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Quantum Mechanics - Finite Square Well - Graphical Solution

  1. Apr 10, 2013 #1
    1. The problem statement, all variables and given/known data

    http://img842.imageshack.us/img842/4917/physp6.jpg [Broken]

    I am trying to solve the above problem. However, I am supposed to solve it with the following values:

    U=54.7eV
    L=0.2nm

    Particle is an electron, so:

    m=9.109E-13kg=0.511eV/c^2

    Essentially I am supposed to find the conditions for which no solution is possible using a graphical method. This is where I am stuck. The book that I have does not discuss the graphical method, so I am not sure where to begin with it or where the values come in to play. I have solved it using a traditional method seen below:

    2. Relevant equations

    Schrodinger Equation

    3. The attempt at a solution

    http://img834.imageshack.us/img834/2686/physw6.png [Broken]

    As can be seen above:

    k=√[(2mE)/h^2]
    α=√[(2m(U-E))/h^2]

    kLcsc(kL)=√[(2mUL^2)/h^2]


    since kLcsc(kL)=kL/sin(kL) can never be smaller than one, (2mUL^2)/h^2 < 1

    Also, since sin(kL) is in the denominator, no solutions would be possible when kL is equal to zero or a multiple of π.


    To do it graphically, would I simply draw a graph of kLcsc(kL) and show where it is undefined? I am confused. Also where would the values I was given come into play?

    Thanks!
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Apr 10, 2013 #2

    vela

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    I didn't check your work completely, but it looked okay. You can rewrite that last expression as
    $$\sin kL = c(kL)$$ where ##c=\sqrt{\frac{\hbar^2}{2mL^2U}}##. Plot the two sides as functions of kL. Solutions are where the two curves intersect.
     
  4. Apr 11, 2013 #3
    Thanks! I plugged the values in for "c" and ended up with the functions sin(kL) and 0.132(kL). I plotted both functions and found that they intersected at kL=0 and kL~0.8rad.
     
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