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Discrete or continious spectrum in QM

  1. Nov 17, 2009 #1
    1. The problem statement, all variables and given/known data

    Here is the question: how can we know that if we have discrete or continuous spectrum just by looking at the potential graph?

    Specifically, let`s consider the potential V(x)=-F*x (F:const) . After we solve, we can conclude wavefunctin is airy function, and so both continious and discrete spectrum. But, without sloving, how can we decide?
     
  2. jcsd
  3. Nov 17, 2009 #2

    physicsworks

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    There are special cases when we can guess what the spectrum would be. For example, when the particle travels in the central field potential it can be shown that: when the energy [tex]E[/tex] of the particle is positive the spectrum is continuous and when [tex]E<0[/tex] the spectrum is discrete.
    For example, if we have repulsive potential energy of the form [tex]U \sim 1/r[/tex], the total energy [tex]E[/tex] of the particle is positive. (Really,
    [tex]E=\frac{1}{2m}\int {\psi^* \hat{\mathbf{P}}^2}\psi dV + \int {\psi^*U \psi}dV[/tex]
    which is positive since [tex]U[/tex] is positive and the eigenvalues of [tex]\hat{\mathbf{P}}^2}[/tex] are positive numbers too).
    So, in this potential field we have [tex]E>0[/tex] and continuous spectrum.
    If [tex]U \sim -1/r[/tex] we have Сoulomb attraction and to possibilities: [tex]E>0[/tex] (continuous spectrum, ionized electron) and [tex]E<0[/tex] (discrete spectrum).
    Finally, in case of 6-12 potential we have continuous spectrum for [tex]E>0[/tex] (dissociated molecule) and discrete spectrum for [tex]E<0[/tex].
    Such examples show that we can guess what the spectrum is without doing anything with Schrödinger equation.
     
  4. Nov 17, 2009 #3
    The way that was explained to me is if there are 2 turning points, then it is discrete. If there is only 1 turning point, it is continuous. I, unfortunately, was never able to ask my professor whether something that had three (or more) turning points would be continuous or not.
     
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