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Calculus with Dirac notation

  1. Jan 2, 2009 #1

    tpg

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    I am working through a problem relating to the conservation of probability in a continuity equation. However, I end up with a contradiction when trying to put the following into the Time-Dependent Schrodinger Equation

    [tex]\frac{\partial\psi(x)}{\partial t}=\frac{\partial\left\langle x|\psi\right\rangle }{\partial t}[/tex]

    where

    [tex]\left|\psi\right\rangle =\intop_{-\infty}^{\infty}dx\,\psi(x)\left|x\right\rangle[/tex]

    If I use the first form, together with

    [tex]i\hbar\frac{\partial\psi(x)}{\partial t} = H\psi(x)[/tex]
    and [tex]-i\hbar\frac{\partial\psi^{*}(x)}{\partial t} = H\psi^{*}(x)[/tex]

    I get a sensible nonzero answer, which I believe to be correct.

    However, if I start with the second form, I rearrange as follows (I'm pretty sure this step is correct)

    [tex]\frac{\partial\left\langle x|\psi\right\rangle }{\partial t}=\frac{\partial\left\langle x\right|}{\partial t}\left|\psi\right\rangle +\left\langle x\right|\frac{\partial\left|\psi\right\rangle }{\partial t}[/tex]

    Then if I use the following form of the TDSE:

    [tex]i\hbar\frac{\partial\left|\psi\right\rangle }{\partial t} = H\left|\psi\right\rangle[/tex]

    Which I believe results in

    [tex]\frac{\partial\left\langle x|\psi\right\rangle }{\partial t}=\frac{i}{\hbar}\left(\left\langle x\right|H\left|\psi\right\rangle -\left\langle x\right|H\left|\psi\right\rangle \right)=0[/tex]

    This doesn't make any sense to me. Can anyone please explain where I've gone wrong? Many thanks in advance.
     
  2. jcsd
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