Dyson Series

Let us now try to find approximate solutions to Equation (8.11) for a general system. It is convenient to work in terms of the time evolution operator, $T(t_0, t)$ , which is defined

$$\vert A, t_0, t\rangle = T(t_0, t) \,\vert A\rangle.$$ (8.37)

(See Section 3.2.) As before, $\vert A, t_0, t\rangle$ 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

$${\rm i}\, \hbar\, \frac{\partial T(t_0, t)}{\partial t} = (H_0 + H_1)\, T(t_0, t),$$ (8.38)

subject to the initial condition

$$T(t_0, t_0 ) = 1.$$ (8.39)

(See Section 3.2.)

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

$$T(t_0, t) = \exp\left[\frac{-{\rm i} \, H_0\,(t-t_0)}{\hbar}\right].$$ (8.40)

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

$$T(t_0, t) = \exp\left[\frac{ -{\rm i} \, H_0\,(t-t_0)}{\hbar}\right] T_I(t_0, t).$$ (8.41)

It is readily demonstrated that $T_I(t_0,t)$ satisfies the differential equation

$${\rm i}\, \hbar\, \frac{\partial T_I(t_0, t)}{\partial t} = H_I(t_0, t)\, T_I(t_0, t),$$ (8.42)

where

$$H_I(t_0,t) = \exp\left[\frac{\,{\rm i} \, H_0\,(t-t_0)}{\hbar}\right] H_1 \exp\left[\frac{ -{\rm i} \, H_0\,(t-t_0)}{\hbar}\right],$$ (8.43)

subject to the initial condition

$$T_I(t_0, t_0) = 1.$$ (8.44)

(See Exercise 3.) Note that $T_I$ specifies that component of the time evolution operator that is generated by the time-dependent perturbation. Thus, we would expect $T_I$ 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 (8.6),

$$\vert i, t_0, t\rangle = \sum_f c_f(t) \exp\left[\frac{ -{\rm i} \, E_f\,(t-t_0)}{\hbar}\right] \vert f\rangle.$$ (8.45)

However, we also have

$$\vert i, t_0, t\rangle = \exp\left[\frac{-{\rm i} \, H_0\,(t-t_0)}{\hbar}\right] T_I(t_0, t)\, \vert i\rangle.$$ (8.46)

It follows that

$$c_f(t) = \langle f\vert\, T_I(t_0, t)\, \vert i\rangle,$$ (8.47)

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

$$P_{i\rightarrow f} (t_0, t) = \vert\langle f\vert\, T_I(t_0, t)\, \vert i\rangle\vert^{\,2}.$$ (8.48)

This quantity is usually termed the transition probability between states $\vert i\rangle$ and $\vert f\rangle$ .

Note that the differential equation (8.45), plus the initial condition (8.47), are equivalent to the following integral equation:

$$T_I(t_0, t) = 1 +\left( \frac{-{\rm i}}{\hbar}\right) \int_{t_0}^t dt' \,H_I(t_0, t')\, T_I(t_0, t') .$$ (8.49)

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

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

$$\simeq 1 +\left(\frac{-{\rm i}}{\hbar}\right) \int_{t_0}^t dt'\,H_I(t_0, t')$$

$$\phantom{=}+ \left(\frac{-{\rm i}}{\hbar}\right)^2 \int_{t_0}^t dt' \int_{t_0}^{t'} dt''\, H_I(t_0, t' )\,H_I(t_0, t'' ) + \cdots.$$ (8.50)

This expansion is known as the Dyson series [34].

Let

$$c_f = c_f^{\,(0)} + c_f^{\,(1)} + c_f^{\,(2)} + \cdots,$$ (8.51)

where the superscript $^{(1)}$ refers to a first-order term in the expansion, et cetera. It follows from Equations (8.50) and (8.53) that

$$c_f^{\,(0)}(t) = \delta_{if},$$ (8.52)

$$c_f^{\,(1)}(t) = \left(-\frac{\rm i}{\hbar}\right) \int_{t_0}^t dt'\,\langle f \vert\,H_I(t_0, t')\,\vert i\rangle,$$ (8.53)

$$c_f^{\,(2)}(t) = \left(\frac{-{\rm i}}{\hbar}\right)^2 \int_{t_0}^t dt' \int_{t_0}^{t'}dt''\, \langle f\vert\, H_I(t_0, t' )\,H_I(t_0, t'' )\,\vert i\rangle.$$ (8.54)

These expressions simplify to give

$$c_f^{\,(0)}(t) = \delta_{if},$$ (8.55)

$$c_f^{\,(1)}(t) = \left(\frac{-{\rm i}}{\hbar}\right) \int_{t_0}^t dt'\, \exp[\,{\rm i} \,\omega_{fi}\, (t'-t_0)]\, H_{fi}(t') ,$$ (8.56)

$$c_f^{\,(2)}(t) = \left(\frac{-{\rm i}}{\hbar}\right)^2 \sum_m \int_{t_0}^t dt'\int_{t_0}^{t' }dt''\,\exp[\,{\rm i} \,\omega_{fm}\,(t'-t_0)]\, H_{fm}(t')$$

$$\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaa} \exp[\,{\rm i} \,\omega_{mi}\,(t''-t_0)]\,H_{mi}(t''),$$ (8.57)

where

$$\omega_{nm} = \frac{E_n -E_m}{\hbar},$$ (8.58)

and

$$H_{nm} (t) = \langle n\vert\, H_1(t)\, \vert m\rangle.$$ (8.59)

Here, use has been made of the completeness relation $\sum_m \vert m\rangle\langle m\vert=1$ , where the sum is over all energy eigenstates. The transition probability between states $i$ and $f$ is simply

$$P_{i\rightarrow f} (t_0, t) = \left\vert c_f^{\,(0)} + c_f^{\,(1)} + c_f^{\,(2)} +\cdots\right\vert^{\,2}.$$ (8.60)

According to the previous analysis, there is no possibility of a transition between the initial state, $\vert i\rangle$ , and the final state, $\vert f\rangle$ , (where $i\neq f$ ) 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 $\langle f\vert\,H_1\,\vert i\rangle$ , weighted by an 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 $\vert f\rangle$ . However, to second order, a transition between states $\vert i\rangle$ and $\vert f\rangle$ is possible even when the matrix element $\langle f\vert\,H_1\,\vert i\rangle$ is zero.