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Normalization of Linear Superposition of ψ States

  1. Sep 19, 2013 #1
    1. The problem statement, all variables and given/known data
    An electron in an infinitely deep potential well of thickness 4 angstroms is placed in a linear superposition of the first and third states. What is the frequency of oscillation of the electron probability density?


    2. Relevant equations
    E=hω


    3. The attempt at a solution
    My main problem right now is with the normalization of the wave function:

    I have ψTotal=Aψ1+ Bψ3
    ∫|ψT|^2=1=∫(|A|^2*sin^2(pi*z/Lz) +|B|^2*sin^2(3pi*z/Lz))dz (the other terms with A*B and B*A become zero after integration)
    =(Lz(|A|^2+|B|^2))/2=1

    So
    |A|^2+|B|^2=2/Lz

    What I'd like to do next is calculate the probability density of the electrons, but I end up with |A|^2, |B|^2, A*B, & B*A terms and don't know how to get actual numbers for all these constants, or otherwise to properly normalize this. I know what to do if the superposition of states was equally split, but it does not say that here.

    Any thoughts on what I'm doing wrong? Thanks
     
    Last edited: Sep 19, 2013
  2. jcsd
  3. Sep 19, 2013 #2

    TSny

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    Gold Member

    You don't need to worry about the values of A and B to answer the question. The frequency of oscillation of the probability distribution does not depend on A or B. So, you can leave A and B unspecified. Since you are interested in the time dependence of the probability distribution, you will need to include the time dependence of each of the two stationary states when you calculate ##|\Psi_{total}(t)|^2##.
     
  4. Sep 19, 2013 #3
    So would the only thing that really matters be the exp(+/-(E3-E1)it/h), where ω=(E3-E1)/h? It just seems strange to me that whether one eigenstate dominates or not does not affect the frequency of electron probability density.

    Also, I have difficulty in general with normalization. If I needed to normalize this sort of function, would that be possible with the information given, or would I need something else?

    Thank you & I appreciate your help.
     
  5. Sep 19, 2013 #4

    TSny

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    Yes, ω=(E3-E1)/[itex]\hbar[/itex].

    When you write out ##|\Psi_{total}(t)|^2##, you should be able to see the effect of changing the relative sizes of A and B. In particular, you should think about the case where A = B and the case where A >> B or A << B.

    From the information given in the problem, you cannot determine the magnitudes of A and B. As you showed, all you can determine is the value of |A|2 + |B|2.
     
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