---
title: "Time-dependent perturbation theory"
firstLetter: "T"
publishDate: 2021-03-07
categories:
- Physics
- Quantum mechanics
- Perturbation

date: 2021-03-07T11:08:14+01:00
draft: false
markup: pandoc
---

# Time-dependent perturbation theory

In quantum mechanics, **time-dependent perturbation theory** exists to deal
with time-varying perturbations to the Schrödinger equation.
This is in contrast to [time-independent perturbation theory](/know/concept/time-independent-perturbation-theory/),
where the perturbation is is stationary.

Let $\hat{H}_0$ be the base time-independent
Hamiltonian, and $\hat{H}_1$ be a time-varying perturbation, with
"bookkeeping" parameter $\lambda$:

$$\begin{aligned}
    \hat{H}(t) = \hat{H}_0 + \lambda \hat{H}_1(t)
\end{aligned}$$

We assume that the unperturbed time-independent problem
$\hat{H}_0 \ket{n} = E_n \ket{n}$ has already been solved, such that the
full solution is:

$$\begin{aligned}
    \ket{\Psi_0(t)} = \sum_{n} c_n \ket{n} \exp(- i E_n t / \hbar)
\end{aligned}$$

Since these $\ket{n}$ form a complete basis, the perturbed wave function
can be written in the same form, but with time-dependent coefficients $c_n(t)$:

$$\begin{aligned}
    \ket{\Psi(t)} = \sum_{n} c_n(t) \ket{n} \exp(- i E_n t / \hbar)
\end{aligned}$$

We insert this ansatz in the time-dependent Schrödinger equation, and
reduce it using the known unperturbed time-independent problem:

$$\begin{aligned}
    0
    &= \hat{H}_0 \ket{\Psi(t)} + \lambda \hat{H}_1 \ket{\Psi(t)} - i \hbar \dv{t} \ket{\Psi(t)}
    \\
    &= \sum_{n}
    \Big( c_n \hat{H}_0 \ket{n} + \lambda c_n \hat{H}_1 \ket{n} - c_n E_n \ket{n} - i \hbar \dv{c_n}{t} \ket{n} \Big) \exp(- i E_n t / \hbar)
    \\
    &= \sum_{n} \Big( \lambda c_n \hat{H}_1 \ket{n} - i \hbar \dv{c_n}{t} \ket{n} \Big) \exp(- i E_n t / \hbar)
\end{aligned}$$

We then take the inner product with an arbitrary stationary basis state $\ket{m}$:

$$\begin{aligned}
    0
    &= \sum_{n} \Big( \lambda c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \frac{d c_n}{dt} \braket{m}{n} \Big) \exp(- i E_n t / \hbar)
\end{aligned}$$

Thanks to orthonormality, this removes the latter term from the summation:

$$\begin{aligned}
    i \hbar \frac{d c_m}{dt} \exp(- i E_m t / \hbar)
    &= \lambda \sum_{n} c_n \matrixel{m}{\hat{H}_1}{n} \exp(- i E_n t / \hbar)
\end{aligned}$$

We divide by the left-hand exponential and define
$\omega_{mn} = (E_m - E_n) / \hbar$ to get:

$$\begin{aligned}
    \boxed{
        i \hbar \frac{d c_m}{dt}
        = \lambda \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} \exp(i \omega_{mn} t)
    }
\end{aligned}$$

So far, we have not invoked any approximation,
so we can analytically find $c_n(t)$ for some simple systems.
Furthermore, it is useful to write this equation in integral form instead:

$$\begin{aligned}
    c_m(t)
    = c_m(0) - \lambda \frac{i}{\hbar} \sum_{n} \int_0^t c_n(\tau) \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau}
\end{aligned}$$

If this cannot be solved exactly, we must approximate it. We expand
$c_m(t)$ in the usual way, with the initial condition $c_m^{(j)}(0) = 0$
for $j > 0$:

$$\begin{aligned}
    c_m(t) = c_m^{(0)} + \lambda c_m^{(1)}(t) + \lambda^2 c_m^{(2)}(t) + ...
\end{aligned}$$

We then insert this into the integral and collect the non-zero orders of $\lambda$:

$$\begin{aligned}
    c_m^{(1)}(t)
    &= - \frac{i}{\hbar} \sum_{n} \int_0^t c_n^{(0)} \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau}
    \\
    c_m^{(2)}(t)
    &= - \frac{i}{\hbar} \sum_{n}
    \int_0^t c_n^{(1)}(\tau) \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau}
    \\
    c_m^{(3)}(t)
    &= - \frac{i}{\hbar} \sum_{n}
    \int_0^t c_n^{(2)}(\tau) \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau}
\end{aligned}$$

And so forth. The pattern here is clear: we can calculate the $(j\!+\!1)$th
correction using only our previous result for the $j$th correction.
We cannot go any further than this without considering a specific perturbation $\hat{H}_1(t)$.


## References
1.  D.J. Griffiths, D.F. Schroeter,
    *Introduction to quantum mechanics*, 3rd edition,
    Cambridge.