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Schrodinger's equation

  1. Apr 11, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex]
    \frac{1}{r}\frac{d}{dr}(r^2\frac{d}{dr}\Psi (r)) + { \frac{2m}{\hbar^2}[E-V(r)] - \frac{l(l+1)}{r^2}}\Psi (r) = 0
    [/tex]
    [tex]
    V(r) = -Vo r\leq a
    [/tex]
    [tex]
    0 r > a
    [/tex]


    Use
    [tex]
    \Psi (r) = \Xi (r)/r
    [/tex]

    Questions asks to Find E l = 0, do I solve the general equation first or should I make l = 0 right away?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 12, 2008 #2

    pam

    User Avatar

    It's much easier to set l=0 first.
     
  4. Apr 12, 2008 #3
    Inside the dot I get

    [tex]

    \Xi (r) = C*Cos[l*r]+D*Sin[l*r]

    [/tex]

    Outside the Dot I get

    [tex]

    \Xi (r) = A*Exp[-k*r]+B[Exp[k*r]

    [/tex]
    Taking r to infinity yields:

    [tex]

    \Xi (r) = A*Exp[-k*r]

    [/tex]


    What other boundary conditions can I apply?
     
  5. Apr 12, 2008 #4
    I took
    [tex]
    \Xi (a)
    [/tex]
    and
    [tex]
    \frac{d\Xi}{dr} [/tex] at r = a

    How would I solve these equations?
    [tex]
    A*Exp[-k*a] = C*Cos[l*a]+D*Sin[l*a]
    [/tex]
    [tex]
    -kAExp[-k*a]=-C*l*Sin[l*a]+D*l*Cos[l*a]
    [/tex]
     
    Last edited: Apr 12, 2008
  6. Apr 13, 2008 #5
    [tex]
    \frac{h^2\pi^2}{8ma^2} < E + V < \frac{h^2\pi^2}{2ma^2}[/tex]

    Now that's the energy for l=0. how would one want to estimtae the min radius to ensure at least one bound e-state?
     
  7. Apr 14, 2008 #6
    NO bound states occur if a^2 < (pi^2 h^2 )/(8*m*Vo)

    take positive root.

    that's the minimum radius . Is that correct? if not I can't continue.

    Now given this:

    [tex]

    \frac{h^2\pi^2}{8ma^2} < E + V < \frac{h^2\pi^2}{2ma^2}
    [/tex]

    letting V = 0.5 eV and m = 0.04*9.11x10^-31,
    how can one find the colour of light in this circle?
     
    Last edited: Apr 14, 2008
  8. Apr 14, 2008 #7

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    what do you mean by: "Inside the dot I get " in post #3

    This Boundary condition is good too:
    [tex}] \Xi (r=0) = 0 [/tex]

    Is the question to find [tex] E{l=0} [/tex]?

    What is your "k" in post #3 ?

    Is it an atomic physics problem? You are speaking about bound e-states
     
  9. Apr 14, 2008 #8
    Take post #6 as a given.

    how would one find the minimum radius
     
  10. Apr 15, 2008 #9

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    This was posted to me via private message T07:54

    "question asks, estimate the min radii required of an InAs spherical quantum dot embedded in a GaAs matrix in order to ensure that there will be at least one bound electron state, or at least two bound states in the dot.

    Choose V = 0.5 eV and effective mass m* = 0.04m in the dot and barrier layers. Does the seperation in the energies between the two bound states tell you about the colour of light the dot would emit when excited?

    If not, why not? What else would you need to do in order to obtain this information?

    So since I found the bound states.
    What do I need to solve for to answer the remaining questions?

    I'm not sure about my minimum radius as I posted in the thread, is it correct?"
     
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