(quantum-mechanics:einstein-coeffs-interaction-picture)= # Interaction picture, perturbation theory and Einstein coefficients (quantum-mechanics:einstein-coeffs-interaction-picture:interaction)= ## Interaction picture Interaction picture is a common and useful description of a system whose Hamiltonian cna be written as the sum of a term $\hat{H}_0$ that's well known and solvable, and $\hat{H}_1$ that's a perturbation to the system $0$, and it's usually either time-dependent and/or hard to solve analytically, $$\hat{H} = \hat{H}_0 + \hat{H}_1 \ .$$ As $\hat{H}_0$ is time-independent, the corresponding unitary evolution operator is $U^{(0)} = \exp\left[ i \dfrac{\hat{H}_0}\hbar t \right]$. A state vector in interaction picture, $| \Psi_I(t) \rangle$ is defined as $$| \Psi_{I} (t) \rangle = U^{(0) \, \dagger} | \Psi_S(t) \rangle \ ,$$ with $| \Psi_S(t) \rangle$ the state vector in Schrodinger picture. If one requires that the expectation value of operators is the same using different pictures, $$\begin{aligned} \langle \hat{A}_I \rangle & = \langle \Psi_I | \hat{A}_I | \Psi_I \rangle = \\ & = \langle \Psi_S | \underbrace{U^{(0)} \hat{A}_I U^{(0) \, \dagger}}_{=\hat{A}_S} | \Psi_S \rangle \ , \end{aligned}$$ it follows that $\hat{A}_I(t) = U^{(0) \, \dagger}(t,t_0) \hat{A}_S U^{(0)}(t,t_0)$. From the commutation of $\hat{H}_0$ with $U^{(0)}$, it follows that it has the same expression in Schrodinger and interaction picture, $H^{(0)}_I = H^{(0)}_S$. **Time evolutions.** ````{dropdown} Time evolution of a state $\ | \Psi_I(t) \rangle$ :open: $$\begin{aligned} i \hbar \dfrac{d}{dt} | \Psi_I \rangle & = \hat{H}^{(1)}_I | \Psi_I \rangle \ . \end{aligned}$$ ```{dropdown} Details $$\begin{aligned} i \hbar \dfrac{d}{dt} | \Psi_I \rangle & = i \hbar \dfrac{d}{dt} \left[ U^{(0) \, \dagger}(t,t_0) | \Psi_S \rangle \right] = \\ & = i \hbar \left[ \partial_t U^{(0) \, \dagger}(t,t_0) | \Psi_S \rangle + U^{(0) \, \dagger} \dfrac{d}{dt} | \Psi_S \rangle \right] = \\ & = - U^{(0) \, \dagger} \hat{H}_0 | \Psi_S \rangle + U^{(0) \, \dagger} \left( \hat{H}^{(0)} + \hat{H}^{(1)} \right) | \Psi_S \rangle = \\ & = \underbrace{U^{(0) \, \dagger} \hat{H}^{(1)} U^{(0)}}_{\hat{H}^{(1)}_I} \, \underbrace{U^{(0) \, \dagger} | \Psi_S \rangle}_{| \Psi_I \rangle} = \\ & = \hat{H}^{(1)}_I | \Psi_I \rangle \ . \end{aligned}$$ recalling that $\hat{H}^{(0)} = i \hbar \, \partial_t U^{(0)} U^{(0) \, \dagger} = - i \hbar U^{(0)} \partial_t U^{(0) \, \dagger}$, and that the Hamiltonian operator is Hermitian, $\hat{H} = \hat{H}^{\dagger}$. ``` ```` ````{dropdown} Time evolution of an operator $\hat{A}_I(t)$ :open: ```{dropdown} Details :open: ``` ```` (quantum-mechanics:einstein-coeffs-interaction-picture:perturbation)= ## Perturbation theory For an isolated system with time-independent Hamiltonian operator $\hat{H}_0$, the state of the system can be written as a linear combination of the eigenstates of the Hamiltonian operator $$| \Psi(t) \rangle = | n \rangle a_n(t) \ ,$$ with $a_n(t) = a_n(0) \exp\left( - i \frac{E_n}{\hbar} t \right) = \langle n | \Psi(0) \rangle \exp\left( - i \frac{E_n}{\hbar} t \right) $. **Time varying perturbation theory.