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Hard Quantum Mechanics Question

  1. Jun 7, 2014 #1
    1. All given variables and known data
    Consider a particle of mass m subject to the infinite square well potential function (with L>0)

    jCTbQHm.gif

    Suppose that you do not know the state function describing the system, but that you are told the expectation value of the position of the particle is given by

    [tex]\left\langle x \right\rangle \left( t \right) = \frac{L}{2} + \alpha L\sin \left( {\Omega t} \right)[/tex]

    where α is some unknown constant less than 1/2, and Ω is some frequency greater than zero.

    2. Relevant equations
    Equations used in Quantum Mechanics​

    Questions:
    a. There are many possible values of Ω - what are the allowed values of Ω? (That is, provide an equation for ℏΩ (h-bar*Ω) ). Explain your answer.

    b. Write down the most general wave function consistent with this expectation value. Explain your answer.​
     
  2. jcsd
  3. Jun 8, 2014 #2

    strangerep

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    So... where is your "attempt at a solution"??

    OK, I'll give you a hint: write down the Schrodinger equation applicable to this situation.
    Then solve it.
     
  4. Jun 8, 2014 #3

    vanhees71

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    2016 Award

    Where is this question from? I find it quite strange, to say it friendly.

    If you look for the state of a system with restricted information like in this example, the best you can do is to look for a statistical operator with maximum (von Neumann) entropy consistent with the given information. This is at least the way to look at this problem from the point of view of Shannon-Jaynes information theory.
     
  5. Jun 8, 2014 #4
    This a great question! I've just spent a lovely hour in the sunshine having a go at it. As it is not a stationary state you need at least two terms in the general solution. As you need to arrive at at an expectation value of x that has a sinΩt dependency it looks like you need just two terms. I got that hΩ will represent the difference between the two energy levels of the two terms.
     
  6. Jun 8, 2014 #5
    :confused:

    Surely it's just a question about pure states.

    I agree! Perhaps we should add that when you have a superposition of two energy eigenstates, the system will "ring" at the characteristic frequency [itex]\Omega[/itex], as a result of "quantum mechanical cross terms". When you couple the system to an electromagnetic field, it might throw out a photon of that frequency.
     
  7. Jun 8, 2014 #6

    strangerep

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    Meanwhile, the OP has still made zero attempt at a solution...
     
  8. Jun 9, 2014 #7
    What's frustrating is that others who put in decent effort get no replies at all...
     
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