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diff --git a/source/know/concept/time-dependent-perturbation-theory/index.md b/source/know/concept/time-dependent-perturbation-theory/index.md index a1c1173..d39e321 100644 --- a/source/know/concept/time-dependent-perturbation-theory/index.md +++ b/source/know/concept/time-dependent-perturbation-theory/index.md @@ -14,24 +14,24 @@ 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 $\lambda$: +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 +$$\hat{H}_0 \Ket{n} = E_n \Ket{n}$$ 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 $c_n(t)$: +Since these $$\Ket{n}$$ form a complete basis, the perturbed wave function +can be written in the same form, but with time-dependent coefficients $$c_n(t)$$: $$\begin{aligned} \Ket{\Psi(t)} = \sum_{n} c_n(t) \Ket{n} \exp(- i E_n t / \hbar) @@ -50,7 +50,7 @@ $$\begin{aligned} &= \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}$$ -We then take the inner product with an arbitrary stationary basis state $\Ket{m}$: +We then take the inner product with an arbitrary stationary basis state $$\Ket{m}$$: $$\begin{aligned} 0 @@ -65,7 +65,7 @@ $$\begin{aligned} \end{aligned}$$ We divide by the left-hand exponential and define -$\omega_{mn} \equiv (E_m - E_n) / \hbar$ to get: +$$\omega_{mn} \equiv (E_m - E_n) / \hbar$$ to get: $$\begin{aligned} \boxed{ @@ -75,7 +75,7 @@ $$\begin{aligned} \end{aligned}$$ So far, we have not invoked any approximation, -so we can analytically find $c_n(t)$ for some simple systems. +so we can analytically find $$c_n(t)$$ for some simple systems. Furthermore, it is useful to write this equation in integral form instead: $$\begin{aligned} @@ -84,14 +84,14 @@ $$\begin{aligned} \end{aligned}$$ 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 $j > 0$: +$$c_m(t)$$ 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 $\lambda$: +We then insert this into the integral and collect the non-zero orders of $$\lambda$$: $$\begin{aligned} c_m^{(1)}(t) @@ -106,9 +106,9 @@ $$\begin{aligned} \int_0^t c_n^{(2)}(\tau) \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau} \end{aligned}$$ -And so forth. The pattern here is clear: we can calculate the $(j\!+\!1)$th -correction using only our previous result for the $j$th correction. -We cannot go any further than this without considering a specific perturbation $\hat{H}_1(t)$. +And so forth. The pattern here is clear: we can calculate the $$(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)$$. ## Sinusoidal perturbation @@ -117,7 +117,7 @@ Arguably the most important perturbation is a sinusoidally-varying potential, which represents e.g. incoming electromagnetic waves, or an AC voltage being applied to the system. -In this case, $\hat{H}_1$ has the following form: +In this case, $$\hat{H}_1$$ has the following form: $$\begin{aligned} \hat{H}_1(\vec{r}, t) @@ -125,7 +125,7 @@ $$\begin{aligned} = \frac{1}{2 i} V(\vec{r}) \: \big( \exp(i \omega t) - \exp(-i \omega t) \big) \end{aligned}$$ -We abbreviate $V_{mn} = \matrixel{m}{V}{n}$, +We abbreviate $$V_{mn} = \matrixel{m}{V}{n}$$, and take the first-order correction formula: $$\begin{aligned} @@ -138,9 +138,9 @@ $$\begin{aligned} + \frac{\exp\!\big(i t (\omega_{mn} \!-\! \omega) \big) - 1}{\omega_{mn} - \omega} \bigg) \end{aligned}$$ -For simplicity, we let the system start in a known state $\Ket{a}$, -such that $c_n^{(0)} = \delta_{na}$, -and we assume that the driving frequency is close to resonance $\omega \approx \omega_{ma}$, +For simplicity, we let the system start in a known state $$\Ket{a}$$, +such that $$c_n^{(0)} = \delta_{na}$$, +and we assume that the driving frequency is close to resonance $$\omega \approx \omega_{ma}$$, such that the second term dominates the first, which can then be neglected. We thus get: @@ -158,8 +158,8 @@ $$\begin{aligned} \end{aligned}$$ Taking the norm squared yields the **transition probability**: -the probability that a particle that started in state $\Ket{a}$ -will be found in $\Ket{m}$ at time $t$: +the probability that a particle that started in state $$\Ket{a}$$ +will be found in $$\Ket{m}$$ at time $$t$$: $$\begin{aligned} \boxed{ @@ -169,22 +169,22 @@ $$\begin{aligned} } \end{aligned}$$ -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 $4$, +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 $\Ket{b}$. +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 $\omega$, +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 $\hbar \omega = E_m \!-\! E_a$. +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 |