** The state of the perturbed system is can be written as a linear combination of the eigenstates of the unperturbed Hamiltonian as well - here explicitly writing a factor $e^{-i E_n t / \hbar}$ in the coefficient, $$| \Psi(t) \rangle = | n \rangle c_n(t) e^{-i E_n t / \hbar} \ ,$$ so that for unperturbed systems $c_n(t) = a_{n,0}$ constant. Inserting this expression in the Schrodinger equation of the system provides dynamical equations for the coefficients $c_n(t)$, i.e. $$\begin{aligned} 0 & = - i \hbar \dfrac{d}{dt} | \Psi \rangle + \left( \hat{H}_0 + \hat{H}_1(t) \right) | \Psi \rangle = \\ & = \sum_n \left[ \left( - i \hbar \dot{c}_n - c_n \, E_n + c_n \, E_n \right) | n \rangle + \hat{H}_1 | n \rangle c_n \right] e^{-i E_n t/\hbar} \\ & = \sum_n \left[ - i \hbar \dot{c}_n | n \rangle + \hat{H}_1 | n \rangle c_n \right] e^{-i E_n t/\hbar} \ . \end{aligned}$$ Projecting on $\langle m |$, a system of an infinite number of equations follows $$\dot{c}_m = - \frac{i}{\hbar} \sum_{n} \langle m | \hat{H}_1 | n \rangle c_n \, e^{i \frac{E_m - E_n}{\hbar} t} \ .$$ (quantum-mechanics:einstein-coeffs-interaction-picture:einstein-coeffs)= ## Einstein coefficients Perturbation theory is applied to an electron around a steady positive nucleus, subject to time-varying electric field. The classical counterpart of the governing equation of a point charge subject to Coulomb potential and a travelling wave in the electric field $$\mathbf{e}(\mathbf{r},t) = 2 \mathbf{E}_0 \cos( \omega t - \mathbf{k} \cdot \mathbf{r} ) = q \mathbf{E}_0 ( e^{i(\omega t - \mathbf{k}\cdot \mathbf{r})} + c.c. ) \ , $$ reads $$m \ddot{\mathbf{r}} + \dfrac{q^2}{4 \pi \varepsilon} \dfrac{\mathbf{r}}{|\mathbf{r}|^3} = 2 q \mathbf{E}_0 \cos(\omega t - \mathbf{k} \cdot \mathbf{r}) \ .$$ **Dipole approximation** assumes that the displacement of the charged particle is small compared to the wave-length of the radiation, $\mathbf{k} \cdot \mathbf{r} = \frac{ \hat{\mathbf{k}} \cdot \mathbf{r} }{2 \pi \lambda} \ll 1$, and thus the dynamic equation becomes $$m \ddot{\mathbf{r}} + \dfrac{q^2}{4 \pi \varepsilon} \dfrac{\mathbf{r}}{|\mathbf{r}|^3} = 2 q \mathbf{E}_0 \cos(\omega t) \ .$$ The total force is the sum of the Coulomb force and the force due to the incoming electric field. This sum can be written as a gradient of a scalar function, a potential $V$, $$\begin{aligned} \mathbf{F} & = - \dfrac{q^2}{4 \pi \varepsilon} \dfrac{\mathbf{r}}{|\mathbf{r}|^3} + 2 q \mathbf{E}_0 \cos(2 \pi \, \nu \, t) = \\ & = \nabla_{\mathbf{r}} \left[ \dfrac{q^2}{4 \pi \varepsilon}\dfrac{1}{|\mathbf{r}|} + 2 q \mathbf{r} \cdot \mathbf{E}_0 \cos(2 \pi \, \nu \, t) \right] = \\ & = - \nabla_{\mathbf{r}} \left( V_0(\mathbf{r}) + V_1(\mathbf{r},t) \right) \ . \end{aligned}$$ that can be written as the stationary potential $V_0$ from Coulomb force and the time-varying perturbation $V_1(\mathbf{r},t)$. The vector $q \mathbf{r}$ is usually defined as the **electric dipole**. Schrodinger equation follows from the promotion of the Hamiltonian function $H = K + V$ to the Hamiltonian operator $$\hat{H} = \hat{H}_0 + \hat{H}_1(t) = \hat{H}_0 + 2 \mathbf{E}_0 \cdot q \hat{\mathbf{r}} \, \cos(2 \pi \nu t ) \ .$$