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+---
+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
+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 $\Ket{m}$:
+
+$$\begin{aligned}
+    0
+    &= \sum_{n} \Big( \lambda c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \dv{c_n}{t} \Inprod{m}{n} \Big) \exp(- i E_n t / \hbar)
+\end{aligned}$$
+
+Thanks to orthonormality, this removes the latter term from the summation:
+
+$$\begin{aligned}
+    i \hbar \dv{c_m}{t} \exp(- i E_m t / \hbar)
+    &= \lambda \sum_{n} c_n \matrixel{m}{\hat{H}_1}{n} \exp(- i E_n t / \hbar)
+\end{aligned}$$
+
+We divide by the left-hand exponential and define
+$\omega_{mn} \equiv (E_m - E_n) / \hbar$ to get:
+
+$$\begin{aligned}
+    \boxed{
+        i \hbar \dv{c_m}{t}
+        = \lambda \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} \exp(i \omega_{mn} t)
+    }
+\end{aligned}$$
+
+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 $\hat{H}_1(t)$.
+
+
+## Sinusoidal perturbation
+
+Arguably the most important perturbation
+is a sinusoidally-varying potential, which represents
+e.g. incoming electromagnetic waves,
+or an AC voltage being applied to the system.
+In this case, $\hat{H}_1$ has the following form:
+
+$$\begin{aligned}
+    \hat{H}_1(\vec{r}, t)
+    \equiv V(\vec{r}) \sin(\omega t)
+    = \frac{1}{2 i} V(\vec{r}) \: \big( \exp(i \omega t) - \exp(-i \omega t) \big)
+\end{aligned}$$
+
+We abbreviate $V_{mn} = \matrixel{m}{V}{n}$,
+and take the first-order correction formula:
+
+$$\begin{aligned}
+    c_m^{(1)}(t)
+    &= - \frac{1}{2 \hbar} \sum_{n} V_{mn} c_n^{(0)}
+    \int_0^t \exp\!\big(i \tau (\omega_{mn} \!+\! \omega)\big) - \exp\!\big(i \tau (\omega_{mn} \!-\! \omega)\big) \dd{\tau}
+    \\
+    &= \frac{i}{2 \hbar} \sum_{n} V_{mn} c_n^{(0)}
+    \bigg( \frac{\exp\!\big(i t (\omega_{mn} \!+\! \omega) \big) - 1}{\omega_{mn} + \omega}
+    + \frac{\exp\!\big(i t (\omega_{mn} \!-\! \omega) \big) - 1}{\omega_{mn} - \omega} \bigg)
+\end{aligned}$$
+
+For simplicity, we let the system start in a known state $\Ket{a}$,
+such that $c_n^{(0)} = \delta_{na}$,
+and we assume that the driving frequency is close to resonance $\omega \approx \omega_{ma}$,
+such that the second term dominates the first, which can then be neglected.
+We thus get:
+
+$$\begin{aligned}
+    c_m^{(1)}(t)
+    &= i \frac{V_{ma}}{2 \hbar} \frac{\exp\!\big(i t (\omega_{ma} \!-\! \omega) \big) - 1}{\omega_{ma} - \omega}
+    \\
+    &= i \frac{V_{ma}}{2 \hbar}
+    \frac{\exp\!\big(i t (\omega_{ma} \!-\! \omega) / 2 \big) - \exp\!\big(\!-\! i t (\omega_{ma} \!-\! \omega) / 2 \big)}{\omega_{ma} - \omega}
+    \: \exp\!\big(i t (\omega_{ma} \!-\! \omega) / 2 \big)
+    \\
+    &= - \frac{V_{ma}}{\hbar}
+    \frac{\sin\!\big( t (\omega_{ma} \!-\! \omega) / 2 \big)}{\omega_{ma} - \omega}
+    \: \exp\!\big(i t (\omega_{ma} \!-\! \omega) / 2 \big)
+\end{aligned}$$
+
+Taking the norm squared yields the **transition probability**:
+the probability that a particle that started in state $\Ket{a}$
+will be found in $\Ket{m}$ at time $t$:
+
+$$\begin{aligned}
+    \boxed{
+        P_{a \to m}
+        = |c_m^{(1)}(t)|^2
+        = \frac{|V_{ma}|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_{ma} - \omega) t / 2 \big)}{(\omega_{ma} - \omega)^2}
+    }
+\end{aligned}$$
+
+The result would be the same if $\hat{H}_1 \equiv V \cos(\omega t)$.
+However, if instead $\hat{H}_1 \equiv V \exp(- i \omega t)$,
+the result is larger by a factor of $4$,
+which can cause confusion when comparing literature.
+
+In any case, the probability oscillates as a function of $t$
+with period $T = 2 \pi / (\omega_{ma} \!-\! \omega)$,
+so after one period the particle is back in $\Ket{a}$,
+and after $T/2$ the particle is in $\Ket{b}$.
+See [Rabi oscillation](/know/concept/rabi-oscillation/)
+for a more accurate treatment of this "flopping" behaviour.
+
+However, when regarded as a function of $\omega$,
+the probability takes the form of
+a sinc-function centred around $(\omega_{ma} \!-\! \omega)$,
+so it is highest for transitions with energy $\hbar \omega = E_m \!-\! E_a$.
+
+Also note that the sinc-distribution becomes narrower over time,
+which roughly means that it takes some time
+for the system to "notice" that
+it is being driven periodically.
+In other words, there is some "inertia" to it.
+
+
+
+## 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.