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Homework Help: Stuck in an Infinite Square Well

  1. Jan 21, 2007 #1

    cepheid

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

    You don't need it verbatim. I'm just trying to solve for the eigenstates and eigenvalues of the Hamiltonian for a one-dimensional infinite square well, with a particle of mass M inside. I'm embarrassed to say it, but the question is throwing me off because the infinite well is centred at zero, ranging from -b < x < b rather than from 0 < x < a.

    2. Relevant equations

    [tex] \mathcal{H}\psi(x) = E\psi(x) [/tex]

    [tex] \mathcal{H} \equiv -\frac{\hbar^2}{2M}\frac{d^2 }{dx^2}\left(\right) + V() [/tex]

    [tex] V = 0 \ \ \mbox{inside the well.} [/tex]


    3. The attempt at a solution

    Immediately from the ODE the solution is obviously:

    [tex] \psi(x) = A \sin \left(\frac{\sqrt{2ME}}{\hbar}x\right) + B \cos \left(\frac{\sqrt{2ME}}{\hbar}x\right) [/tex]

    [tex] \psi(b) = \psi(-b) = 0 [/tex]

    [tex] \Rightarrow A \sin \left(\frac{\sqrt{2ME}}{\hbar}b\right) + B \cos \left(\frac{\sqrt{2ME}}{\hbar}b\right) =0 [/tex]

    [tex] \Rightarrow -A \sin \left(\frac{\sqrt{2ME}}{\hbar}b\right) + B \cos \left(\frac{\sqrt{2ME}}{\hbar}b\right) =0 [/tex]

    I'm really not sure how to proceed. There is no value of b for which both the sine and cosine will be zero, suggesting that for some eigenstates, A is zero, and for others, B is zero. Griffiths even hints that even states are given by only cosines and odd states by sines, but I can't figure out how to arrive at this result systematically.
     
    Last edited: Jan 21, 2007
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  3. Jan 21, 2007 #2

    Meir Achuz

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    I recommend you use a variable z=x+b. Then the well will be between
    z=0 and z=2b, with the efunction sin(n\pi z/2b).
     
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