Semiclassical approximation

A semiclassical approximation (literally semi-classical approximation ) in quantum physics stands for an approximation to a system in which the lowest quantum mechanical correction of the classical treatment of the system is considered; It is therefore meant that this approximation or correction is still relatively close to the classical treatment of the system compared to other possible corrections which are further away from the classical treatment.

The term is used in two different ways:

• only part of the system is considered quantum mechanically, while external fields are described as constant or as classically changing. The outer field can e.g. B. be an electromagnetic field or the gravitational field , for example the quantum field theory of particles in the classical Einstein's general theory of relativity curved but classical spacetime .
• the system is described by a series of perturbations , usually with an expansion according to powers of Planck's constant . Classical physics results in the power ; the first nontrivial approximation to the power is called the semiclassical approximation, an example of this is the WKB approximation of quantum mechanics. In quantum field theory , only Feynman diagrams with a maximum of one closed loop are taken into account in the semiclassical approximation (this corresponds precisely to the powers of Planck's constant given above). ${\displaystyle \hbar }$ ${\displaystyle 0}$ ${\displaystyle 1}$

Semiclassical approximation in perturbation calculation according to ℏ

The transition from quantum mechanics to classical mechanics should take place in the limit value . In the path integral formulation in the time-space points each with the over all paths between two probability amplitude (with the effect , the Lagrangian is summed), which leads to strong singular behavior at . The main contribution, however, comes from paths close to the classical solution, for which the variation in the effect when the path is varied in the spatial space , indicated by the position variable , is minimal (solutions to the Lagrangian equations ). ${\displaystyle \hbar \rightarrow 0}$ ${\displaystyle \exp {\left(i{\frac {S}{\hbar }}\right)}}$ ${\displaystyle S=\int L\,dt}$ ${\displaystyle L}$ ${\displaystyle \hbar \rightarrow 0}$ ${\displaystyle \delta S}$ ${\displaystyle \delta q}$ ${\displaystyle q}$

In the Hamilton-Jacobi formalism , the classical one satisfies the Hamilton-Jacobi equation for a non-relativistic one - particle problem with potential and mass : ${\displaystyle S}$ ${\displaystyle V}$ ${\displaystyle m}$

${\displaystyle {\frac {\partial S}{\partial t}}+{\frac {1}{2m}}{\left({\frac {\partial S}{\partial q}}\right)}^{2}+V(q)=0}$

If one takes the following approach in the Schrödinger equation :

${\displaystyle \Psi =\exp {\left(i{\frac {S(q,t)}{\hbar }}\right)}=\cos {\frac {S(q,t)}{\hbar }}+i\cdot \sin {\frac {S(q,t)}{\hbar }}}$

With

• Wave function ${\displaystyle \Psi }$
• location- and time-dependent complex phase whose real part corresponds to the frequency and whose imaginary part corresponds to damping ${\displaystyle S(q,t)}$
• imaginary unit , ${\displaystyle i}$

one obtains in quantum mechanics (see WKB method) the equation:

${\displaystyle {\frac {\partial S}{\partial t}}+{\frac {1}{2m}}{\left({\frac {\partial S}{\partial q}}\right)}^{2}+V(q)=-{\frac {\hbar }{i}}\cdot {\frac {1}{2m}}{\frac {\partial ^{2}S}{\partial ^{2}q}}}$

So you have the classical Hamilton-Jacobi equation on the left and a quantum mechanical diffusion term on the right , which also vanishes and is also complex. ${\displaystyle \hbar \rightarrow 0}$

If you develop according to : ${\displaystyle S}$ ${\displaystyle {\frac {\hbar }{i}}}$

${\displaystyle S=S_{0}+\left({\frac {\hbar }{i}}\right)S_{1}+{\left({\frac {\hbar }{i}}\right)}^{2}S_{2}+\dots }$

with the classical effect , a good approximation is if: ${\displaystyle S_{0}}$ ${\displaystyle \Psi =\exp {\left(i{\frac {S_{0}}{\hbar }}\right)}}$

${\displaystyle {\left|{\frac {\partial S}{\partial q}}\right|}^{2}\gg \hbar \left|{\frac {\partial ^{2}S}{\partial ^{2}q}}\right|}$

is or with the local (location-dependent) De Broglie wavelength , defined by : ${\displaystyle \lambda (q)}$ ${\displaystyle p(q)={\frac {\hbar }{\lambda (q)}}={\frac {\partial S}{\partial q}}}$

${\displaystyle \left|{\frac {\partial \lambda }{\partial q}}\right|\ll 1}$

New interest arose in the semi-classical approximation of quantum mechanics in the theory of quantum chaos from the 1970s ( Michael Berry , Martin Gutzwiller and others). In the case of classically chaotic systems, the chaotic behavior is actually suppressed in the quantum mechanical version if one considers isolated systems (the energy levels are discrete ). The highly singular transition , however , leads to quantum chaos in non-isolated systems, even if they only interact weakly with the environment ( decoherence ). B. is reflected in the statistics of the energy levels for highly excited states. ${\displaystyle \hbar \rightarrow 0}$

literature

• Michael Berry, KE Mount: Semiclassical approximations in wave mechanics, Reports Progress Physics, 35, 1972, 315–397, online, Berry website
• M. Berry, Semiclassical Mechanics of regular and irregular motion, in: G. Iooss, RHG Helleman, R. Stora (eds.): Les Houches Lecture Series Session XXXVI, North Holland, Amsterdam, 1983, pp. 171-271.
• M. Gutzwiller: Chaos in classical and quantum mechanics, Springer 1990
• M. Gutzwiller: Resource Letter ICQM-1: The Interplay between Classical and Quantum Mechanics, American Journal of Physics, Volume 66, 1998, p. 304 (literature review)

As well as many textbooks on path integrals such as von Hagen Kleinert (Path Integrals, 2nd edition, World Scientific 1995).

Individual evidence

1. ↑ Illustration based on Kurt Gottfried, Yan: Quantum Mechanics, Springer 2003, p. 102ff
2. ↑ Berry, Chaos and the semiclassical limit of quantum mechanics, 2001, pdf