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Physical chemistry: Energy operator and eigenfunction

  1. Jan 20, 2017 #1
    1. The problem statement, all variables and given/known data
    The energy operator for a time-dependent system is iħ d/dt. A possible eigenfunction for the system is
    Ψ(x,y,z,t)=ψ(x,y,z)e-2πiEt/h
    Show that the probability density is independent of time


    2. Relevant equations
    ĤΨn(x) = EnΨn

    3. The attempt at a solution

    I understand the concept of eigenfuntions but I don't really know which side of the equation I would apply the operator to or how it would prove anything about probability density
     
  2. jcsd
  3. Jan 20, 2017 #2

    BvU

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    Hi,

    Your relevant equation isn't :smile:. At least not here.

    You need an expression for the probability density in terms of the probability amplitude ##\psi## and when you work that out you'll see the time dependence cancels; that's all.
     
  4. Jan 20, 2017 #3

    blue_leaf77

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    I don't think a time-dependent system (e.g. time dependent potential) always has such a separable eigenfunction, let alone the existence of the eigenfunctions. Are you sure it was not "time-independent"?
     
  5. Jan 20, 2017 #4

    BvU

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    Agree with BL. Could you render the problem statement exactly as is ?
     
  6. Jan 20, 2017 #5
    That was taken directly from the assignment.
     
  7. Jan 21, 2017 #6

    BvU

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    Well, then I would claim that the given function is a solution and write down the probability density.
     
  8. Jan 21, 2017 #7
    I found something in the textbook that might be relevant

    Â(x) Ψn(x,t) = an Ψn(x,t)
    Â(x) Ψn(x) e-it(E/ħ)= an Ψn(x) e-it(E/ħ)
    Â(x) Ψn(x) = an Ψn(x)
    The time cancels out
     
  9. Jan 21, 2017 #8

    Borek

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  10. Jan 21, 2017 #9

    BvU

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    Yes, fine. First thing you need is an expression for the probability density in terms of the probability amplitude ##\psi##. This is the third time I ask. Please answer by posting that expression.
     
  11. Jan 21, 2017 #10
    Probability density= / ψ/2dx
     
  12. Jan 21, 2017 #11

    BvU

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    Now fill in ##\psi## as given

    PS wasn't the probability density just ##|\psi^2|## ?
     
  13. Jan 21, 2017 #12
    That's from my notes but you're right. The dx makes no sense. It's not written in a comprehensible way in my textbook.
     
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