From Classical to Quantum

So far everything is still completely classical. We obtain an effective Hamiltonian for $N$ molecules by writing

$${\mathcal H}_{\rm eff} \equiv {\mathcal H}_0 + V,\quad {\mathcal H}_0=\sum_n H_0^{(n)}$$ (2.10)

$$V = \frac{1}{2} \sum_{nn'}E_{nn'}^{\rm d-d},$$ (2.11)

with $H_0^{(n)}$ the individual Hamiltonian of molecule $n$, and $V$ the interaction between all the molecules.

From the Hamiltonian ${\mathcal H}_{\rm eff}$, Eq. (IX.2.11), a semi-classical theory can be constructed as follows: for given internal states of the molecules, we derive effective classical interaction potentials that eventually lead to a classical dynamics of the molecule positions ${\bf R}_n$ and momenta $\hat{{\bf P}}_n$ in the phase space of the $(\hat{{\bf P}}_n,{\bf R}_n)$ of the molecules.

If $V$ is regarded as a small perturbation, the interaction potentials are obtained most easily by calculating the $T$-matrix of the system of $N$ molecules with respect to the decomposition Eq. (IX.2.11).