Time-Dependent Perturbation Theory

Time dependent perturbation theory is the main analytical tool for treating the transitions of QM systems from one energy state to another.

Table of Contents:

  1. Formulation
  2. Harmonic Perturbation and Fermi’s Golden Rule

FORMULATION

We can first assume that we have a time-independent Hamiltonian of some system $\hat{H_0}$ that satisfies the time-independent Schrodinger equation. If such a system at a certain time, say t=0, possesses energy $E_m$, then it would stay at that energy for all subsequent time t>0. This follows from the Schrodinger equation for the system:

\begin{equation} \hat{H_0}\psi(r,t) = i\hbar\frac{\partial\psi(r,t)}{\partial{t}} \end{equation}

whose general solution, given in the last subsection, is given as:

\begin{equation} \psi(r,t) = \sum_ic_i\psi_i(\vec{r})e^{(-\frac{i}{\hbar}E_it)} \end{equation}

Here, we can directly see that if at t = 0 we have $c_m$ = 1 and $c_i$=0 for all $i\neqm$, then the system will indeed stay at the initial energy $E_m$ for all t>0.

We can now consider that the system is weakly perturbed with a time-varying external perturbation $\hat{H’}(t)$. The system Hamiltonian then becomes:

\begin{equation} \hat{H}(t) = \hat{H_0}+\hat{H'(t)} \end{equation}

and the state vector now has to satisfy the equation:

\begin{equation} [\hat{H_0}+\hat{H'(t)}]\psi = i\hbar\frac{\partial\psi}{\partial{t}} \end{equation}

Since the eigenvectors of a Hermitian operator (for example, $\hat{H_0}$) form a complete orthonormal basis, we can expand $\psi(r,t)$ in terms of $\psi_i(\vec{r})$:

\begin{equation} \psi(\vec{r},t) = \sum_ic_i(t)\cdot\psi_i(\vec{r})e^{-iE_it/\hbar} \end{equation}

where now the expansion coefficients are time-dependent. Let’s then assume that at t=0, when the external field is turned on, we have $c_m(0)=1$ i.e. the system is in the eigenstate with the eigenvalue $E_m$ $c_i(0)=0$ for all $i\neq{m}$.

Then, if an energy measurement is made at some time t>0 it may yield some value $E_k$ with $k\neq{m}$. The probability of this event is $|c_k(t)|^2$, which gives the probability of finding the system in the state k at time t given that at t-0 it occupied state m – i.e. the probability of the transition from m to k occuring by time t, due to the influence of the perturbation $\hat{H’}(t)$.

%d bloggers like this: