Stationary and Quasi-Classical Approximations

It is interesting to know when to expect the wave nature of a particle propagation to be well approximated by the properties of the trajectory of a classical particle. For the propagation of light, say, the wavelength X is much smaller than the characteristic physical size L of the system. In this case, the diffraction phenomena are unimportant, and we can employ geometric optics. Similar results are expected in quantum mechanics when the wavelength X is short enough compared to the typical length L over which the potential varies. We can then expect a wave packet with size within X and L, with the motion approximated by that of a classical particle.

Usually, whenever there is an integral expression for a given quantity, it is often easier to employ approximate methods compared to a differential equation. Good examples include the perturbative expansion and the steepest descent method. We could equally employ change of variables to simplify the problem. Most of the quantum mechanical systems with the potential, at most quadratic in the coordinate are exactly solvable. However, in most cases of interest, the potential is more complicated and apart from a few exceptions an exact calculation of the path integral may be impossible. Such systems may be solvable only via approximate schemes. We will limit ourselves to the most important approximations valid when the quantum fluctuations are small or, equivalently, when the actions involved are large compared to Planck’s constant, ft:

It is simply a matter of convenience representing a path by a classical path and fluctuations around it. It is really immaterial how to express a path satisfying the boundary conditions for the exactly solvable problem of a driven harmonic oscillator, say. For the quasi-classical approximation, however, it is critical to expand around the path leading to the dominant contribution, i.e. the classical path. From a mathematical approach, we have to evaluate a path integral over exp jysj for small ft and this can be done

systematically by the stationary phase method where the exponent has to be expanded about the extrema of the action S.