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Commutator of two operators

  1. Sep 8, 2011 #1
    1. The problem statement, all variables and given/known data
    Let [itex]U[/itex] and [itex]V[/itex] be the complementary unitary operators for a system of [itex]N[/itex] eingenstates as discussed in lecture. Recall that they both have eigenvalues [itex]x_n=e^{2\pi in/N}[/itex] where [itex]n[/itex] is an integer satisfying [itex]0\leq n\leq N[/itex]. The operators have forms
    [tex]
    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}
    [/tex]
    These operators can be expressed as exponentials of complementary self-adjoint operators [itex]A[/itex] and [itex]B[/itex]:
    [tex]
    U=e^{i2\pi A}\quad\quad V=e^{i2\pi B}
    [/tex]
    where the operators [itex]A[/itex] and [itex]B[/itex] are
    [tex]
    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 |
    [/tex]
    Calculate the commutator [itex][A,B][/itex].


    2. Relevant equations
    [tex]
    \mathbb{I}=\sum |n\rangle\langle n|
    [/tex]

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

    3. The attempt at a solution
    First I have tried to work out AB and BA separately then combine them. Here is AB
    [tex]
    \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}
    [/tex]
    for BA:
    [tex]
    \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}
    [/tex]
    Then
    [tex]
    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)
    [/tex]
    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,
     
  2. jcsd
  3. Sep 11, 2011 #2

    diazona

    User Avatar
    Homework Helper

    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 [itex]AB - BA[/itex] in either basis, and see if the result suggests anything.
     
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