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 have

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 :

We can compare this with 11 inserted into 14:

Setting 32 equal to 29 we have

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.