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What does Solve for the time dependence of mean?

  1. Apr 24, 2012 #1
    What does "Solve for the time dependence of" mean?

    1. The problem statement, all variables and given/known data
    Use the Heisenberg equation of motion to solve for the time dependence of [itex]x(t)[/itex] given the Hamiltonian

    [itex]H = \frac{p^{2}(t)}{2m} + mgx(t)[/itex]


    2. Relevant equations

    The Heisenberg equation of motion is
    [itex]\frac{dA(t)}{dt} = \frac{i}{\hbar}\left[H,A(t)\right][/itex]

    3. The attempt at a solution

    As I said, I'm not sure what is meant by "Solve for the time dependence of [itex]x(t)[/itex]". Do they just want [itex]\frac{d x(t)}{dt}[/itex]?

    I already have a value for that. [itex]\frac{d x(t)}{dt} = \frac{\hbar}{im}\frac{d}{dx}[/itex]
     
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  3. Apr 24, 2012 #2
    Re: What does "Solve for the time dependence of" mean?

    I think it means x(t) for all t.
     
  4. Apr 24, 2012 #3

    Fredrik

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    Re: What does "Solve for the time dependence of" mean?

    That's how I would interpret it too.
     
  5. Apr 24, 2012 #4
    Re: What does "Solve for the time dependence of" mean?

    Please explain. I'm sure x(t) = x in real space. What do I need a Hamiltonian and such for?
     
  6. Apr 24, 2012 #5

    HallsofIvy

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    Re: What does "Solve for the time dependence of" mean?

    ?? What do you mean by "x(t)= x"??
     
  7. Apr 24, 2012 #6
    Re: What does "Solve for the time dependence of" mean?

    [itex]\hat{x}(t)\left|ψ\right\rangle = x\left|ψ\right\rangle[/itex]

    As in, the time dependent position operator's eigenfunction is [itex]x[/itex]
     
  8. Apr 25, 2012 #7

    Fredrik

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    Re: What does "Solve for the time dependence of" mean?

    You clearly won't be able to use Heisenberg's equation without it.

    This still doesn't make sense. If anything in this equation should be called "eigenfunction", it's |ψ>, not x. (The terms "eigenvector" and "eigenket" are more common when the ket notation is used). And if this equality would hold, then ##\hat x## would be a constant function, since the right-hand side is independent of t.
     
    Last edited: Apr 25, 2012
  9. Apr 25, 2012 #8
    Re: What does "Solve for the time dependence of" mean?

    In the Heisenberg picture, it is the operators that evolve with time. The equation of motion, which is an operator equation, can be solved for the time evolution of the operator x as a function of t. (Hope this helps, it's been years since I last looked at QM problems)
     
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