Harmonic oscillators, quantum dynamics relaxation

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y -> 0 limit of the Kramers model we are dealing with energy relaxation of a classical anharmonic oscillator. One may justifiably question the use of Markovian classical dynamics in this part of the problem, and we will come to this issue later. For now we focus on the solution of the mathematical problem posed by the low friction limit of the Kramers problem. 

---