I read that RWA in terms of quantum optics is used to find an approximate solution for time-dependent Schrodinger equation? What is the physical meaning of this technique? What or/and When do we need it? And more important: do we use RWA technique in terms of classical mechanics? i.e. Can we use RWA technique to get an approximate solution for "classical" time-dependent differential equation (equation describing classical dynamics )?
Rotating wave approximation is useful when the Hamiltonian of the quantum system has explicit time dependence, such that the system Hamiltonian at two arbitrary different times do not necessarily commute, as a result the Unitary time evolution operator associated cannot be expressed in a simple, closed form, the Dyson series expansion is employed in this situation to construct the zero, first, second and higher order terms of the operator.
Rotating wave approximation is used in NMR spectroscopy, here, the RF linearly oscillating field is visualized as consisting of two circularly oscillating fields, oscillating with the same frequency ( that of the linear field) but in opposite sense, in presence of this RF pulse field, the total NMR spin Hamiltonian has explicit time dependence and is not commuting at different times. By switching over to a rotating frame of reference rotating at the frequency of the Rf feild, one of the circular components becomes stationary in this reference frame and the other appears to oscillate twice as fast, therefore, its contribution to spin evolution is negligible, therefore, the time dependant Scrodinger equation in the rotating reference frame is associated with an Hamiltonian that effectively time independent and hence, the equation for time evolution can be expressed in closed, analytic form.
Dear Debopam, Thank you for your useful details.
Now I believe I understand little bit the physical meaning but want to make sure I am not missing some important mathematical aspect of the precise definition.
What is the mathematical meaning of RWA?
Can we use RWA to solve classical problems? For instance, Can we use RWA to get an approximate solution of nonlinear differential equations (NDEs) describing the dynamic of many body interacting systems?
Thanks.
Did you see this?
https://en.wikipedia.org/wiki/Rotating_wave_approximation
About other problems, the question is not whether the problem is cassical or quantum. Look at the mathematics developped in this Wikipedia article and see whether you can bring your other problem to a similar mathematical formulation.
As to many-body interacting systems, take into atention that the RWA is for objects in rapidly oscillating fields. So, the question is whether the interactions you speak of, are time-dependent fields oscillating with high frequences.
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