Skip to main content
SpringerLink
Log in

Perturbation Theory, Semiclassical

  • Reference work entry
Encyclopedia of Complexity and Systems Science

  • 157 Accesses

Definition of the Subject

Several kinds of perturbation methods are commonly used in quantum mechanics Perturbation Theory and Molecular Dynamics, Perturbation Theory in Quantum Mechanics, Perturbation Theory. This chapter deals with semiclassical approximations where expressions for energy levels and wave functions are obtained in the limiting case of smallvalues of Planck's constant.

We emphasize that wave functions are highly singular as the parameter ? goes to zero. In fact the semiclassical limit is a singularperturbation problem; namely, Eq. (1) suffers a reduction of order setting ? equal tozero and the resulting equation is not a differential equation and does not give the classical limit correctly. Therefore, ordinary perturbationmethods, which usually give energy levels and wave-functions as convergent power series of the small parameter, cannot be applied.

The main goal of semiclassical methods consists ofobtaining asymptotic series, in the limit of small ?, for the...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Abbreviations

Agmon metric :

In the classically forbidden region where the potential energy \( { V(x) } \) is larger than the total energy E, i.?e. \( { V(x) > E } \), we introduce a notion of distance based on the Agmon metric defined as \( { [ V(x) - E ] \text{d} x^2 } \), where \( { \text{d} x^2 } \) is the usual Riemann metric. We emphasize that such an Agmon metric is the “semiclassical” equivalent of the “classical” Jacobi metric \( { [ E - V(x) ] \text{d} x^2 } \) introduced in classical mechanics in the classically permitted region where \( { V(x) < E } \).

Asymptotic series:

The notion of asymptotic series goes back to Poincaré. We say that a formal power series \( { \sum_r a_r z^{-r} } \) is taken to be the asymptotic power series for a function \( { f(z) } \), as \( { |z|\to \infty } \) in a given infinite sector \( { S= \{ z\in \mathbb{C} \colon \alpha \le \arg z \le \beta \} } \), if for any fixed N the remainder term

$$ R_N (z) = f(z) - \sum_{r=0}^{N-1} a_r z^{-r} $$

is such that \( { R_N (z) = \mathcal{O} (z^{-N}) } \), that is

$$ \left | R_N (z) \right | \le C_N \left | z^{-N} \right |\:,\enskip \forall z \in S\:, $$

for some positive constant C N depending on N.

Classically allowed and forbidden regions – turning points:

We distinguish two different regions: the region \( { V(x) < E } \) where the classical motion is allowed and the region \( { V(x) > E } \) where the classical motion is forbidden. The points that separate these two regions, that is such that \( { V(x)=E } \), will play a special role and they are named turning points.

Semiclassical limit :

Cornerstone of Quantum Mechanics is the time-dependent Schrödinger equation

$$ i \hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m} \Delta \psi + V(x) \psi\:, $$
(1)

where \( { V(x) } \) is assumed to be a real-valued function, usually it represents the potential energy, and m is the mass of the particle. The solution of this equation defines the density of probability \( { |\psi (x,t)|^2 } \) to find the particle in some region of space. The parameter ? in Eq. (1) is related to Planck's constant h

$$ \begin{aligned} \hbar = \frac{h}{2\pi } &= 1.0545 \times 10^{-27} \text{erg sec} \\ &= 6.5819 \times 10^{-22} \enskip \text{MeV sec}\:. \end{aligned} $$

According to the correspondence principle, when Planck's constant can be considered small with respect to the other parameters, such as masses and distances, then quantum theory approaches classical Newton theory. Thus, roughly speaking, we expect that classical mechanics is contained in quantum mechanics as a limiting form (i.?e. \( { \hbar \to 0 } \)). The limit of small ?, when compared with the other parameters, is the so-called semiclassical limit. We should emphasize that making Planck's constant small in Eq. (1) is a rather singular limit and many difficult mathematical problems occur.

