User:Espen180/Quantum Mechanics/Time-Dependent Perturbation Theory

As the title suggests, this chapter deals with the situation where we have a solvable system with a small, time-dependent perturbation.

Let $$H_0$$ be the unperturbed Hamiltonian, with known energy levels and eigenstates $$H_0|n\rangle = E_n |n\rangle$$. We are interested in the system


 * $$i\hbar\partial_t|\Psi(t)\rangle=H|\Psi(t)\rangle=(H_0+H_1(t))|\Psi(t)\rangle$$,

where $$H_1(t)$$ is a small perturbation. Of course, we can forget about finding stationary states of the system, since it is time-dependent. However, since the $$|n\rangle$$'s are an orthonormal basis set, we can write


 * $$|\Psi(t)\rangle=\sum_{n=1}^\infty a_n(t)e^{-iE_nt/\hbar}|n\rangle$$.

Inserting this into the Schrödinger equation and multiplying by $$\langle n|e^{iE_nt/\hbar}$$, we obtain


 * $$i\hbar \frac{\partial a_n(t)}{\partial t}+a_n(t)E_n=a_n(t)\langle n|H_0|n\rangle+\sum_{m=1}^{\infty} a_m(t)e^{-i(E_m-E_n)t/\hbar}\langle n|H_1(t)|m\rangle$$.

We see that $$a(t)\langle n|H_0|n\rangle=a(t)E_n$$ cancels with the $$a(t)E_n$$ term on the right side, and we are left with


 * $$i\hbar \frac{\partial a_n(t)}{\partial t}=\sum_{m=1}^{\infty} a_m(t)e^{i(E_n-E_m)t/\hbar}\langle n|H_1(t)|m\rangle$$.

So far we have made no approximations, so this equation is exact and equivalent to the initial Schrödinger equation.

Now, if the perturbation is small, we can assume that the $$a_i(t)$$'s vary slowly with time. Therefore we can, to first order, neglect the time dependence of the $$a_m(t)$$'s on the right side of the equation. The resulting differential equation is trivial, with solution


 * $$a_n(t)=a_n(t_0)+\frac{1}{i\hbar}\sum_{m=1}^{\infty}a_k(t_0)\int_{t_0}^{t}e^{i(E_n-E_m)t^\prime /\hbar}\langle n|H_1(t^\prime)|m\rangle \mathrm{d}t^\prime$$.