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
<br /> 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}<br />
These operators can be expressed as exponentials of complementary self-adjoint operators A and B:
<br /> U=e^{i2\pi A}\quad\quad V=e^{i2\pi B}<br />
where the operators A and B are
<br /> 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 |<br />
Calculate the commutator [A,B].

Homework Equations

<br /> \mathbb{I}=\sum |n\rangle\langle n|<br />

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

The Attempt at a Solution

First I have tried to work out AB and BA separately then combine them. Here is AB
<br /> \begin{align}<br /> AB &amp;= \sum_{n}|n_u\rangle\langle n_u |\frac{n}{N}\sum_{m}|m_v\rangle\langle m_v |\frac{m}{N} \\<br /> &amp;= \sum_{n}\sum_{m}|n_u\rangle\langle n_u|m_v\rangle\langle m_v| \frac{nm}{N^2} \\<br /> &amp;= \sum_{n}\sum_{m}|n_u\rangle \frac{1}{\sqrt{N}}e^{i 2\pi nm/N}\langle m_v |\frac{nm}{N^2}<br /> \end{align}<br />
for BA:
<br /> \begin{align}<br /> BA &amp;=\sum_{m}\sum_{n}|m_v\rangle\langle m_v | n_u\rangle\langle n_u |\frac{nm}{N^2} \\<br /> &amp;= \sum_{m}\sum_{n}|m_v\rangle \frac{1}{\sqrt{N}}e^{-i2\pi nm/N}\langle n_u |\frac{nm}{N^2}<br /> \end{align}<br />
Then
<br /> 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)<br />
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.