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Homework Help: # bound states in a given system?

  1. Nov 29, 2008 #1
    # bound states in a given system??

    1. The problem statement, all variables and given/known data

    An electron is confined to a potential well of finite depth and width, 10^-9 cm. The eigenstate of highest energy of this system corresponds to the value ¥=3.2.

    a) How many bound states does the system have?
    b) Estimate the energy of the ground state with respect to the zero energy line at the bottom of the well. Express answer in eV.

    2. Relevant equations

    For the non-dimensional wavenumbers ¥=k*a, W=K*a:

    We know that ¥^2 + W^2 = (2*m*|V|)/hbar^2 = p^2, which is a circle of radius p in ¥W space, and the states obey

    ¥*tan(¥) = W for even energy eigenstates

    ¥cot(¥) = -W for odd eigenstates.

    The intersection of the even/odd eigenstate equation and the circle give the number of bound states, somehow (I guess).

    3. The attempt at a solution

    I am just confused in general. Our text says that you can locate the number of bound states by finding the intersection of the circle in ¥W space and the ¥tan¥ or ¥cot¥ term, but I don't see how to know how many there are without drawing a picture explicitly. I don't really understand the relationship here and how it translates to bound states or bands.

    I thought that since we knew |V| = a = 10^-11 m we could compute the radius of the circle, but my answer just didn't seem reasonable. And I really don't see where to go to get to bound states from here. I tried setting W = sqrt(p^2 - ¥^2)= ¥tan¥, but I am not getting anywhere. Help to even know where to start would be great as I am totally lost!
  2. jcsd
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