8.5: Dyson Series

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:

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

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

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

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)

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