Semiclassical Description Basic Equations

In the semiclassical description, the radiation incident upon an atom is described by a classical electromagnetic (EM) plane wave 

The atom, on the other hand, is treated quantum-mechanically. In order to simplify the equations, we restrict ourselves to a two-level system with the eigenstates Ea and Eb (Fig. 2.24). 

The radiation field causes transitions in the atom. This means that the eigenfunctions of the atom become time-dependent. The general solution, t) of the time-dependent Schrodinger equation 

The spatial parts u (r) of these eigenfunctions are solutions of the time-independent Schrodinger equation 

The coefficients a(t) and 6(t) are the time-dependent probability amplitudes of the atomic states a) and fe). This means that the value a(t)P gives the probability of finding the system in level a) at timet. Obviously, the relation a(t)p-l- ( (t)p = 1 must hold at aU times t, if decay into other levels is neglected. 

Until now laser spectroscopy was performed in spectral regions where the wavelength A was large compared to the diameter d of an atom (e.g., in the visible spectrum X is 500 nm, but d is only about 0.5 nm). For d, the phase of the EM wave does not change much within the volume of an atom because kz = (2it/X)z 1 for z d. We can therefore neglect the spatial derivatives of the field amplitude (dipole approximation). In a coordinate system with its origin in the center of the atom, we can assume 0 within the 

The general solution V r,t) of the time-dependent SchrSdinger equation 

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