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Assumptions and Approximations

Before proceeding any further, let us make a few simplifying assumptions about the experimental situation. Assume that the electric dipole moment and the electric field are parallel and that the amplitude and frequency of the electric field are real. The second assumption limits the radiation to a linearly polarized form. may now be redefined as a real quantity

Notice that is directly proportional to the amplitude of the electric field .

Most scholars would simplify eq:T5 and eq:T6 as well. The first step is to move and into a new frame of reference via a simple transformation

This transformation defines a new system called the rotating frame which is circulating at the electromagnetic field's frequency of oscillation. In the rotating frame, eq:T5 and eq:T6 become

subtracted from is the detuning and is represented by . The detuning is a measure of how close and are, i.e., how close is to the resonant frequency, . If we assume that is small and that has the same order of magnitude, then the terms in eq:T8 and eq:T9 fall into different groups. One group has phasor terms that oscillate at , while the other will oscillate at a slower rate defined by the order of magnitude of and .

If a theorist wishes to study systems which have small (electric field amplitudes) and small , then these fast--moving terms may be neglected because over a number of periods they average out. If the fast oscillating terms are removed, both eq:T8 and eq:T9 significantly simplify:

This approximation is used commonly in physics. It is called the Rotating Wave Approximation (RWA). Often this approximation is quite good; however, it is an approximation. There are many situations when one may desire a large or a which is far from . In both of these situations, one must not make this approximation. We shall reexamine the meaning of the RWA in a later section.

Next: The Density Matrix Up: Theory Previous: Schrödinger Derivation of

Andy Antonelli
Wed May 17 14:34:24 EDT 1995