Calculating Commutator of Two Operators

[jfy4]

Homework Statement

Let $U$ and $V$ be the complementary unitary operators for a system of $N$ eingenstates as discussed in lecture. Recall that they both have eigenvalues $x_n=e^{2\pi in/N}$ where $n$ is an integer satisfying $0\leq n\leq N$. The operators have forms
$$U=\sum_{n}|n_u\rangle\langle n_u |e^{2\pi in/N}\quad\quad V=\sum_{m}|m_v\rangle\langle m_v |e^{2\pi i m/N}$$
These operators can be expressed as exponentials of complementary self-adjoint operators $A$ and $B$:
$$U=e^{i2\pi A}\quad\quad V=e^{i2\pi B}$$
where the operators $A$ and $B$ are
$$A=\frac{n}{N}\sum_{n}|n_u\rangle\langle n_u| \quad\quad B=\frac{m}{N}\sum_{m}|m_v\rangle\langle m_v |$$
Calculate the commutator $[A,B]$.

Homework Equations

$$\mathbb{I}=\sum |n\rangle\langle n|$$

for complementary observables
$$\frac{1}{\sqrt{N}}e^{i2\pi mn/N}=\sum_{n}\sum_{m}\langle n_u |m_v\rangle$$

The Attempt at a Solution

First I have tried to work out AB and BA separately then combine them. Here is AB
$$\begin{align} AB &= \sum_{n}|n_u\rangle\langle n_u |\frac{n}{N}\sum_{m}|m_v\rangle\langle m_v |\frac{m}{N} \\ &= \sum_{n}\sum_{m}|n_u\rangle\langle n_u|m_v\rangle\langle m_v| \frac{nm}{N^2} \\ &= \sum_{n}\sum_{m}|n_u\rangle \frac{1}{\sqrt{N}}e^{i 2\pi nm/N}\langle m_v |\frac{nm}{N^2} \end{align}$$
for BA:
$$\begin{align} BA &=\sum_{m}\sum_{n}|m_v\rangle\langle m_v | n_u\rangle\langle n_u |\frac{nm}{N^2} \\ &= \sum_{m}\sum_{n}|m_v\rangle \frac{1}{\sqrt{N}}e^{-i2\pi nm/N}\langle n_u |\frac{nm}{N^2} \end{align}$$
Then
$$AB-BA=\sum_{m}\sum_{n}\frac{nm}{N^2}\frac{1}{\sqrt{N}}\left( e^{i2\pi nm/N}|n_u\rangle\langle m_v |-|m_v\rangle\langle n_u |e^{-i2\pi nm/N}\right)$$
I'm stuck here more or less. I can put either the u basis vectors into the v basis or visa versa, but I don't know if that is right. Where should I go from here?

Thanks,

 

[diazona]

Hm... well, I haven't done the problem myself so I can't guarantee that this will work, but in general the commutator of operators is itself an operator. So I would suggest calculating the matrix elements of $AB - BA$ in either basis, and see if the result suggests anything.