diff options
Diffstat (limited to 'source/know/concept/time-dependent-perturbation-theory')
| -rw-r--r-- | source/know/concept/time-dependent-perturbation-theory/index.md | 83 |
1 files changed, 0 insertions, 83 deletions
diff --git a/source/know/concept/time-dependent-perturbation-theory/index.md b/source/know/concept/time-dependent-perturbation-theory/index.md index 841c219..b0ffa42 100644 --- a/source/know/concept/time-dependent-perturbation-theory/index.md +++ b/source/know/concept/time-dependent-perturbation-theory/index.md @@ -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, |
