19. Interaction picture, perturbation theory and Einstein coefficients

19.1. 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,

\[\begin{split}\begin{aligned} \langle \hat{A}_I \rangle & = \langle \Psi_I | \hat{A}_I | \Psi_I \rangle = \\ & = \langle \Psi_S | \underbrace{U^{(0)} \hat{A}_I U^{(0) \, \dagger}}_{=\hat{A}_S} | \Psi_S \rangle \ , \end{aligned}\end{split}\]

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.

Time evolution of a state \(\ | \Psi_I(t) \rangle\)

\[\begin{aligned} i \hbar \dfrac{d}{dt} | \Psi_I \rangle & = \hat{H}^{(1)}_I | \Psi_I \rangle \ . \end{aligned}\]

Details

\[\begin{split}\begin{aligned} i \hbar \dfrac{d}{dt} | \Psi_I \rangle & = i \hbar \dfrac{d}{dt} \left[ U^{(0) \, \dagger}(t,t_0) | \Psi_S \rangle \right] = \\ & = i \hbar \left[ \partial_t U^{(0) \, \dagger}(t,t_0) | \Psi_S \rangle + U^{(0) \, \dagger} \dfrac{d}{dt} | \Psi_S \rangle \right] = \\ & = - U^{(0) \, \dagger} \hat{H}_0 | \Psi_S \rangle + U^{(0) \, \dagger} \left( \hat{H}^{(0)} + \hat{H}^{(1)} \right) | \Psi_S \rangle = \\ & = \underbrace{U^{(0) \, \dagger} \hat{H}^{(1)} U^{(0)}}_{\hat{H}^{(1)}_I} \, \underbrace{U^{(0) \, \dagger} | \Psi_S \rangle}_{| \Psi_I \rangle} = \\ & = \hat{H}^{(1)}_I | \Psi_I \rangle \ . \end{aligned}\end{split}\]

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, \(\hat{H} = \hat{H}^{\dagger}\).

Time evolution of an operator \(\hat{A}_I(t)\)

Details

19.2. 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.

\[\begin{split}\begin{aligned} 0 & = - i \hbar \dfrac{d}{dt} | \Psi \rangle + \left( \hat{H}_0 + \hat{H}_1(t) \right) | \Psi \rangle = \\ & = \sum_n \left[ \left( - i \hbar \dot{c}_n - c_n \, E_n + c_n \, E_n \right) | n \rangle + \hat{H}_1 | n \rangle c_n \right] e^{-i E_n t/\hbar} \\ & = \sum_n \left[ - i \hbar \dot{c}_n | n \rangle + \hat{H}_1 | n \rangle c_n \right] e^{-i E_n t/\hbar} \ . \end{aligned}\end{split}\]

Projecting on \(\langle m |\), a system of an infinite number of equations follows

\[\dot{c}_m = - \frac{i}{\hbar} \sum_{n} \langle m | \hat{H}_1 | n \rangle c_n \, e^{i \frac{E_m - E_n}{\hbar} t} \ .\]

19.3. 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 \(V\),

\[\begin{split}\begin{aligned} \mathbf{F} & = - \dfrac{q^2}{4 \pi \varepsilon} \dfrac{\mathbf{r}}{|\mathbf{r}|^3} + 2 q \mathbf{E}_0 \cos(2 \pi \, \nu \, t) = \\ & = \nabla_{\mathbf{r}} \left[ \dfrac{q^2}{4 \pi \varepsilon}\dfrac{1}{|\mathbf{r}|} + 2 q \mathbf{r} \cdot \mathbf{E}_0 \cos(2 \pi \, \nu \, t) \right] = \\ & = - \nabla_{\mathbf{r}} \left( V_0(\mathbf{r}) + V_1(\mathbf{r},t) \right) \ . \end{aligned}\end{split}\]

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

\[\hat{H} = \hat{H}_0 + \hat{H}_1(t) = \hat{H}_0 + 2 \mathbf{E}_0 \cdot q \hat{\mathbf{r}} \, \cos(2 \pi \nu t ) \ .\]