# 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.

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:

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

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

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

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:

$$\hat{H}(t) = \hat{H_0}+\hat{H'(t)}$$

and the state vector now has to satisfy the equation:

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

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 $\psi_i(\vec{r})$:

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

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 $\hat{H’}(t)$.