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''quasi-classical approximation''

An asymptotic representation, that is, the asymptotics as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839901.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839902.png" /> is Planck's constant), of the solutions of the equations of quantum mechanics. The [[Schrödinger equation|Schrödinger equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839903.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

describes the motion of a quantum-mechanical particle in a potential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839904.png" />. The motion of a classical particle is described by the Hamilton–Jacobi equation (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

or by the [[Hamiltonian system|Hamiltonian system]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

The [[Cauchy problem|Cauchy problem]] for the Schrödinger equation,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839907.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

is compared with the Cauchy problem for the system (3):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839908.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

(here the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s0839909.png" /> are smooth, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399010.png" /> are real-valued and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399011.png" /> is of compact support). The asymptotics of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399012.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399014.png" />, and for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399015.png" /> have the form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399017.png" /> is the solution of (2) with Cauchy data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399018.png" /> (the classical action), while

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399021.png" /> is the solution of (3), (5). The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399022.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399023.png" /> are defined from the recurrent system of transport equations (these are ordinary differential equations along the trajectories of the system (3)), so that all terms of the asymptotics are expressed in the terminology of classical mechanics. The Bohr correspondence principle asserts: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399024.png" /> tends to zero, then the quantum laws must go over to the classical laws. The method of seeking asymptotics in the form (6) was proposed by P. Debye and has been widely applied in quantum mechanics.

The asymptotic solution of the problem (1), (4) in the large (that is, for any finite time) is constructed via the canonical operator of V.P. Maslov [[#References|[3]]]. The Cauchy data (5) fill out an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399025.png" />-dimensional Lagrangian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399026.png" /> in the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399027.png" />. Its translates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399028.png" /> along the trajectories of the system (3) are also Lagrangian manifolds; their union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399029.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399030.png" />-dimensional Lagrangian manifold in the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399031.png" /> with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399032.png" />. For the canonical operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399033.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399034.png" />, the following commutation relation holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399036.png" /> is the derivative along the system (3) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399037.png" /> is the Schrödinger operator. The asymptotics of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399038.png" /> in the large are given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399040.png" /> are defined from the Cauchy data (4) by means of the transport equations and can be expressed in terms of classical mechanics. At a non-focal point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399041.png" /> the asymptotics have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399042.png" /></td> </tr></table>

where the sum is taken over all rays arriving at this point, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399044.png" /> are the action and Jacobian for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399045.png" />-th ray and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399046.png" /> is the Morse index of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399047.png" />-th ray. For the stationary Schrödinger equation in the semi-classical approximation the problem of scattering has been studied, as well as the problem of the field of a point source, and the classical series (of Balmer type) of eigenvalues has been obtained.

Semi-classical approximations in the wide sense (synonyms: high-frequency asymptotics, short-wave approximations, approximations of geometrical optics, WKB-method, eikonal method, quasi-classical approximations in the wide sense) are the asymptotics of solutions of partial differential equations with real characteristics of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

as well as of differential and pseudo-differential equations. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399049.png" /> is a large parameter and the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399050.png" /> depends weakly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399051.png" />. Corresponding to (9) are the equations of classical mechanics, namely the Hamilton–Jacobi equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399052.png" /></td> </tr></table>

and the Hamiltonian system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083990/s08399054.png" />. The semi-classical approximation is constructed by means of the canonical operator corresponding to the Lagrangian manifolds that are invariant with respect to the dynamical system (10) and having a form similar to (8).

The semi-classical approximation is widely applied in modern physics in problems of the propagation of sound, elastic and electromagnetic waves, in non-relativistic and relativistic quantum mechanics and other questions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Brillouin,   "La théorie des quanta et l'atome de Bohr" , Blanchard  (1922)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "Quantum mechanics" , Pergamon  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.P. Maslov,   "Théorie des perturbations et méthodes asymptotiques" , Dunod  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.P. Maslov,   M.V. Fedoryuk,   "Quasi-classical approximation for the equations of quantum mechanics" , Reidel  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.A. Fok,   "Problems of diffraction and propagation of electromagnetic waves" , Moscow  (1970)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.M. Babich,   V.S. Buldyrev,   "Asymptotic methods in the diffraction of short waves" , Moscow  (1972)  (In Russian)  (Translation forthcoming: Springer)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.P. Maslov,   "Operational methods" , MIR  (1976)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Simon,   "Functional integration and quantum mechanics" , Acad. Press  (1979)</TD></TR></table>