What are coupled equations?

In quantum scattering, coupled equations are sets of differential equations that describe dynamics of colliding molecules and/or atoms under the influence of the interaction potential. These equations are "coupled" because the solution for one equation depends on the solutions of the others. In the context of diatom-atom collisions, these equations account for rotational and vibrational states of the diatom and the relative motion of the colliding partners.

Coupled equations are derived from time-independent Schrodinger equation, by expanding the scattering wave function in the chosen basis, $|\gamma \rangle$

$$\begin{equation} \label{eq:schrodinger} | \Psi \rangle = R^{-1} \sum_{\gamma} f_{\gamma} (R) |\gamma \rangle \end{equation}$$ which leads to a set of coupled differential equations on the expansion coefficients, $f_{\gamma} (R)$. The basis states $|\gamma \rangle$ define the collision channels. In the total angular momentum ($J$) representation, the coupled equations are block-diagonal with respect to total angular momentum and parity: $$\begin{equation} \label{eq:CC} \frac{d^{2}}{dR^{2}} {f}^{Jp}_{\gamma} (R) = \sum_{\gamma'} {W}^{Jp}_{\gamma, \gamma'}(R) {f}^{Jp}_{\gamma'} (R). \end{equation}$$

Here, $W$ is the coupling matrix $$\begin{equation} \label{eq:coupling_matrix} {W}^{Jp}_{\gamma, \gamma'}(R) = 2\mu {V}^{Jp}_{\gamma, \gamma'}(R) + \frac{1}{R^{2}}\mathbf{L}^{2\,Jp}_{\gamma, \gamma'} - \delta_{\gamma, \gamma'} k_{\gamma}^{2}, \end{equation}$$ which involves the contribution from the atom - molecule interaction energy, ${V}^{Jp}_{\gamma, \gamma'}(R)$, relative motion of atom with respect to the molecule, quantified by the square relative angular momentum operator, $\mathbf{L}^{2\,Jp}_{\gamma, \gamma'}$, and relatvie kinetic energy of the colliding pair, expressed using wavevector, $k_{\gamma} = \sqrt{2\mu(E - E_{\mathrm{mol}})}$. $\mu$ is the reduced mass of the atom - molecule system, $E$ is the total energy and $E_{\mathrm{mol}}$ is the internal (rovibrational) energy of the molecule.