new momenta do depend on the old coordinates, so we need to actually do some calculation.
Finally, we need to work out the mixed brackets.
Thus all the Poisson brackets are correct, so the transformation is canonical.