Stokes lines :

There is some misunderstanding in the literature concerning the name of the curves \( { \Im \left[ \xi (z) \right]=0 } \), where \( { \Im (\xi) } \) denotes the imaginary part of

$$ \xi (z) = \frac{i}{\hbar} \int\limits_{x_E}^z \sqrt {E-V(q)} \text{d} q \:,\quad z\in \mathbb{C} \:, $$

and \( { x_E \in \mathbb{R} } \) is a turning point (i.?e. \( { V(x_E)=E } \)), the potential V is assumed to be an analytic function in the complex plane. As usually physicists do, and in agreement with Stokes' original treatment, we adopt here the convention to name as Stokes line any path coming from the turning point x E such that \( { \Im [ \xi (z) ]=0 } \); reserving the name of anti-Stokes lines, or regular lines, for the curves such that \( { \Re [ \xi (z) ]=0 } \), where \( { \Re (\xi) } \) denotes the real part of ? (we should emphasize that most mathematicians adopt the opposite rule, that is they call Stokes lines the paths such that \( { \Re [ \xi (z) ]=0 } \)).

Tunnel effect:

At a real (simple) turning point x E where \( { E=V(x) } \) (and \( { V^{\prime}(x) \neq 0 } \)), classical particles incident from an accessible region [where \( { E > V(x) } \)] reverse their velocity and return back; the adjacent region [where \( { E < V(x) } \)] is forbidden to these particles according to classical mechanics. In fact, quantum mechanically exponentially growing or damped waves can exist in forbidden regions and a quantum particle can pass through a potential barrier. This is the so-called tunnel effect.

WKB:

The WKB method, named after the contributions independently given by Wentzel, Kramers and Brillouin, consists of connecting the approximate solutions of the time independent Schrödinger equation across turning points.

Bibliography

  1. Ben Abdallah N, Pinaud O (2006) Multiscale simulation of transport in an openquantum system: Resonances ans WKB interpolation. J Comp Phys 213:288–310

    MathSciNet  ADS  MATH  Google Scholar 

  2. Berezin FA, Shubin MA (1991) The Schrödinger equation. Kluwer,Dordrecht

    MATH  Google Scholar 

  3. Berry MV, Mount KE (1972) Semiclassical approximation in wave mechanics. RepProg Phys 35:315–397

    ADS  Google Scholar 

  4. Bonnaillie-Noël V, Nier F, Patel Y (2006) Computing the steady states foran asymptotic model of quantum transport in resonant heterostructures. J Comp Phys 219:644–670

    Google Scholar 

  5. Claviere P, Jona Lasinio G (1986) Instability of tunneling and the concept ofmolecular structure in quantum mechanics: The case of pyramidal molecules and the enantiomer problem. Phys Rev A33:2245–2253

    ADS  Google Scholar 

  6. Dimassi M, Sjöstrand J (1999) Spectral asymptotics in the semiclassicallimit. In: London Math Soc Lecture Note Series 268. Cambridge University Press, Cambridge

    Google Scholar 

  7. Dingle RB (1973) Asymptotic expansion: Their derivation andinterpretation. Academic, London

    MATH  Google Scholar 

  8. Egorov YV (1971) Canonical transformation of pseudo-differentialoperators. Trans Moscow Math Soc 24:1–24

    Google Scholar 

  9. Folland G (1988) Harmonic analysis in phase space. Princeton University Press,Princeton

    Google Scholar 

  10. Fröman N, Fröman PO (1965) JWKB approximation. North Holland,Amsterdam

    Google Scholar 

  11. Fröman N, Fröman PO (2002) Physical problems solved by the phaseintegral methods. Cambridge University Press, Cambridge

    Google Scholar 

  12. Graffi S, Grecchi V, Jona-Lasinio G (1984) Tunneling instability viaperturbation theory. J Phys A: Math Gen 17:2935–2944

    MathSciNet  ADS  MATH  Google Scholar 

  13. Grecchi V, Martinez A, Sacchetti A (1996) Splitting instability: The unstabledouble wells. J Phys A: Math Gen 29:4561–4587

