Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.2.1.
As a first look at rotational invariance in quantum mechanics, we’ll look at two-dimensional rotations about the axis. Classically, a rotation by an angle about the axis is given by the matrix equation for the coordinates
The momenta transform the same way, since we are merely changing the direction of the and axes. Thus we have also
The rotation matrix can be written as an operator, defined as
In quantum mechanics, due to the uncertainty principle we cannot specify position and momentum precisely at the same time, so as with the case of translational invariance, we deal with expectation values. As usual, a rotation is represented by a unitary operator so that a quantum state transforms according to
Dealing with expectation values means that the rotation operator must satisfy
The expectation values on the LHS of these equations are calculated using the rotated state, so that
and so on.
In two dimensions, the position eigenkets depend on the two independent coordinates and , and each of these eigenkets transforms under rotation in the same way the position variables above. Operating on such an eigenket with the unitary rotation operator thus must give
As with the translation operator, we try to construct an explicity form for by considering an infinitesimal rotation about the axis. We propose that the unitary operator for this rotation is given by
where is, at this stage, an unknown operator called the generator of infinitesimal rotations (although, as the notation suggests, it will turn out to be the component of angular momentum). Under this rotation, we have, to first order in :
Note that we’ve omitted a possible phase factor in this rotation. That is, we could have written
for some real function . Dropping the phase factor has the effect of making the momentum expectation values transform in the same way as the position expectaton values, as shown by Shankar in his equation 12.2.13, so we’ll just take the phase factor to be 1 from now on.
We can now find the position space form of a general state vector under an infinitesimal rotation by following a similar procedure to that for a translation.
We can now change integration variables if we define
The differentials transform by considering terms only up to first order in infinitesimal quantities, so we have
Also, to first order in infinitesimal quantities, we can invert the variables to get
The ranges of integration are still , so we end up with
Multiplying on the left by the bra we have
This can now be expanded in a 2-variable Taylor series to give, to first order in :
Using the position-space forms of the momenta
we see that is given by
which is the quantum equivalent of the component of angular momentum, as promised.