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Finding the eigen function for an infinite square well (quantum mechanics)

  1. Nov 6, 2012 #1

    ElijahRockers

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

    Quantum mechanics is absolutely confusing me.

    A proton is confined in an infinite square well of length 10-5nm.

    Calculate the wavelength and energy associated with the photon that is emitted when the proton undergoes a transition from the first excited state (n=2) to the ground state (n=1).

    In what region of the electromagnetic spectrum does this wavelength belong?

    3. The attempt at a solution

    I'm not really sure what I'm doing at all.

    But I started with the time independent Schrodinger equation. For region I and III, (where the potential is infinite) then the eigenfunction must be 0.

    So I put in a potential of zero for the second region and got

    [itex]\frac{d^2\psi}{dx^2} + k^2\psi = 0[/itex] where [itex]k^2 = \frac{2mE}{\hbar}[/itex]

    Looks like a second order diff eq, so I tried to solve it. Solutions to the characteristic equation were ±ik...

    From here I am a little stumped, I didn't take notes, and I can't remember what the solution was that he used in class.

    But anyway if I'm just going by my normal diff eq understanding,

    [itex]\psi = Acos(kx)+Bsin(kx)[/itex]

    What's throwing me off is that I recall he had exponential solutions with imaginary components in them. These would oscillate, and so does my solution, but even if I had my solution in his form, what next?

    Where does the length of the well come into play? What does n=2 -> n=1 mean?
    Am I making this more complicated than it should be?
     
    Last edited: Nov 6, 2012
  2. jcsd
  3. Nov 7, 2012 #2

    ehild

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