---
title: "Time-dependent perturbation theory"
sort_title: "Time-dependent perturbation theory"
date: 2021-03-07
categories:
- Physics
- Quantum mechanics
- Perturbation
layout: "concept"
---

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 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 general solution for the full $$\hat{H}$$ can be written as:

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

These time-dependent coefficients are then governed by
the [amplitude rate equations](/know/concept/amplitude-rate-equations/):

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

So far, we have not made any approximations at all.
We rewrite this in integral form:

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

If this cannot be solved exactly, we must approximate it.
We expand $$c_m(t)$$ as a power series,
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 nonzero 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 our result for the $$j$$th correction.
The only purpose of $$\lambda$$ was to help us collect its orders;
in the end we simply set $$\lambda = 1$$ or absorb it into $$\hat{H}_1$$.
Now we have the essence of time-dependent perturbation theory,
and we cannot go any further without considering a specific $$\hat{H}_1$$.



## References
1.  D.J. Griffiths, D.F. Schroeter,
    *Introduction to quantum mechanics*, 3rd edition,
    Cambridge.
2.  R. Shankar,
    *Principles of quantum mechanics*, 2nd edition,
    Springer.