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Quantum hamiltonian with an expoenntial potetial.

  1. Nov 24, 2012 #1
    given the Schroedinger equation with an exponential potential

    [tex] -D^{2}y(x)+ae^{bx}y(x)-E_{n}y(x)= 0 [/tex]

    with the boudnary conditons [tex] y(0)=0=y(\infty) [/tex]

    is this solvable ?? what would be the energies and eigenfunction ? thanks.
  2. jcsd
  3. Nov 24, 2012 #2

    Simon Bridge

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    ##V(x)=ae^{bx}## represents an infinite barrier when approached from the left ... if b>0.
    Where did you get those boundary conditions from?

    ##V(x=0)=ae^{0}=a## if E>a, then y(0) need not be zero.
    You do need y(x) to be continuous where it crosses the barrier.
  4. Nov 25, 2012 #3
    um i forgot .. [tex] y(0)=0 [/tex] assume there is an infinite potential barrier so the wave function must be 0 at the origin.
  5. Nov 25, 2012 #4

    Simon Bridge

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    Oh well, that one will have solutions that look a lot like the 1/x potential - at least, for the lower energies.

    For simplicity, measure energy from the bottom of the well so ##V(0<x)=a(e^{bx}-1)## ... positive values of E will include bound states - so the answer to your question is: yes - solutions exist, and the SE for this potential should be solvable.

    If you want a rigorous proof of solvability you'll have to ask a mathematician ;)
    Getting the analytical solution will probably be a bit of a pain... but it usually is.
    How did this come up?
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