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Homework Help: Stuck on a 'normalisation' step

  1. May 29, 2009 #1
    Hey guys,

    1. The problem statement, all variables and given/known data
    I have an assignment, which is to Solve Schrodinger's equations, for a certain potential distribution, which can be divided up into three regions.

    A solution for one of the regions is of the form: Ae[tex]^{kx}[/tex]

    If you substitute this into Schrodinger's equation (time independant, one dimension) and solve for k, you get this:

    [tex]\frac{-h}{2m} \frac{d^{2}}{dx^{2}} \Psi (x) + Vo \Psi (x) = E \Psi (x)[/tex]

    Solve for k:
    k = [tex]\frac{\sqrt{2mE}}{h}[/tex]

    I know this part is right because I've seen it written on the board a couple of times, and it's also what I get on paper.

    But then there's the next bit, which I don't get. Apparently it's 'normalising k' which I just don't get..

    k = [tex]\frac{\sqrt{2mE}}{h}[/tex]

    [tex]k\overline{^}\overline{}[/tex] = [tex]\frac{\sqrt{2mE}}{h}.\frac{L}{2}[/tex]

    [E] = [tex]\frac{\pi^{2}h^{2}}{2ML^{2}}[/tex]

    k_hat = [tex]\frac{L}{2} . \frac{\sqrt{2m}}{h} . \sqrt{E}[/tex]

    k_hat = [tex]\frac{L}{2} . \frac{\sqrt{2m}}{h} . \sqrt{\frac{E[E]}{[E]}}[/tex]

    k_hat = [tex]\frac{L}{2} . \frac{\sqrt{2m}}{h} . \sqrt{[E]} \sqrt{Ehat} [/tex]

    k_hat = [tex]\frac{\pi}{2} \sqrt{Ehat}[/tex]
    2. Relevant equations


    3. The attempt at a solution

    If I rearrange k = [tex]\frac{\sqrt{2mE}}{h}[/tex] and make E the subject,

    I get ..

    E = [tex]\frac{h^{2}khat^{2}}{2m}[/tex]

    and maybe this is where I go wrong.. because I assume E = [E] ?

    Subtituting [E] into the second last step in section 2 above yields:

    [tex]khat = \frac{L}{2} khat \sqrt{Ehat}[/tex]

    and that doesn't equal the last step >_<

    EDIT: ahh.. if I use their definition of [E], [E] = [tex]\frac{\pi^{2}h^{2}}{2ML^{2}}[/tex]
    I arrive at the right answer..

    so how did they come up with [E] = [tex]\frac{\pi^{2}h^{2}}{2ML^{2}}[/tex]??
    Last edited: May 29, 2009
  2. jcsd
  3. May 29, 2009 #2


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    Homework Helper

    What is k_hat and what are you trying to achieve, an expression for E?
  4. May 29, 2009 #3
    Sorry!! I've edited my above post.. Now it makes more sense :)

    What I'm really after is how did they (my teacher) come up with [E] and what is it?

    k_hat is apparently a normalised k
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