1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook