There are few systems where the quantum mechanical equations can be ex-actly solved and the spectrum and eigenfunctions explicitly calculated. In fact, there is a whole branch of the mathematical physics dedicated to extend our knowledge about the class of exactly solvable systems [5]. Methods of ap-proximation are required and have been available since the birth of quantum mechanics.
Following Berry and Mount [6], we classify the methods of approximation in quantum mechanics into three broad types:
• perturbation theory, where the quantities of interest are expressed as an infinite (and often divergent) expansion in some small parameter,
• variational methods, where the best approximate solution is selected out from a given set of trial functions, and
• semiclassical techniques, where the quantum mechanical quantities are expressed as asymptotic series in the (effective) Planck’s constant.
The domain of validity of the semiclassical methods is bounded by the domain of validity of the stationary phase approximation involved in the derivation of the semiclassical approximation to the propagator [7]. Without going into details, usually such domain is given by the condition
Scl
= ef f 1 (1.6)
whereScl is an -independent characteristic classical action (a rough estimate forScl ishpiLwithhpithe average momenta for given energy andLthe linear system size). Except for pathological examples, this condition is achieved in the regime of high energies and/or high quantum numbers, a domain that we will refer to in the following as the semiclassical regime.
At this point a comment about semantics is in order. Even when the early attempts to construct a theory of atomic spectra were rooted into classical mechanics (Bohr’s quantization rules), the first use of the classical dynamics as an approximation to the real quantum evolution in the framework of the modern theory was formalized by Ehrenfest [1]. The Ehrenfest theorem can be summarized by stating that for a wave packet the equations of motion of the expectation values of position and momenta are given by the classical equations of Hamilton. Pictorially, such quantum-classical correspondence breaks down at the time quantum fluctuations around the average behavior are large enough to produce interference among different parts of the wave packet (the so-called Ehrenfest time). Modern semiclassical techniques are, however, based on a representation of the quantum propagator which takes into account interference effects, and this approximation goes far beyond the limitations of the simple quantum-classical correspondence encoded in the Ehrenfest theorem.
Sadly, this misconception about semiclassical techniques, considering them more as a naive use of classical mechanics rather than as a full quantum me-chanical scheme, has been taken as granted for years and even in very respect-ful textbooks [8]. All along this work we will stress the fact that semiclassical methods provide a consistent scheme to calculate quantum mechanical quan-tities by means of classical information only, but incorporating interference effects.
In order to stress this point we can consider the following question. Since the semiclassical regime is defined by the existence of a small parameter, namely the effective Planck’s constant ef f, are then the semiclassical ex-pressions perturbative in ef f? The answer is no. The reason is that, as can be easily seen, the dependence on ef f in the quantum propagator is non analytical. In fact, the value ef f = 0 is an essential singularity and then its vicinity can not been studied using any kind of finite-order perturbative treatment. The power of the semiclassical approximation lies in the fact that such a singular behavior, the one being responsible for the interference ef-fects, is respected when the main approximation tool, the stationary phase approximation, is used. After the singularity is properly taken into account,
what is left can be treated with standard perturbative techniques.
After this short turn into semantics, we come back now to the formalism of the semiclassical methods. Historically, the first attempt to deal with the singular character of the ef f → 0 limit was proposed by Wentzel, Kramers and Brillouin for one-dimensional problems and is the well known WKB ap-proximation [1]. Shortly after, Van Vleck tried to generalize the method to deal with multi-dimensional systems where the Schr¨odinger equation is non-separable (the separable case is formally identical to a collection of one-dimensional problems). The so-called Van Vleck propagator faced two prob-lems [7]. First, it is divergent at the points where the classical trajectories have turning points, making it valid only for extremely short times. This alone was not a reason to make the approach useless, since one can always consider the propagator far away form the turning points, as it was already in use within the WKB method. The second problem was that there was no known way to connect the different solutions corresponding to classical paths before and after the turning points. This is the so-called connection problem and it was responsible for putting the Van Vleck propagator into oblivion for years.
This was the state of the affair when in a series of classic papers, Gutzwiller [9] successfully applied the method of stationary phase approximation to the Feynman propagator. The divergences at the turning points were still there, but the connection problem was solved by using an extension of classical me-chanics dealing with the behavior of non-classical paths around the classical ones. Morse theory [7] finally gives the recipes to add suitable phases to the semiclassical expressions and connect correctly the regions before and after the divergences.
The result of this analysis is the semiclassical approximation to the prop-agator, or simplysemiclassical propagatorgiven as a sum over classical paths,
Ksc(~ri, ti;~rf, tf) =X
p
Ap(~ri, ti;~rf, tf)eiRp(~ri,ti;~rf,tf)+iupπ4. (1.7) After a Fourier transform to the energy domain one obtains the most impor-tant result of the semiclassical analysis, the semiclassical (Gutzwiller) Green function [10]:
Gsc(~ri, ~rf, e) =X
p
q
|Dp(~ri, ti, e)|eiSp(~ri,~rf,e)+iνpπ4, (1.8)
where the sum runs over all classical pathspjoining~ri and~rf at given energy e,Sp(~ri, ~rf, e) =R
p~p.d~ris the corresponding action,Dp(~ri, ti, e) an amplitude depending on the stability properties of the trajectory, andνp is a topological index that solves the connection problem.
It is impossible to overestimate the importance of this expression and the huge amount of understanding and developments it has produced. In the semiclassical regime, all information about the quantum system can be recovered using this sum, including interference effects, in terms of pure classical information encoded in the action and stability properties of the classical trajectories.