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8.2: General Analysis

  • Page ID
    1227
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    Suppose that at \( \vert A\rangle = \sum_n c_n\, \vert n\rangle,\) \ref{741}

    where the \( \vert A, t_0, t\rangle = \sum_n c_n \exp[-{\rm i}\,E_n \,(t-t_0)/\hbar]\,\vert n\rangle.\) \ref{742}

    Now, the probability of finding the system in state \( t\) is

    \( H_1= 0\) , the probability of finding the system in state \( t\) is exactly the same as the probability of finding the system in this state at the initial time \( H_1\neq 0\) , we expect \( \vert A, t_0, t\rangle = \sum_n c_n(t) \exp[-{\rm i}\,E_n\,(t-t_0)/\hbar]\,\vert n\rangle,\) \ref{744}

    where \( c_n(t)\) , which depends entirely on the perturbation (i.e., \( H_1= 0\) ). Note that the eigenkets \( H_0\) evaluated at the time \( {\rm i}\,\hbar\, \frac{\partial}{\partial t}\,\vert A, t_0, t\rangle = H\,\vert A,t_0,t\rangle= (H_0+H_1) \,\vert A,t_0,t\rangle.\) \ref{745}

    It follows from Equation \ref{744} that

    \( (H_0+H_1)\, \vert A,t_0,t\rangle = \sum_m c_m(t) \exp[-{\rm i}\,E_m\, (t-t_0)/\hbar]\, (E_m + H_1)\,\vert m\rangle.\) \ref{746}

    We also have

    $ {\rm i}\,\hbar\, \frac{\partial}{\partial t}\,\vert A,t_0,t\rangl...
...{dt}+ c_m(t)\, E_m\right) \exp[-{\rm i}\,E_m \,(t-t_0)/\hbar]\, \vert m\rangle,$ \ref{747}

    where use has been made of the time-independence of the kets \( \vert m\rangle\) . According to Equation \ref{745}, we can equate the right-hand sides of the previous two equations to obtain

    $ \sum_m {\rm i}\,\hbar\, \frac{d c_m}{dt}\exp[-{\rm i}\,E_m \,(t-t...
...gle = \sum_m c_m(t) \exp[-{\rm i}\,E_m \,(t-t_0)/\hbar]\, H_1\, \vert m\rangle.$ \ref{748}

    Left-multiplication by \( {\rm i}\,\hbar\, \frac{d c_n}{dt} = \sum_m H_{nm}(t)\, \exp[\,{\rm i}\,\omega_{nm}\, (t-t_0)]\, c_m(t),\) \ref{749}

    where

    \( \omega_{nm} = \frac{E_n - E_m}{\hbar}.\) \ref{751}

    Here, we have made use of the standard orthonormality result, $ \langle n\vert m\rangle
=\delta_{nm}$ . Suppose that there are \( c_n\) , which specify the probability of finding the system in state \( t\) , is determined by \( N=2\) ), it is actually possible to solve Equation \ref{749} without approximation. This solution is of great practical importance.

    Contributors

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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    This page titled 8.2: General Analysis is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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