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@@ -78,89 +78,6 @@ and we cannot go any further without considering a specific $$\hat{H}_1$$.
-## 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,