14.2: Time-Dependent Perturbation Theory

The mathematical machinery needed to compute the rates of transitions among molecular states induced by such a time-dependent perturbation is contained in time dependent perturbation theory (TDPT). The development of this theory proceeds as follows. One first assumes that one has in-hand all of the eigenfunctions {\(\Phi_k\)} and eigenvalues {\(E_k^0\)} that characterize the Hamiltonian \(H^0\) of the molecule in the absence of the external perturbation:

\[ H^0 \Phi_k = E_k^0 \Phi_k. \nonumber \]

One then writes the time-dependent Schrödinger equation

\[ i\hbar\dfrac{\partial \Psi}{\partial t} = (H^0 + H_{int}) \Psi \nonumber \]

in which the full Hamiltonian is explicitly divided into a part that governs the system in the absence of the radiation field and \(H_{int}\) which describes the interaction with the field.

Perturbative Solution

By treating \(H^0\) as of zeroth order (in the field strength |\(\textbf{A}_0\)|), expanding \( \Psi \) order-by order in the field-strength parameter:

\[ \Psi = \Psi^0 + \Psi^1 + \Psi^2 + \Psi^3 + ..., \nonumber \]

realizing that Hint contains terms that are both first- and second- order in \(|\textbf{A}_0|\)

\[ H^1_{int} = \sum\limits_j \left[ \left(\dfrac{ie\hbar}{m_ec}\right) \textbf{A}(r_j,t)\cdot{\nabla_j} \right] + \sum\limits_a \sum\limits_a \left[ \left(\dfrac{iZ_ae\hbar}{m_ac}\right) \textbf{A}(R_a,t)\cdot{\nabla_a} \right], \nonumber \]

\[ H^2_{int} = \sum\limits_j \left[ \left(\dfrac{e^2}{2m_ec^2}\right) |\textbf{A}(r_j,t)|^2\right] + \sum\limits_a \left[ \left( \dfrac{Z_a^2e^2}{2m_ac^2}\right) |\textbf{A}(R_a,t)|^2 \right], \nonumber \]

and then collecting together all terms of like power of \(|\textbf{A}_0|\), one obtains the set of time dependent perturbation theory equations. The lowest order such equations read:

\[ i\hbar \dfrac{\partial \Psi^0}{\partial t} = H^0 \Psi^0 \nonumber \]

\[ i\hbar\dfrac{\partial \Psi^1}{\partial t} = (H^0 \Psi^1 + H^1_{int} \Psi^0) \nonumber \]

\[ i\hbar\dfrac{\partial \Psi^2}{\partial t} = (H^0 \Psi^2 + H^2_{int}\Psi^0 + H^1_{int}\Psi^1). \nonumber \]

The zeroth order equations can easily be solved because \(H^0\) is independent of time. Assuming that at \(t = - \infty, \Psi = \psi_i\) (we use the index i to denote the initial state), this solution is:

\[ \Psi^0 = \Phi_i e^{\dfrac{-iE_i^0t}{\hbar}}. \nonumber \]

The first-order correction to \(\Psi^0, \Psi^1\) can be found by (i) expanding \(\Psi^1\) in the complete set of zeroth-order states {\(\Phi_f\)}:

\[ \Psi^1 = \sum\limits_f\Phi_f<\Phi_f|\Psi^1> = \sum\limits_f\Phi_fC_f^1, \nonumber \]

(ii) using the fact that

\[ H^0\Phi_f = E_f^0 \Phi_f \nonumber \],

and (iii) substituting all of this into the equation that Y1 obeys. The resultant equation for the coefficients that appear in the first-order equation can be written as

\[ i\hbar \dfrac{\partial C_f^1}{\partial t} = \sum\limits_k [E_k^0 C_k^1 \delta_{f,k}] + <\Phi_f| H^1_{int}|\Phi_i> e^{\dfrac{-iE_i^0t}{\hbar}}, \nonumber \]

or

\[ i\hbar\dfrac{\partial C_f^1}{\partial t} = E_f^0C_f^1 + <\Phi_f|H^1_{int}|\Phi_i> e^{\dfrac{-iE_i^0t}{\hbar}}. \nonumber \]

Defining

\[ C_f^1(t) = D_f^1(t)e^{\dfrac{-iE_f^0t}{\hbar}}. \nonumber \]

his equation can be cast in terms of an easy-to-solve equation for the \(D_f^1\) coefficients:

\[ i\hbar\dfrac{\partial D_f^1}{\partial t} = <\Phi_f|H^1_{int}|\Phi_i> e^{\dfrac{i[E_f^0-E_i^0]t}{\hbar}}. \nonumber \]

Assuming that the electromagnetic field \(\textbf{A}(\textbf{r},t)\) is turned on at t=0, and remains on until t = T, this equation for \(D_f^1\) can be integrated to yield:

\[ D_f^1(t) = \dfrac{1}{(i\hbar)}\int\limits_0^T <\Phi_f|H^1_{int}|\Phi_i> e^{\dfrac{i[E_f^0-E_i^0]t'}{\hbar}}dt'. \nonumber \]