    MathSciNet  ADS  MATH  Google Scholar 

  14. Grecchi V, Martinez A, Sacchetti A (2002) Destruction of the beating effectfor a non-linear Schrödinger equation. Comm Math Phys 227:191–209

    MathSciNet  ADS  MATH  Google Scholar 

  15. Grigis B, Sjöstrand J (1994) Microlocal analysis for differentialoperators. An introduction. In: London Math. Soc. Lecture Note Series 196. Cambridge University Press, Cambridge

    Google Scholar 

  16. Harrell EM (1980) Double wells. Commun Math Phys75:239–261

    Google Scholar 

  17. Helffer B (1988) Semi-classical analysis for the Schrödingeroperator and applications. Lecture Notes in Mathematics 1336. Springer, Berlin

    MATH  Google Scholar 

  18. Helffer B, Sjöstrand J (1984) Multiple wells in the semiclassical limitI. Comm Part Diff Eq 9:337–408

    Google Scholar 

  19. Helffer B, Sjöstrand J (1986) Resonances en limitesemi-classique. Mém Soc Math France (N.S.) 24–25

    Google Scholar 

  20. Hislop P, Sigal IM (1996) Introduction to spectral theory. In: Appl Math Sci,vol. 113. Springer, New York

    Google Scholar 

  21. Landau LD, Lifshitz EM (1959) Quantum mechanics. Course of theoreticalphysics. Pergamon, Oxford

    Google Scholar 

  22. Martinez A (2002) An introduction to semiclassical and microlocalanalysis. Springer, New York

    MATH  Google Scholar 

  23. McHugh JAM (1971) An historical survey of ordinary linear differentialequations with a large parameter and turning points. Arch Hist Exact Sci 7:277–324

    MathSciNet  MATH  Google Scholar 

  24. Merzbacher E (1970) Quantum mechanics, 2nd edn. Wiley, NewYork

    Google Scholar 

  25. Olver FWJ (1974) Asymptotics and Special Functions. Academic, NewYork

    Google Scholar 

  26. Presilla C, Sjöstrand J (1996) Transport properties in resonant tunnelingheterostructures. J Math Phys 37:4816–4844

    Google Scholar 

  27. Raghavan S, Smerzi A, Fantoni S, Shenoy SR (1999) Coherent oscillationsbetwene two weakly coupled Bose–Einstein condensates: Josephson effects, p oscillations, and macroscopic quantum self-trapping. Phys Rev A59:620–633

    ADS  Google Scholar 

  28. Robert D (1987) Autour de l'Approximation Semiclassique. Birkhäuser,Basel

    Google Scholar 

  29. Robert D (1988) Semi-classical approximation in quantummechanics. A survey of old and recent mathematical results. Helv Phys Acta 71:44–116

    Google Scholar 

  30. Sacchetti A (2005) Nonlinear double well Schrödinger equations in thesemiclassical limit. J Stat Phys 119:1347–1382

    MathSciNet  ADS  MATH  Google Scholar 

  31. Simon B (1983) Semiclassical limit of low lying Eigenvalues I: Non degenerateminima. Ann H Poincaré 38:295–307

    MATH  Google Scholar 

  32. Simon B (1985) Semiclassical limit of low lying Eigenvalues IV: The flea ofthe elephant. J Funct Anal 63:123–136

    MathSciNet  MATH  Google Scholar 

  33. Voros A (1982) Spectre de l'Équation de Schrödinger et MéthodeBKW. Publications Mathmatiques d'Orsay 81.09

    Google Scholar 

  34. Wilkinson M, Hannay JH (1987) Multidimensional tunneling between excitedstates. Phys D: Nonlin Phenom 27:201–212

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Universitá di Modena e Reggio Emilia, Modena, Italy

    Andrea Sacchetti

Authors
  1. Andrea Sacchetti
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

Sacchetti, A. (2009). Perturbation Theory, Semiclassical. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_403

Download citation

Publish with us

Policies and ethics

Search

Navigation