1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Solving an equation for a unitary matrix

  1. Jan 7, 2009 #1
    1. The problem statement, all variables and given/known data

    I have an equation for a unitary matrix [tex]U[/tex],
    [tex]\sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} - \tilde{\epsilon}_k \delta_{qm}} \right)} = 0[/tex]
    I need to solve this equation for [tex]U[/tex]

    2. Relevant equations

    The property of unitarity requires that [tex]U\bar{U} = \hat{I}[/tex]

    3. The attempt at a solution
    If [tex]q \neq m[/tex] then
    [tex]\sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} } \right)} = 0[/tex]
    so that
    [tex]\sum_k \left(\varepsilon_k - \mu\right)\bar{U}_{qk} U_{km} = - \gamma \sum_{kp} \bar{U}_{qk} U_{pm}[/tex]

    If [tex]q = m[/tex] then
    [tex]\sum_k \left(\varepsilon_k - \mu\right) = - \sum_{kp} \left(U_{mk} U_{pm} - \tilde{\epsilon_k}\right)[/tex]

    How do I combine these two results in one equation for [tex]U[/tex]?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted