The harmonic oscillator propagator

American Journal of Physics

Volumn 66, Issue 7, 1998, Pages 583-589

  Holstein, Barry R ^a  

^a UNIVERSITY OF MASSACHUSETTS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032335820     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18910     Document Type: Article

References (29)

1. See, e.g., R. W. Robinett, "Quantum Mechanical Time-Development Operator for the Uniformly Accelerated Particle," Am. J. Phys. 64, 803-808 (1996); L. S. Brown and Y. Zhang, "Path Integral for the Motion of a Particle in Linear Potential," ibid. 62, 806-808 (1994); G. P. Arrighini, N. L. Durante, and C. Guidotti, "More on the Quantum Propagator of a Particle in a Linear Potential," ibid. 64, 1036-1041 (1996); B. R. Holstein, "The Linear Potential Propagator," ibid. 65, 414-418 (1997); P. Nardone, "Heisenberg Picture in Quantum Mechanics and Linear Evolutionary Systems," ibid. 61, 232-237 (1993).
2. See, e.g., R. W. Robinett, "Quantum Mechanical Time-Development Operator for the Uniformly Accelerated Particle," Am. J. Phys. 64, 803-808 (1996); L. S. Brown and Y. Zhang, "Path Integral for the Motion of a Particle in Linear Potential," ibid. 62, 806-808 (1994); G. P. Arrighini, N. L. Durante, and C. Guidotti, "More on the Quantum Propagator of a Particle in a Linear Potential," ibid. 64, 1036-1041 (1996); B. R. Holstein, "The Linear Potential Propagator," ibid. 65, 414-418 (1997); P. Nardone, "Heisenberg Picture in Quantum Mechanics and Linear Evolutionary Systems," ibid. 61, 232-237 (1993).
3. See, e.g., R. W. Robinett, "Quantum Mechanical Time-Development Operator for the Uniformly Accelerated Particle," Am. J. Phys. 64, 803-808 (1996); L. S. Brown and Y. Zhang, "Path Integral for the Motion of a Particle in Linear Potential," ibid. 62, 806-808 (1994); G. P. Arrighini, N. L. Durante, and C. Guidotti, "More on the Quantum Propagator of a Particle in a Linear Potential," ibid. 64, 1036-1041 (1996); B. R. Holstein, "The Linear Potential Propagator," ibid. 65, 414-418 (1997); P. Nardone, "Heisenberg Picture in Quantum Mechanics and Linear Evolutionary Systems," ibid. 61, 232-237 (1993).
4. See, e.g., R. W. Robinett, "Quantum Mechanical Time-Development Operator for the Uniformly Accelerated Particle," Am. J. Phys. 64, 803-808 (1996); L. S. Brown and Y. Zhang, "Path Integral for the Motion of a Particle in Linear Potential," ibid. 62, 806-808 (1994); G. P. Arrighini, N. L. Durante, and C. Guidotti, "More on the Quantum Propagator of a Particle in a Linear Potential," ibid. 64, 1036-1041 (1996); B. R. Holstein, "The Linear Potential Propagator," ibid. 65, 414-418 (1997); P. Nardone, "Heisenberg Picture in Quantum Mechanics and Linear Evolutionary Systems," ibid. 61, 232-237 (1993).
5. See, e.g., R. W. Robinett, "Quantum Mechanical Time-Development Operator for the Uniformly Accelerated Particle," Am. J. Phys. 64, 803-808 (1996); L. S. Brown and Y. Zhang, "Path Integral for the Motion of a Particle in Linear Potential," ibid. 62, 806-808 (1994); G. P. Arrighini, N. L. Durante, and C. Guidotti, "More on the Quantum Propagator of a Particle in a Linear Potential," ibid. 64, 1036-1041 (1996); B. R. Holstein, "The Linear Potential Propagator," ibid. 65, 414-418 (1997); P. Nardone, "Heisenberg Picture in Quantum Mechanics and Linear Evolutionary Systems," ibid. 61, 232-237 (1993).
6. See, e.g., R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
7. P. Nardone, Ref. 1.
8. D. S. Saxon, Elementary Quantum Mechanics (Holden-Day, San Francisco, 1968).
9. E. Merzbacher, private communication.
10. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (USGPO, Washington, DC, 1968, #4.3.89; reprinted by Dover Publications, New York).
11. B. R. Holstein, Topics in Advanced Quantum Mechanics (Addison-Wesley, Reading, MA, 1992); B. Felsager, Geometry, Particles and Fields (Odense U.P., Odense, 1981); J. V. Narlikar and T. Padmanabhan, Gravity, Gauge Theories and Quantum Cosmology (Reidel, New York, 1986).
12. B. R. Holstein, Topics in Advanced Quantum Mechanics (Addison-Wesley, Reading, MA, 1992); B. Felsager, Geometry, Particles and Fields (Odense U.P., Odense, 1981); J. V. Narlikar and T. Padmanabhan, Gravity, Gauge Theories and Quantum Cosmology (Reidel, New York, 1986).
13. B. R. Holstein, Topics in Advanced Quantum Mechanics (Addison-Wesley, Reading, MA, 1992); B. Felsager, Geometry, Particles and Fields (Odense U.P., Odense, 1981); J. V. Narlikar and T. Padmanabhan, Gravity, Gauge Theories and Quantum Cosmology (Reidel, New York, 1986).
