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Dirac Notation question

  1. Feb 19, 2016 #1
    1. The problem statement, all variables and given/known data

    A particle in a harmonic oscillator potential in the following state after a time t:

    ## | ψ(t) > = \frac{1}{\sqrt{2}} [e^{(-iE_0 t/\hbar)} |ψ_0> + e^{(-iE_1 t/\hbar)} |ψ_1> ] ##

    I want to write an expression for ## <ψ(t)| \hat{x} | ψ(t) > ##.

    2. Relevant equations

    The answer is meant to be:

    ## <ψ(t)| \hat{x} | ψ(t) > = \frac{1}{2} [ <ψ_0| \hat{x} | ψ_1> e^{-i(E_1 - E_0)t/\hbar)} + <ψ_1| \hat{x} | ψ_0> e^{-i(E_0 - E_1)t/\hbar)}] ##


    3. The attempt at a solution

    ## <ψ(t)| \hat{x} | ψ(t) > = \int{ψ^{*}(t) \hat{x} ψ(t)}##
    ## = \int{\frac{1}{\sqrt{2}} [ e^{(iE_0 t/\hbar)} ψ^{*}_0 + e^{(iE_1 t/\hbar)} ψ^{*}_1}] \hat{x} \frac{1}{\sqrt{2}}[ e^{-(iE_0 t/\hbar)} ψ_0 + e^{-(iE_1 t/\hbar)} ψ_1] ##

    ## = \int{\frac{1}{2} [ e^{(iE_0 t/\hbar)} ψ^{*}_0 \hat{x} e^{-(iE_0 t/\hbar)} ψ_0 + e^{(iE_0 t/\hbar)} ψ^{*}_0 \hat{x} e^{-(iE_1 t/\hbar)} ψ_1 + e^{(iE_1 t/\hbar)} ψ^{*}_1} \hat{x} e^{-(iE_0 t/\hbar)} ψ_0 + e^{(iE_1 t/\hbar)} ψ^{*}_1} \hat{x} e^{-(iE_1 t/\hbar)} ψ_1 ##

    ## = \frac{1}{2} [<ψ_0| \hat{x} | ψ_0 > + e^{-i(E_1 - E_0)t/\hbar} <ψ_0| \hat{x} | ψ_1 > + e^{i(E_1 - E_0)t/\hbar} <ψ_1| \hat{x} | ψ_0 > + <ψ_1| \hat{x} | ψ_1> ]##

    This is a different answer than it should be, where am I going wrong?
     
  2. jcsd
  3. Feb 20, 2016 #2

    Orodruin

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    No, it is correct. You are just missing one step of simplification.
     
  4. Feb 20, 2016 #3
    Do you say that ##<ψ_0| \hat{x} | ψ_0 >## and ##<ψ_1| \hat{x} | ψ_1 >## are expectation values of position, which for the simple harmonic oscillator, are zero?
     
  5. Feb 21, 2016 #4

    Orodruin

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    I did not say it, I wanted you to do it.
     
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