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Linear Potential Question

  1. Dec 25, 2008 #1

    A particle in one dimension is subjected to a constant force derivable from

    [tex]V = \lambda x[/tex]​

    Is the energy spectrum continuous or discrete? Write down an approximate expression for the energy eigenfunction specified by [tex]E[/tex].

    Solution attempt
    We have an unbound state, so we would have a continuous energy spectrum. Well, I was thinking of something along the lines of [tex]\psi(x)=e^{-f(\lambda) x} \sin (x-e^{-h(\lambda)})}[/tex] for the region [tex] x < E/\lambda[/tex] (of course, for [tex]x > E/\lambda[/tex], we need an exponentially decreasing function); I chose this function based on the following:

    1. The function needs to increase exponentially as we go farther left, because the energies are greater than the linear potential by a greater degree.
    2. The zeroes need to be bunched together closer as [tex]x\rightarrow -\infty[/tex], because the particle is more energetic here.

    Above, [tex]f(\lambda), g(\lambda)[/tex] are positive functions for [tex] \lambda > 0 [/tex].

    Is the above correct?
  2. jcsd
  3. Dec 26, 2008 #2


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    Science Advisor

    The energy spectrum is continuous, but your guess as to the form of the solution is wrong.

    Try writing down the solution for a constant potential (in each region, V<E and V>E), and then plugging in V(x) in place of the constant V.

    Also, if you've studied the WKB approximation, this is a good place to apply it. (If not, ignore this comment.)

    (I'm going offline for a week, so won't reply further, good luck!)
  4. Dec 26, 2008 #3
    Oh yeah, heh, I forgot about the WKB. Thanks for the help!
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