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Eigenfunction this!

  1. Jan 7, 2009 #1
    Hi there,

    I don't have Mathematica and the standard books didn't work. I am looking for the eigenfunctions to this operator:

    [tex]\hat{O} = (\sin \alpha) \, \partial_x +(\cos \alpha) \, x \qquad \alpha \in \mathbb{R}[/tex]
     
  2. jcsd
  3. Jan 7, 2009 #2

    Mute

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    Homework Helper

    Well, it's a first order equation, so you can solve it using an integrating factor.

    [tex][(\sin \alpha) \, \partial_x +(\cos \alpha) \, x]\psi(x) = \lambda \psi(x)[/tex]

    and so moving the RHS over to the LHS and multiplying by [itex]\mu(x)[/itex], and dividing the [itex]\sin \alpha[/itex] through,

    [tex] \mu(x)\partial_x \psi(x) + \left[(\cot \alpha) \, x - \frac{\lambda}{\sin\alpha}\right] \psi(x)\mu(x) = 0[/tex]

    Hence through the usual arguments let

    [tex]\frac{d\mu}{dx} = \left[(\cot \alpha) \, x - \frac{\lambda}{\sin\alpha}\right] \mu[/tex]

    and the above expression is just the product rule expanded and you can solve for [itex]\psi[/itex]. Then apply the initial condition, and possibly any other conditions (such as demanded the function not diverge as x gets large) that might put restrictions on the values of [itex]\lambda[/itex].
     
  4. Jan 7, 2009 #3
    Thanks a lot!
     
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