7.6 Time Dependent Perturbation Theory

7.6.1 The Coupled Equations

We'd like to consider transitions between the stationary states of a Hamiltonian $\hat{H}_o$ ,

$\begin{displaymath} \hat{H}_o \Psi_n = E_n \Psi_n \end{displaymath}$

(7.115)

with

$\begin{displaymath} \Psi_n(x,t) = \psi_n(x) e^{-i \omega_n t} \end{displaymath}$

and

$\begin{displaymath} E_n = \hbar \omega_n \end{displaymath}$

We will assert, without proof, that the stationary states of the Hamiltonian are orthogonal to each other,^7.9

$\begin{displaymath} \int_{-\infty}^\infty \psi^*_n \psi_m dx = \delta_{nm} \end{displaymath}$

(7.116)

We add to the ``unperturbed system'' a time dependent perturbation $\hat{U}(x,t)$ to the potential energy corresponding to an external disturbance of the system (an electromagnetic field, for example).

We write a general wavefunction as a linear superposition of the stationary states,

$\begin{displaymath} \Psi(x,t) = \sum_{n=0}^\infty c_n(t) \psi_n(x) e^{-i \omega_n t} \end{displaymath}$

(7.117)

where we account for a time dependent response to the perturbation with time dependent coefficients.

The modified Hamiltonian

$\begin{displaymath} \hat{H} = \hat{H_o} + \hat{U}(x,t) \end{displaymath}$

(7.118)

acting on the wavefunction yields^7.10

$\begin{displaymath} \hat{H} \Psi = \sum_{n=0}^\infty c_n [E_n + \hat{U}] \psi_n e^{-i \omega_n t} \end{displaymath}$

(7.119)

Now, we take advantage of the orthogonality relation, Eq. 7.116, by taking the inner product of $\hat{H} \Psi$ with one of the stationary states,

$\begin{displaymath} \int_{-\infty}^\infty \psi^*_m \hat{H} \Psi dx = \sum_{n=0}^... ... \int_{-\infty}^\infty \psi^*_m \hat{U} \psi_n dx \right] \end{displaymath}$

(7.120)

We also have the (time dependent) Schrödinger equation

$\begin{displaymath} \hat{H} \Psi = i \hbar \frac{\partial \Psi}{\partial t} \end{displaymath}$

which transforms the left side of Eq. 7.120 into

$\begin{displaymath} i \hbar \sum_{n=0}^\infty \frac{\partial}{\partial t} c_n e... ...psi_n dx = [E_m c_m + i \hbar \dot{c}_m] e^{-i \omega_m t} \end{displaymath}$

Adopting the shorthand

$\begin{displaymath} U_{mn} = \int_{-\infty}^\infty \psi^*_m \hat{U} \psi_n dx \end{displaymath}$

(7.121)

we obtain a set of coupled first order differential equations in the

$\begin{displaymath} \dot{c_m} = -\frac{i}{\hbar} \sum_{n \neq m} c_n U_{mn} e^{-i (\omega_n-\omega_m) t} \end{displaymath}$

(7.122)

Note that the $U_{mn}$ are just numbers, and they are numbers we can calculate if $\hat{U}$ is known. They are called the matrix elements of $\hat{U}$ .

The $U_{nn}$ are called diagonal matrix elements and are only non-zero if $\hat{U}$ contains the unperturbed potential. This is generally not the case, and we will assume $U_{nn}=0$ . Hence the sum in Eq. 7.122 is over $n \neq m$ .

It is generally the case that only a few of the $U_{mn}$ are non-zero for a given state

. Hence, the infinite sum in Eq. 7.122 is actually finite, even for systems with an infinite number of states.

7.6.2 First Order Corrections

Consider a system initially in state i

$\begin{displaymath} c_i(0) = 1, c_{n\neq1} = 0 \end{displaymath}$

(7.123)

States

and

have nonzero matrix elements $U_{if}$ and $U_{fi}$ connecting them. For the moment, we will assume all other matrix elements are zero, giving a pair of coupled equations

$\displaystyle \dot{c_i} = -\frac{i}{\hbar} c_f U_{if} e^{-i (\omega_f-\omega_i) t}$
$\displaystyle \dot{c_f} = -\frac{i}{\hbar} c_i U_{fi} e^{-i (\omega_i-\omega_f) t}$			(7.124)

Note that

$\begin{displaymath} U_{nm} = U^*_{mn} \end{displaymath}$

(7.125)

so that we only really need to calculate one matrix element here.

Provided the perturbation $\hat{U}$ is small, we can consider approximations in which we only consider corrections proportional to the first few powers of the $U_{nm}$ . To first order, we apply the initial conditions (Eq. 7.123) to the right hand sides of Eq.'s 7.124, giving

$\displaystyle c^{(1)}_i$	$\textstyle =$	$\displaystyle 1$
$\displaystyle c^{(1)}_f$	$\textstyle =$	$\displaystyle -\frac{i}{\hbar} \int_0^t U_{fi} e^{-i (\omega_i-\omega_f) t'} dt'$	(7.126)

Eq.'s 7.126 imply that the wavefunction is no longer normalized. We can correct this if we wish. We also have the probability that a transition has occurred from state

to state

$\begin{displaymath} P_{i\rightarrow f} (t) = \vert c_f(t)\vert^2 = \frac{1}{\hba... ...t U_{fi} e^{-i (\omega_i-\omega_f) t'} dt' \right\vert^2 \end{displaymath}$

(7.127)

We will use this first order result to calculate transition probabilities in Sec. 7.7.1.

The corrections of various orders

$c_n^{(N)}$ are added together to give the corrected coefficients,

$\begin{displaymath} c_n(t) = c_n(0) + c_n^{(1)} + c_n^{(2)} + ... \end{displaymath}$

(7.128)

7.6.3 Second Order Corrections

Higher order corrections involve inserting the coefficients of the previous order into the right hand sides of Eq.'s 7.124.

$\displaystyle \dot{c_i}$	$\textstyle =$	$\displaystyle -\frac{1}{\hbar^2} U_{if} e^{-i (\omega_f-\omega_i) t} \int_0^t U_{fi} e^{-i (\omega_i-\omega_f) t'} dt'$
$\displaystyle \dot{c_f}$	$\textstyle =$	$\displaystyle -\frac{i}{\hbar} U_{fi} e^{-i (\omega_i-\omega_f) t}$	(7.129)

Hence, $c_f^{(2)} = c_f^{(1)}$ , and the second order correction is only to

$\begin{displaymath} c^{(2)}_i = -\frac{1}{\hbar^2} \int_0^t U_{if} e^{-i (... ...} U_{fi} e^{-i (\omega_i-\omega_f) t'} dt' \right] dt'' \end{displaymath}$

(7.130)

Note that this is indeed a second order correction, proportional to $\hat{U}^2$ and is therefore much smaller than the first order correction to

The second order correction involves both $U_{if}$ and $U_{fi}$ . We can think of it as expressing the probability that the system makes transitions there ( $i\rightarrow f$ ) and back again ( $f \rightarrow i$ ).

Updated Wed Jan 18 09:51:28 2006