# Principal Bundle

## Angles in Quantum Mechanics

Posted in Physics, Quanum Mechanics by principalbundle on February 28, 2009

We know from intro quantum mechanics the positions $x$ and $y$ of a particle should be interpreted as operators. What happens when we transform to polar coordinates, $x=r \cos\theta$, $y=r\sin\theta$? Since $\theta$ classically becomes a dynamical variable, we expect it to become an operator quantum mechanically. What happens when we do this? The result is a very basic consequence of operators on a Hilbert space, but comes as a bit of a surprise, since these types of issues are not normally discussed in quantum mechanics classes or texts.

We can define the angular momentum operator, $L_z = -i \frac{\partial}{\partial\theta}$. And we can find the commuter $[L_z,\theta]=-i$. We also know $L_z$ has integer eigenvalues. Hitting each side of the commuter with eigenstates, we have

$(m-m') \langle m | \theta | m' \rangle = -i\delta_{m,m'}$.

Simple, right? No! This equation is false. When $m=m'$, it reads $0=1$.

What has gone wrong?

To answer this, let’s look at another question. From the commutation relation above, we can, by using the generalized uncertainty principal, deduce,

$\Delta L_z \Delta\theta \geq \hbar/2$.

However, eigenstates of $L_z$ have $\Delta L_z = 0$. Another contradiction! How to we resolve this?

Let’s recall our definition of $L_z$. It is an operator in a Hilbert space which acts on functions. But which functions does it act on? It only acts on functions which are periodic in $\theta$.1 So we have to be careful when acting on a function like $\theta$, which is either multi-valued, or nonanalytic.

Because multivalued and nonanalytic operators are not allowed as observables, $\theta$ should not be thought of an observable over its whole range.

The obvious resolution to this is to only consider the analytic functions made from $\theta$. So we can find the commutation relations

$[L_z,\sin\theta] = -i \cos\theta$
$[L_z,\cos\theta] = +i \sin\theta$

We can then find the uncertainty relations,

$\Delta L_z \Delta \sin\theta \geq \frac{1}{2} \langle\cos\theta\rangle$
$\Delta L_z \Delta \cos\theta \geq \frac{1}{2} \langle\sin\theta\rangle$

which is not a contradiction, because eigenstates of $L_z$ have $\langle\sin\theta\rangle = \langle\cos\theta\rangle = 0$.

1 How do we phrase this in terms of compactness? Functions that $L_z$ acts on take their values from a compact set, however the domain isn’t the set of functions whose domain is compact. So something else about the topology of the domain of functions that $L_z$ acts on is needed, but I don’t know enough functional analysis to have an obvious more elegant way to write this. We can phrase this in terms of when $L_z$ is self-adjoint, but I don’t know that this leads to a nicer description offhand.