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Determine if a wavefunction is sharp or fuzzy in energy

  1. May 18, 2015 #1
    1. The problem statement, all variables and given/known data

    For a particle of mass m in a one-dimensional infinite square well 0 < x < a, the normalised energy eigenfunctions ψn and eigenvalues En (integer n = 1, 2, 3, ...) are

    $$ \psi_n(x) =\sqrt{\frac{2}{a}} sin \left( \frac{n \pi x}{a} \right) \;$$ inside the well otherwise Ψn(x)=0

    Consider a state A with wave function ΨA(x,t) which at time t = 0 is given by
    $$\Psi_A(x,0) = \frac {4}{\sqrt{5a}} sin^{3}\left( \frac{\pi x}{a} \right) for\; 0<x<a\; otherwise\; 0. $$

    1. Is the state A sharp or fuzzy in energy?
    2. Check that ΨA(x, 0) is normalised to one particle.
    3. If you think A is sharp in energy, what energy does it have? Otherwise, what are the possible results of repeated energy measurements for the state A, and with what probability do they occur?
    4. What is the formula for ΨA(x,t) for all times t?
    2. Relevant equations


    3. The attempt at a solution
    The only part of this question I can confidently do is 2. I have checked this and it is normalised to one particle.

    Going back to 1 I am pretty sure that the state A is fuzzy in energy. To check this I think I need ΨA(x, 0) to be an eigenfunction of the K energy operator. Without doing the second derivative I can be pretty sure that function is not coming back anytime soon and is therefore not an eignefunction...

    If I am on the right path I think I then need to find <E>A for part 3.

    I would really appreciate it is anyone can help point me in the right direction on this.

    Thanks
     
  2. jcsd
  3. May 18, 2015 #2

    Orodruin

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    Correct. If it has a definite energy, it will be an eigenstate of the Hamiltonian.

    No, you do not need to compute the expected energy and in fact it will tell you very little about the mix of eigenvalues. Instead, you should be writing the wave function as a linear combination of different eigenfunctions.
     
  4. May 18, 2015 #3
    Thanks. I'm not sure how to write my wave function as a linear combination of eigenfunctions, not sure if we have seen that. Is there another way perhaps?

    Ps nice quote in your sig.
     
  5. May 18, 2015 #4

    Orodruin

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    You have a function ##\sin^3(\pi x/a)##. You need to write it as a linear combination of the functions ##\sin(\pi n x/a)##. How would you go about doing this?

    Edit: Of course, you will have to care about the constants too, but let us start here.
     
  6. May 18, 2015 #5
    Are you saying I just separate the ^3 terms into three sin(nπx/a)
     
  7. May 18, 2015 #6

    Orodruin

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    Yes ... or whatever number of sin(nπx/a) would be the correct one ...
     
  8. May 18, 2015 #7
    And that's where the constants come in?
     
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