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Asymmetric potential well problem

  1. Oct 30, 2008 #1
    The task is to find a wavefunction (doesn't need to be normalised) and the energy levels of a particle in an asymmetric potential well (a Schrodinger problem). i.e V = V1 for x<0, 0 for 0<x<d and V = V2 for x>d.

    What I've got so far is

    Let α^2= 2m (E-V1)/ℏ^2
    β^2= 2mE/ℏ^2
    T^2= 2m (E-V2)/ℏ^2

    Using these substitutions in Shrodinger's Equations and keeping Ψ finite yields

    Ψ = A exp (αx) x<0
    C exp (iβx) + D exp (-iβx) 0<x<d
    G exp (-Tx)

    Boundary conditions give

    A = C + D
    Aα = (C - D) iβ
    Gexp(-Td) = C exp(iβd) + D exp(-iβd)
    -TGexp(-Td) = Ciβexp(iβd) - Diβexp(-iβd)

    Eliminating A and G:
    (C + D) α = (C - D) iβ
    -T(Cexp(iβd) + Dexp(-iβd)) = iβ(Cexp(iβd) - Dexp(-iβd))

    Shifting all these terms of these side onto one side ( so get 0 = .....) and putting into a matrix and setting determinant = 0 yields:

    (αT - αiβ - Tiβ - β^2)(exp(-iβd)) = (αT + αiβ + Tiβ - β^2)(exp(iβd))

    exp(-2iβd) = (αT - αiβ - Tiβ - β^2)/(αT + αiβ + Tiβ - β^2)

    cos(2βd) -i sin(2βd) = (αT - αiβ - Tiβ - β^2)/(αT + αiβ + Tiβ - β^2)

    at which point I am stuck. Hints??
     
  2. jcsd
  3. Oct 30, 2008 #2

    Avodyne

    User Avatar
    Science Advisor

    This seems a like a very messy problem, but it would probably be easier to use a linear combination of sin and cos in the middle, since we know that the eigenfunctions are real.
     
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