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8.5: Dyson Series

  • Page ID
    1230
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    Let us now try to find approximate solutions of Equation \ref{749} for a general system. It is convenient to work in terms of the time evolution operator, \( U(t_0, t)\) , which is defined

    \( \vert A, t_0, t\rangle = U(t_0, t) \,\vert A\rangle.\) \ref{777}

    Here, \( \vert A, t_0, t\rangle\) is the state ket of the system at time \( t\) , given that the state ket at the initial time \( t_0\) is \( \vert A\rangle\) . It is easily seen that the time evolution operator satisfies the differential equation

    \( {\rm i}\, \hbar\, \frac{\partial U(t_0, t)}{\partial t} = (H_0 + H_1)\, U(t_0, t),\) \ref{778}

    subject to the initial condition

    \( U(t_0, t_0 ) = 1.\) \ref{779}

    In the absence of the external perturbation, the time evolution operator reduces to

    \( U(t_0, t) = \exp[-{\rm i} \, H_0\,(t-t_0)/\hbar].\) \ref{780}

    Let us switch on the perturbation and look for a solution of the form

    \( U(t_0, t) = \exp[ -{\rm i} \, H_0\,(t-t_0)/\hbar]\, U_I(t_0, t).\) \ref{781}

    It is readily demonstrated that \( U_I\) satisfies the differential equation

    \( {\rm i}\, \hbar\, \frac{\partial U_I(t_0, t)}{\partial t} = H_I(t_0, t)\, U_I(t_0, t),\) \ref{782}

    where

    \( H_I(t_0,t) = \exp[ +{\rm i} \, H_0\,(t-t_0)/\hbar] \, H_1\, \exp[ -{\rm i} \, H_0\,(t-t_0)/\hbar],\) \ref{783}

    subject to the initial condition

    \( U_I(t_0, t_0) = 1.\) \ref{784}

    Note that \( U_I\) specifies that component of the time evolution operator which is due to the time-dependent perturbation. Thus, we would expect \( U_I\) to contain all of the information regarding transitions between different eigenstates of \( H_0\) caused by the perturbation.

    Suppose that the system starts off at time \( t_0\) in the eigenstate \( \vert i\rangle\) of the unperturbed Hamiltonian. The subsequent evolution of the state ket is given by Equation \ref{744},

    \( \vert i, t_0, t\rangle = \sum_m c_m(t) \exp[ -{\rm i} \, E_m\,(t-t_0)/\hbar]\, \vert m\rangle.\) \ref{785}

    However, we also have

    \( \vert i, t_0, t\rangle = \exp[-{\rm i} \, H_0\,(t-t_0)/\hbar]\, U_I(t_0, t)\, \vert i\rangle.\) \ref{786}

    It follows that

    \( c_n(t) = \langle n\vert\, U_I(t_0, t)\, \vert i\rangle,\) \ref{787}

    where use has been made of \( \langle n\vert m \rangle = \delta_{n\,m}\) . Thus, the probability that the system is found in state \( \vert n\rangle\) at time \( t\) , given that it is definitely in state \( \vert i\rangle\) at time \( t_0\) , is simply

    \( P_{i\rightarrow n} (t_0, t) = \vert\langle n\vert\, U_I(t_0, t)\, \vert i\rangle\vert^{\,2}.\) \ref{788}

    This quantity is usually termed the transition probability between states \( \vert i\rangle\) and \( \vert n\rangle\) .

    Note that the differential equation \ref{782}, plus the initial condition \ref{784}, are equivalent to the following integral equation,

    \( U_I(t_0, t) = 1 - \frac{\rm i}{\hbar} \int_{t_0}^t dt' \,H_I(t_0, t')\, U_I(t_0, t') .\) \ref{789}

    We can obtain an approximate solution to this equation by iteration:

    \( U_I(t_0, t)\) \( \simeq 1 - \frac{\rm i}{\hbar} \int_{t_0}^t H_I(t_0, t') \left[ 1 - \frac{\rm i}{\hbar} \int_{t_0}^{t'} dt'\,H_I(t_0, t'')\, U_I(t_0, t'')\right]\) $ \simeq 1 - \frac{\rm i}{\hbar} \int_{t_0}^t H_I(t_0, t')\,dt' + \...
...\int_{t_0}^t dt' \int_{t_0}^{t'} dt''\, H_I(t_0, t' )\,H_I(t_0, t'' ) + \cdots.$ \ref{790}

    This expansion is known as the Dyson series. Let

    \( c_n = c_n^{\ref{0}} + c_n^{\ref{1}} + c_n^{\ref{2}} + \cdots,\) \ref{791}

    where the superscript \( ^{\ref{1}}\) refers to a first-order term in the expansion, etc. It follows from Equations \ref{787} and \ref{790} that

    \( c_n^{\ref{0}}(t)\) \( = \delta_{i\,n},\) \ref{792} \( c_n^{\ref{1}}(t)\) \( = -\frac{\rm i}{\hbar} \int_{t_0}^t dt'\,\langle n \vert\,H_I(t_0, t')\,\vert i\rangle,\) \ref{793} \( c_n^{\ref{2}}(t)\) \( = \left(\frac{-{\rm i}}{\hbar}\right)^2 \int_{t_0}^t dt' \int_{t_0}^{t'}dt''\, \langle n\vert\, H_I(t_0, t' )\,H_I(t_0, t'' )\,\vert i\rangle.\) \ref{794}

    These expressions simplify to

    \( c_n^{\ref{0}}(t)\) \( = \delta_{in},\) \ref{795} \( c_n^{\ref{1}}(t)\) \( = -\frac{\rm i}{\hbar} \int_{t_0}^t dt'\, \exp[\,{\rm i} \,\omega_{ni}\, (t'-t_0)]\, H_{ni}(t') ,\) \ref{796} \( c_n^{\ref{2}}(t)\) $ = \left(\frac{-{\rm i}}{\hbar}\right)^2 \sum_m \int_{t_0}^t dt'\i...
...t'-t_0)]\, H_{nm}(t') \, \exp[\,{\rm i} \,\omega_{mi}\,(t''-t_0)]\,H_{mi}(t''),$ \ref{797}

    where

    \( \omega_{nm} = \frac{E_n -E_m}{\hbar},\) \ref{798}

    and

    \( H_{nm} (t) = \langle n\vert\, H_1(t)\, \vert m\rangle.\) \ref{799}

    The transition probability between states \( i\) and \( n\) is simply

    \( P_{i\rightarrow n} (t_0, t) = \vert c_n^{\ref{0}} + c_n^{\ref{1}} + c_n^{\ref{2}} +\cdots\vert^{\,2}.\) \ref{800}

    According to the above analysis, there is no chance of a transition between states \( \vert i\rangle\) and \( \vert n\rangle\) (where \( i\neq n\) ) to zeroth order (i.e., in the absence of the perturbation). To first order, the transition probability is proportional to the time integral of the matrix element \( \langle n\vert\,H_1\,\vert i\rangle\) , weighted by some oscillatory phase-factor. Thus, if the matrix element is zero then there is no chance of a first-order transition between states \( \vert i\rangle\) and \( \vert n\rangle\) . However, to second order, a transition between states \( \vert i\rangle\) and \( \vert n\rangle\) is possible even when the matrix element \( \langle n\vert\,H_1\,\vert i\rangle\) is zero.

    Contributors

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

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    This page titled 8.5: Dyson Series is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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