Il Nuovo Cimento A (1971-1996)

, Volume 112, Issue 10, pp 1195–1228 | Cite as

Relations between low-lying quantum wave functions and solutions of the Hamilton-Jacobi equation

  • R. Friedberg
  • T. D. Lee
  • W. Q. Zhao


We discuss a new relation between the low-lying Schrödinger wave function of a particle in a one-dimensional potential V andthe solution of the corresponding Hamilton-Jacobi equation with —V as its potential. The functionV is ≥ 0, andcan have several minima (V = 0). We assume the problem to be characterized by a small anharmonicity parameterg-1 anda much smaller quantum tunneling parameter ɛ between these different minima. Expanding either the wave function or its energy as a formal double power series ing-1 and ɛ, we show how the coefficients ofg-mɛn in such an expansion can be expressedin terms of definite integrals, with leading-order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potentialV =12g2(x2 -a2)2.

PACS 11.10.Ef

Lagrangian andHamiltonian approach 

PACS 03.65.Ge

Solutions of wave equations: boundstates 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Balavin A. A., Polyakov A. M., Schwartz A. S. andTyupkin Yu S.,Phys. Lett. B,59 (1975) 85.CrossRefMathSciNetADSGoogle Scholar
  2. [2]
    ’tHooft G.,Phys. Rev. Lett.,37 (1979) 8.ADSGoogle Scholar

Copyright information

© Società Italiana di Fisica 1999

Authors and Affiliations

  • R. Friedberg
    • 1
  • T. D. Lee
    • 1
    • 2
    • 3
  • W. Q. Zhao
    • 1
    • 4
  1. 1.Physics DepartmentColumbia UniversityNew YorkUSA
  2. 2.China Center of Advanced Science and Technology (CCAST)World LaboratoryBeijingChina
  3. 3.RIKEN BNL Research Center (RBRC)Brookhaven National Lab.UptonUSA
  4. 4.Institute of High Energy PhysicsAcademia SinicaBeijingChina

Personalised recommendations