Tunnel splittings for one-dimensional potential wells revisited

American Journal of Physics

Volumn 68, Issue 5, 2000, Pages 430-437

  Garg, Anupam ^a  

^a NORTHWESTERN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034386550     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19458     Document Type: Article

References (39)

1. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1977), 3rd ed., Chap. VII, Sec. 50, Problem 3.
2. Indeed, some of the confusion about the WKB results is perhaps due to the great authority of Ref. 1. For example, many workers in the field of macroscopic quantum tunneling, including the author, have often been puzzled and frustrated that the final formula in this book does not yield the same ground state splitting as the instanton approach for the quartic double well. An erudite colleague who studies chaos assisted tunneling once stated to the author that "WKB is known to be wrong for the ground state splitting."
3. J. S. Langer, "Theory of the Condensation Point," Ann. Phys. (N.Y.) 41, 108-157 (1967).
4. S. Coleman, "Fate of the False Vacuum: Semiclassical Theory," Phys. Rev. D 15, 2929-2936 (1977); Aspects of Symmetry (Cambridge U.P., Cambridge, 1985), Chap. 7.
5. S. Coleman, "Fate of the False Vacuum: Semiclassical Theory," Phys. Rev. D 15, 2929-2936 (1977); Aspects of Symmetry (Cambridge U.P., Cambridge, 1985), Chap. 7.
6. C. M. Bender and T. T. Wu, "Anharmonic Oscillator," Phys. Rev. 184, 1231-1260 (1969); "Large Order Behavior of Perturbation Theory," Phys. Rev. Lett. 27, 461-465 (1971); "Anharmonic Oscillator II. A Study of Perturbation Theory in Large Order," Phys. Rev. D 7, 1620-1636 (1973).
7. C. M. Bender and T. T. Wu, "Anharmonic Oscillator," Phys. Rev. 184, 1231-1260 (1969); "Large Order Behavior of Perturbation Theory," Phys. Rev. Lett. 27, 461-465 (1971); "Anharmonic Oscillator II. A Study of Perturbation Theory in Large Order," Phys. Rev. D 7, 1620-1636 (1973).
8. C. M. Bender and T. T. Wu, "Anharmonic Oscillator," Phys. Rev. 184, 1231-1260 (1969); "Large Order Behavior of Perturbation Theory," Phys. Rev. Lett. 27, 461-465 (1971); "Anharmonic Oscillator II. A Study of Perturbation Theory in Large Order," Phys. Rev. D 7, 1620-1636 (1973).
9. E. Gildener and A. Patrascioiu, "Pseudoparticle Contributions to the Energy Spectrum of a One-Dimensional System," Phys. Rev. D 16, 423-430 (1977).
10. E. Brézin, G. Parisi, and J. Zinn-Justin, "Perturbation Theory at Large Orders for a Potential with Degenerate Minima," Phys. Rev. D 16, 408-412 (1977).
11. E. B. Bogomolny and V. A. Fateyev, ®Large Order Calculations in Gauge Theories," Phys. Lett. B 71, 93-96 (1977).
12. E. B. Bogomolny, "Calculation of Instanton-Antiinstanton Contributions in Quantum Mechanics," Phys. Lett. B 91, 431-435 (1980).
13. J. Zinn-Justin, "Multi-instanton Contributions in Quantum Mechanics," Nucl. Phys. B 192, 125-140 (1981); "Multi-instanton Contributions in Quantum Mechanics. II," Nucl. Phys. B 218, 333-348 (1983).
14. J. Zinn-Justin, "Multi-instanton Contributions in Quantum Mechanics," Nucl. Phys. B 192, 125-140 (1981); "Multi-instanton Contributions in Quantum Mechanics. II," Nucl. Phys. B 218, 333-348 (1983).
15. B. R. Holstein, "Semiclassical Treatment of the Double Well," Am. J. Phys. 56, 338-345 (1988).
16. B. R. Holstein and A. R. Swift, "Path Integrals and the WKB Approximation," Am. J. Phys. 50, 829-832 (1982).
17. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978). See especially Chaps. 3 and 10.
18. M. V. Berry and K. E. Mount, Rep. Prog. Phys. 35, 315-397 (1972).
19. We should note that the complete discussion of quadratic turning point formulas is far more complex than needed for our purpose. For us, the discussion in Sec. II is sufficient, and only draws upon knowledge of harmonic oscillator eigenfunctions.
20. Although we have not seen this formula elsewhere it seems very likely that the sophisticated mathematical physicists who have studied the relationship between WKB methods, instantons, and large order perturbation theory (Refs. 4-10) would find it familiar or obvious. Hence, we hereby explicitly disavow any claims to originality.
21. R. D. Carlitz and D. A. Nicole, "Classical Paths and Quantum Mechanics," Ann. Phys. (N.Y.) 164. 411-462 (1985).
22. -1/2.
23. W. H. Furry, "Two Notes on Phase-Integral Methods," Phys. Rev. 71, 360-371 (1947).
24. 0 is replaced by the mean energy of the pair.
25. C. Herring, "Critique of the Heitler-London Method of Calculating Spin Couplings at Large Distances," Rev. Mod. Phys. 34, 631-645 (1962).
26. C. Herring and M. Flicker, "Asymptotic Exchange Coupling of Two Hydrogen Atoms," Phys. Rev. A 134, 362-366 (1964).
27. S is defined in the usual way, i.e., such that the eigenvalue of S-S is S(S+1).
28. M. Enz and R. Schilling, "Spin Tunnelling in the Semiclassical Limit," J. Phys. C 19, 1765-1770 (1986); "Magnetic Field Dependence of the Tunnelling Splitting of Quantum Spins," J. Phys. C 19, L711-L715 (1986).
29. M. Enz and R. Schilling, "Spin Tunnelling in the Semiclassical Limit," J. Phys. C 19, 1765-1770 (1986); "Magnetic Field Dependence of the Tunnelling Splitting of Quantum Spins," J. Phys. C 19, L711-L715 (1986).
30. J. L. van Hemmen and A. Süto, "Tunneling of Quantum Spins," Europhys. Lett. 1, 481-490 (1986); "Tunneling of Quantum Spins," Physica B 141, 37-75 (1986).
31. J. L. van Hemmen and A. Süto, "Tunneling of Quantum Spins," Europhys. Lett. 1, 481-490 (1986); "Tunneling of Quantum Spins," Physica B 141, 37-75 (1986).
32. V. I. Belinicher, C. Providencia, and J. da Providencia, "Instanton Picture of the Spin Tunneling in the Lipkin-Meshov-Glick Model," J. Phys. A 30, 5633-5643 (1997).
33. A. Garg, "Application of the Discrete Wentzel-Kramers-Brillouin Method to Spin Tunneling," J. Math. Phys. 39, 5166-5179 (1998).
34. G. Scharf, W. F. Wreszinski, and J. L. van Hemmen, "Tunnelling of a Large Spin: Mapping onto a Particle Problem," J. Phys. A 20, 4309-4319 (1987).
35. 2).
36. Amit Goswami, Quantum Mechanics (Brown, Dubuque, IA, 1992), Chap. 4, Problem 9.
37. Gordon Baym, Lectures on Quantum Mechanics (Benjamin-Cummings, Menlo Park, 1969). See Chap. 4, Problems 7 and 11.
38. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), 2nd ed., pp. 66-74.
39. David Park, Introduction to the Quantum Theory (McGraw-Hill, New York, 1974), 2nd ed. See pp. 116-120. Park uses connection formulas to obtain the Gamow factor, but appeals to a heuristic argument for the prefactor. This is good physics in its own right, but the prefactor is precisely the object we are after in this article.