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A simple differential equation!

  1. Mar 29, 2008 #1
    Just out of pure curiosity, can anyone here give me any advice on the problem of solving the following differential equation

    [tex]\left\{\sum_i \vec{\alpha}_i \cdot \nabla_i + \sum_{i neq j}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\psi} = K\frac{\partial}{\partial t}\vec{\psi} = iKm\vec{\psi}.[/tex]

    where,

    [tex][\alpha_{ix} , \alpha_{jx}] \equiv \alpha_{ix} \alpha_{jx} + \alpha_{jx}\alpha_{ix} = \delta_{ij}[/tex]

    [tex][\alpha_{i\tau} , \alpha_{j\tau}] = \delta_{ij}[/tex]

    [tex][\beta_{ij} , \beta_{kl}] = \delta_{ik}\delta_{jl}[/tex]

    [tex][\alpha_{ix}, \beta_{kl}]=[\alpha_{i\tau}, \beta_{kl}] = 0 \text{ where } \delta_{ij} =
    \left\{\begin{array}{ c c }
    0, & \text{ if } i \neq j \\
    1, & \text{ if } i=j
    \end{array} \right.
    [/tex]

    Any advice on approaching this problem would be greatly appreciated.

    Thank you very much!
     
  2. jcsd
  3. Mar 30, 2008 #2
    It doesn't look simple to me ...lol...
     
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