Here we continue with the previous section on time dependent perturbation theory

💼 Case: lets consider an oscillatory perturbation to our system of the form

$$ \hat V(t)=\left \{ \begin{matrix} 0 & \text{for} & t\le 0 \\ \hat V_0 e^{-i\omega t} & \text{for} & t>0 \end{matrix} \right . $$

where $\hat V_0$ is not time dependent. 🧽 Assume: initially the state is $\ket{i}$

⚽ Goal: find the probability of transitioning to a final state $\ket{f}$ so calculating $P_{i\to f}(t)=|c_f^{(1)}(t)|^2$

• Lets start with $c_f^{(1)}(t)$

  $$ \begin{aligned} c_f^{(1)}(t) & = - \frac{i}{\hbar} \int ^t_0\text dt ' \, e^{i \omega {fi} t'}\bra{f} \hat V(t') \ket{i} \\ &=-\frac{i}{\hbar} \bra{f} \hat V_0 \ket{i} \int ^t_0 \text dt' \, e^{i(\omega{fi}-\omega)t'} \\ &=\frac{1}{\hbar} \bra{f}\hat V_0 \ket{i} \frac{1-e^{i (\omega_{fi}-\omega )t}}{\omega_{fi}-\omega}

  \end{aligned} $$

  where $\hbar \omega_{fi}=E_f-E_i$

• From there we directly calculate the probability to be

  $$ \begin{aligned} P_{i \to f}(t) &= \left| c_f^{(1)}(t) \right|^2 = c_f^{(1)}(t) \left( c_f^{(1)}(t) \right)^* \\ &= \frac{4}{\hbar^2} \left| \langle f | \hat{V}0 | i \rangle \right|^2 \frac{\sin^2 \left( \frac{(\omega{fi} - \omega) t}{2} \right)}{(\omega_{fi} - \omega)^2} \\ &= \frac{1}{\hbar^2} \left| \langle f | \hat{V}_0 | i \rangle \right|^2 \frac{\sin^2 \left( \frac{x t}{2} \right)}{(x/2)^2}\end{aligned} $$

  where $x=(\omega_{fi}-\omega)$

💎 Conclusion: we have our probability lets now analyse it for different parameters

• For $\omega \ne \omega_{fi}$ , when the perturbation frequency $\omega$ does not coincide with the transition frequency $\omega_{fi}$ the probability oscillated in time with a small amplitude
• For $\omega=\omega_{fi}$, when the perturbation frequency $\omega$ is resonant with the transition frequency $\omega_{fi}$ we get the following

  • In the long-time limit we get

    $$ \lim_{t\to \infin}\left [ P_{i\to f} (t)\right ]=\frac{2\pi t}{\hbar^2} |\bra{f} \hat V_0 \ket{i}|^2 \delta (\omega_{fi}-\omega) $$

    • Where we used (expand for details)

    In this limit the transition rate is

    $$ \lim_{t\to\infin} \left [R_{1\to f}(t)\right ]= \lim_{t\to\infin} \left [ \frac{\text d P_{i\to f}(t)}{\text dt} \right ]=\frac{2\pi }{\hbar^2} |\bra{f} \hat V_0 \ket{i}|^2 \delta (\omega_{fi}-\omega) $$

    Which we call Fermi’s golden rule, using $\delta (\hbar \omega)=\delta (\omega)/\hbar$ we can write it as

    <aside> 🏆

    Fermi’s golden rule: In the long term limit only a perturbation with frequency that matches the system transition frequency $\omega_{fi}$ can induce a transition from $\ket{i}$ to $\ket{f}$ such that the energy difference between the initial and final states is supplied by the perturbation

    $$ \lim_{t\to\infin} \left [ R_{i\to f}(t))\right ]= \frac{2\pi}{\hbar} |\bra{f} \hat V_0 \ket{i} |^2 \,\delta (E_{fi}-\hbar \omega) $$

    </aside>

    🗒️ Note: energy is absorbed by the perturbing field $E_f>E_i$. If instead $\hat V(t)=V_0e^{i\omega t}$ we would get $\delta (E_{fi}+\hbar \omega)$ meaning energy is given through emission $E_f<E_i$. For a real field (ours is not Hermitian, non physical) $\hat V(t)=\hat V_0 \cos(\omega t)$, both processes occur

    ---

    If the process is constant we get $\omega=0$

    $$ \lim_{t\to\infin} \left [ R_{i\to f}(t))\right ]= \frac{2\pi}{\hbar} |\bra{f} \hat V_0 \ket{i} |^2 \,\delta (E_{fi}) $$

    💎 Conclusion: if $\omega=0$ then $E_f=E_i$

Fermi’s golden rule

Lets take a closer look at Fermi’s golden rule

• if $\hbar \omega \ne E_{fi}$ then there is no change, so processes involving perturbation frequencies that do not coincide with the system transition frequency do not occur in the long-time limit within first-order perturbation theory

• if $\hbar \omega = E_{fi}$ then the transition rate is infinite, a consequence of our case being non-physical

  Not Hermitian

  Perfectly monochromatic radiation

  perfectly sharp $E$ levels

  Instead we should consider a spread of frequency with a weighting function $\rho(\omega)$

💼 Case: lets do it, an oscillating electric field $E_0\vec \epsilon \cos(\omega t)$ where $\vec \epsilon$ is a unit vector field direction

• The corresponding perturbation is

  $$ \hat V(t)=\left \{ \begin{matrix} 0 & \text{for } & t\le 0 \\ eE_0 \cos(\omega t) \vec \epsilon \cdot \hat r =\frac{eE_0}{2} (e^{i\omega t}+e^{-i\omega t})\vec \epsilon \cdot \hat r & \text{for} & t>0 \end{matrix} \right . $$

  🧽 Assume: long-wavelength limit with no spatial variation of $E$ and ignore magnetic effects

• Applying fermis golden rule to this perturbation we get

  $$ \lim_{t\to\infin} \left [ R_{i\to f}(t))\right ]= \frac{\pi}{2\hbar^2}e^2 E_0^2 |\bra{f} \vec \epsilon \cdot \hat r \ket{i} |^2 \,[\delta (\omega_{fi}-\omega)+ \delta (\omega_{fi}+\omega)] $$

To avoid unphysical results, consider $E_0\equiv E_0(\omega)$, representing the monochromaticity of our field