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2 questions

  1. Sep 8, 2004 #1
    let's say..

    there is a particle, with mass m, in a 2-dimensions x-y plane. in a region
    0 < x < 3L ; 0 < y < 2L

    how to calculate the energy eigenvalues and eigenfunctions of the particle?

    thx :smile:

    and.. 2nd question..

    there is a particle of kinetic energy E is incident from the left on the potential barrier, height U, situated at the origin. The barrier is infinitely wide and E>U.

    how to get an expression for the reflection coefficient R of the particle, as a function of the ratio ε= E/U
    and how would the sketch look like?


    :cry: i dont really know how to work them out...
     
  2. jcsd
  3. Sep 8, 2004 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Basically, you are talking about a "potential well" in which the energy is taken to be 0 inside the rectangle, infinite outside. You need to solve Schrodinger's equation with Energy 0 inside the rectangle. This turns out to be one of the few situations where it can be solved exactly. The solution consists of "standing waves" (just like waves on a rubber rectangular sheet fixed at the boundaries).
     
  4. Sep 8, 2004 #3
    i dont understand when it comes into 2 dimension :(
     
  5. Sep 9, 2004 #4
    can someone help me see if these look right?

    π = pi

    0 < x < 3L ; 0 < y < 2L
    E1 + E2 = E
    Eigenvalue :
    ψ(x,y) = A sin [(n1 π x) / (3 L)] sin [(n2 π y) / (2 L)]
    Eigenfunction :
    E = E1 + E2 = [(n1^2 π^2 ħ^2) / (6 m L^2)] + [(n2^2 π^2 ħ^2) / (4 m L^2)]
    E = [(π^2 ħ^2) / (2 m L^2)] * [(n1^2 / 3) + (n2^2 / 2)]

    for the 3 lowest energy
    E11 = [(π^2 ħ^2) / (2 m L^2)] * [(1^2 / 3) + (1^2 / 2)]
    = (5 π^2 ħ^2) / (12 m L^2)
    E12 = [(π^2 ħ^2) / (2 m L^2)] * [(1^2 / 3) + (2^2 / 2)]
    = (7 π^2 ħ^2) / (6 m L^2)
    E21 = [(π^2 ħ^2) / (2 m L^2)] * [(2^2 / 3) + (1^2 / 2)]
    = (11 π^2 ħ^2) / (12 m L^2)
     
  6. Sep 10, 2004 #5
    http://electron6.phys.utk.edu/qm1/modules/m2/step.htm

    Nor do I, but here's a jumping point if it'll help. . . my brain doesn't work after 1:00am EST :biggrin:
     
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