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3D Schrodinger Equation of a Wire for Energy Levels - Quantum Physics

  1. Dec 6, 2011 #1
    1. The problem statement, all variables and given/known data

    Given a wire with length a and square base b x b (where a >> b), show that the first 1700 (approximately) levels of the electron in the wire are identical for the one dimensional box, when a = 1m and b = 1mm.



    2. Relevant equations

    I know that the allowed energies of a mass M in a 3D rectangular rigid box with sides a, b, and c are:

    E1 = ħ^2 * (Pi)^2 / 2M * (nx^2 / a^2 + ny^2 / b^2 + nz^2 / c^2)

    The energy of a 1-D box with length a is similar, being

    E2 = ħ^2 * (Pi)^2 / 2M * (n^2 / a^2)

    3. The attempt at a solution

    For the purpose of this problem the energy of my wire is:

    E = ħ^2 * (Pi)^2 / 2M * (nx^2 / a^2 + ny^2 / b^2 + nz^2 / b^2)

    Now, I know by inspection and common sense that a long and very thing wire can be considered as a one dimensional system, which is used by the 1D Schrodinger equation.

    The problem is that I'm not quite sure how it works out mathematically. If a >> b, then doesn't this mean we generally ignore the a component of E1, since the b part basically dominates the energy? But all the same, you can't do that for the equation for the 1D box.

    It's all just a bit confusing for me.
     
  2. jcsd
  3. Dec 7, 2011 #2
    Assuming your formula for E is correct calculate the energy level for

    nx, ny, nz = 1700, 1, 1 and nx, ny, nz = 1, 2, 1

    they should be roughly equal?
     
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