Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.1.1.
In preparation for an examination of rotation invariance, we’ll have a look at translational invariance in two dimensions. We can apply much of what we did with translation in one dimension, where we showed that the momentum is the generator of translations. In particular, the translation operator for an infinitesimal translation is
In two dimensions, we can write an infinitesimal translation as where
In one dimension, we showed earlier that
The analogous relation in two dimensions is
We can verify that the correct form for is
Using the representation of momentum in the position basis, which is
the LHS of 4 is, using :
The last line is also what we get if we expand the RHS of 4 to first order in , which verifies that 5 is correct, so that the two-dimensional momentum is the generator of two-dimensional translations.
We can apply the exponentiation technique we used in the one-dimensional case to obtain the translation operator for a finite translation in two dimensions. We need to be careful that we don’t run into problems with non-commuting operators, but in view of 7 and 8 and the fact that derivatives with respect to different independent variables commute, we see that
We can divide a finite translation into small steps, each of size , so that the translation is
Because the two components of momentum commute, we can take the limit of this expression to get the exponential form:
Again, because the two components of momentum commute, we can combine two translations, by and then by , to get