Dear Manoj,

the choice of the unperturbed Hamiltonian depends on the problem you are dealing with, in particular it also depends on whether you are considering time dependent or time independent Hamiltonians. The interaction hamiltonian can be exploited on the same foot of the free energy when you use the symmetric split decomposition. For further comments you can look at the paper (I believe it is on research gate) by Dattoli, Ottaviani, Torre and Vazquez Rivista del Nuovo Cimento 1997.

Regards

Pino

H=H0+H1

In the language of perturbation theory, Ho is known as the unperturbed Hamiltonian and describes a system of interest which has an exact solution  H1 is known as the perturbation, and it often describes the system that is affected by an external field, and it is in most cases is not depend on time

 Dattoli and abu sir,

for two level atom interacting with the single mode field will the free part must commute with the interaction part?

When you are dealing with perturbations you may have different problems associated with non commutativity.

In the case of time depending Hamiltonians you should also consider the problems associated with the commutation of the Hamiltonian with itself at different times, in this case the problem is further complicated by time ordering.

In general if the interaction and the free parts commute each other and the free part has a known set of eigenfunctions and eigenvalues the solution of the problem with the total hamiltonian does not present any difficulty. In practice we have no perturbation at all.

When the two pieces of the Hamiltonian do not commute, then you should start to study a strategy, namely what solutions are you looking for? time dependent solutions, eigenvalues perturbations...?

Regarding the specific problem of two level system is the external field classical or should it be treated as quantized?

If the external field is classical (time dependent or not) exact solution exist (according to the classical work by Feynman et al. The problem is reduced to a Bloch type rotation).

For a quantized system the problem is slightly more complicated but can also be treated exactly.

The best

Pino

The best

G. Dattoli