1. Not finding help here? Sign up for a free 30min 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!

Quantum Dot Problem (Griffiths 7.20)

  1. Feb 7, 2005 #1
    Hello,

    Damn Griffiths 3-star problems... The setup for the potential is a 2D cross basically, so there's a column of zero potential 2a wide (straddling the y-axis) going to +/- infinity, and there's the same thing on the x-axis, so they cross over the origin. Everywhere else the potential is infinity. It's Griffith's 7.20 in the 2nd Edition. The goal is to find the lowest energy which propagates to infinity (doesn't decay to zero along the "arms" of the cross). The provided hint is to "Go way out one arm (say, x>>a) and solve the Schroedinger equation through Sep of Vars; if the wave function goes out to infinity the dependance on x must take the form e^ikx, k>0". Here's what I did:
    [tex]\frac{-\hbar^2}{2m}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\psi=E\psi[/tex]

    sub in [tex]\psi=X(x)Y(y)[/tex]

    [tex]\frac{X''}{X}=-\frac{Y''}{Y}-\frac{2mE}{\hbar^2}=-K^2[/tex]

    [tex]X=Ae^{iKx}+Be^{-iKx}, Y=Ce^{iTy}+De^{-iTy}[/tex]
    where [tex]T^2=-K^2+\frac{2mE}{\hbar^2}[/tex]

    Now I'm having trouble with the boundary conditions, as well as what Griffiths means by "Go way out one arm". Assuming we're already down this arm, Y(a)=Y(-a)=0 due to the infinite potential. That makes C=D, so
    [tex]Y=\frac{C}{2}\cos(Ta)=0[/tex]
    Therefore [tex]T=\frac{(n+1/2)\pi}{a}[/tex]

    That's pretty much as far as I can get. I don't know what the boundary conditions for X would be, and that's the only way I can think of proceeding. I'm really stumped on this one, so any pointers would be nice!

    - AL
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Quantum Dot Problem (Griffiths 7.20)
  1. Thermodynamic problem (Replies: 0)

Loading...