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Expectation value of total energy in QM

  1. Feb 12, 2015 #1
    1. The problem statement, all variables and given/known data
    The problem asks me to find the expectation value of W.

    2. Relevant equations
    The given ψ[x,t] is Asin(πx/a) e^((-i Eot)/ħ).
    By QM postulate 2 the QM operator of W is: iħ δ/δt or equivalently -ħ/i δ/δt.
    3. The attempt at a solution

    <w>=∫ψ*iħδ/δtψ= iħδ/δt 1/(2e^(-iEot/)ħ) sin(πx/a) e^(iEot/ħ) dx

    I am trying to wrap my head around what is going on so far. Solving for A in the wave equation would require you to normalize and integrate. You then put the normalized wave function times the QM operator and solve for the expectation value. When solving for position this is no problemo. I hit a snag when solving for W and W^2. I thought that for energy you would be integrating with respect to time, which i was informed was incorrect.
    We happened to glaze over this problem in class, but the time term from inside the wave function was brought out to cancel the partial wrt time.

    I know that for my above attempt I messed up my U substitution pretty badly. I would just like some clarification on how the treatment of solving for W differs from position.

    Thanks ahead of time helping this QM neophyte.
     
  2. jcsd
  3. Feb 12, 2015 #2

    BvU

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    Hello Ben, welcome to PF :)

    The idea is that from part 1 of the template a complete problem is formulated. Plus a list of variables, given/known data. In part 2 you provide the relevant equations and in part 3 your attempt at solution. Can't be all that hard.

    So you are given the ##\psi(x, t) = A\; \sin(\pi x/a)\; e^{-i{E_0t\over \hbar}}## and they ask for ##<E>## ?

    Your relevant formula is ##<Y> = \int \; \psi^*\; Y\psi ## for an operator Y

    and the operator W for the Energy is ##i\hbar\; {\partial \over \partial t}##

    and your attempt starts off filling that in: ## <W> = \int \ \psi^*\; i\hbar {\partial \over \partial t} \psi ## which is good.

    Then I detect a slight derailment:
    • the ##A^*\;A## are missing
    • one ##\sin## is missing
    • the differentiation kind of went wrong. ##{de^{-kx}\over dx} = ## . . .
    • the ##{\partial \over \partial t}## has been moved to the left of ##\psi^*## which is really bad: ##\psi^*\; {\partial \over \partial t} \psi \ne {\partial \over \partial t} \psi^* \psi ## !
    and yes, the integration gives an expectation value of the energy as a function of time. The integration is over x.
     
  4. Feb 12, 2015 #3

    BvU

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    PS the ##A^*\;A## are really useful: look at that expression and at the expression for <E> and realize that you don't have to do any work (integrating) at alll here !

    (bit of a spoiler, but a gentle intro into QM is a good thing)
     
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