14. R. P. Feynman and A. R. Hibbs, Ref. 2; M. Swanson, Path Integrals and Quantum Processes (Academic, New York, 1992); B. Felsager, Ref. 7. See also S. Coleman, Aspects of Symmetry (Cambridge U.P., New York, 1985), Chap. 7, for an ingenious derivation of the ratio of determinants.
15. R. P. Feynman and A. R. Hibbs, Ref. 2; M. Swanson, Path Integrals and Quantum Processes (Academic, New York, 1992); B. Felsager, Ref. 7. See also S. Coleman, Aspects of Symmetry (Cambridge U.P., New York, 1985), Chap. 7, for an ingenious derivation of the ratio of determinants.
16. R. P. Feynman and A. R. Hibbs, Ref. 2; M. Swanson, Path Integrals and Quantum Processes (Academic, New York, 1992); B. Felsager, Ref. 7. See also S. Coleman, Aspects of Symmetry (Cambridge U.P., New York, 1985), Chap. 7, for an ingenious derivation of the ratio of determinants.
17. R. P. Feynman and A. R. Hibbs, Ref. 2; M. Swanson, Path Integrals and Quantum Processes (Academic, New York, 1992); B. Felsager, Ref. 7. See also S. Coleman, Aspects of Symmetry (Cambridge U.P., New York, 1985), Chap. 7, for an ingenious derivation of the ratio of determinants.
18. This procedure is similar to that used by L. C. Albuquerque, C. Farina, and S. Rabello, "Schwinger's Method and the Computation of Determinants," Sao Paulo, preprint (1997).
19. J. F. Donoghue, E. Golowich, and B. R. Holstein, Dynamics of the Standard Model (Cambridge U.P., New York, 1992), Appendix B.
20. E. T. Whittaker and G. N. Watson, A Course in Modern Analysis (Cambridge U.P., New York, 1973).
21. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, New York, 1965), #3.471.9 and 3.381.4.
22. C. Farina, H. Boschi-Filho, and A. de Souza Dutra, "Green Function Approach for Computing Non-Relativistic Determinants," Rio de Janeiro, preprint (1994).
23. n-2, (70) cf. H. Kleinert, Path Integrals in Quantum Mechanics, Statistical and Polymer Physics (World Scientific, Singapore, 1995); see also L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981); and C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980), Chap. 9.
24. n-2, (70) cf. H. Kleinert, Path Integrals in Quantum Mechanics, Statistical and Polymer Physics (World Scientific, Singapore, 1995); see also L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981); and C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980), Chap. 9.
25. n-2, (70) cf. H. Kleinert, Path Integrals in Quantum Mechanics, Statistical and Polymer Physics (World Scientific, Singapore, 1995); see also L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981); and C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980), Chap. 9.
26. The continued fraction form of the prefactor which results herein was first noted by L. Q. English and R. R. Winters, "Continued Fractions and the Harmonic Oscillator using Feynman's Path Integral," Am. J. Phys. 65, 390-393 (1997). Alternative means of evaluation were noted by K. Unnikrishnan, "Comment on 'Continued Fractions and the Harmonic Oscillator using Feynman's Path Integral," by L. Q. English and R. R. Winters [Am. J. Phys. 65, 390-393 (1997)]," ibid. 65, 1212 (1997) and by B. R. Holstein, "Answer to Question #4. Is there a physics application that is best analyzed in terms of Continued Fractions?" ibid. 65, 1133-1135 (1997).
27. The continued fraction form of the prefactor which results herein was first noted by L. Q. English and R. R. Winters, "Continued Fractions and the Harmonic Oscillator using Feynman's Path Integral," Am. J. Phys. 65, 390-393 (1997). Alternative means of evaluation were noted by K. Unnikrishnan, "Comment on 'Continued Fractions and the Harmonic Oscillator using Feynman's Path Integral," by L. Q. English and R. R. Winters [Am. J. Phys. 65, 390-393 (1997)]," ibid. 65, 1212 (1997) and by B. R. Holstein, "Answer to Question #4. Is there a physics application that is best analyzed in terms of Continued Fractions?" ibid. 65, 1133-1135 (1997).
28. The continued fraction form of the prefactor which results herein was first noted by L. Q. English and R. R. Winters, "Continued Fractions and the Harmonic Oscillator using Feynman's Path Integral," Am. J. Phys. 65, 390-393 (1997). Alternative means of evaluation were noted by K. Unnikrishnan, "Comment on 'Continued Fractions and the Harmonic Oscillator using Feynman's Path Integral," by L. Q. English and R. R. Winters [Am. J. Phys. 65, 390-393 (1997)]," ibid. 65, 1212 (1997) and by B. R. Holstein, "Answer to Question #4. Is there a physics application that is best analyzed in terms of Continued Fractions?" ibid. 65, 1133-1135 (1997).
29. The continued fraction form of the prefactor which results herein was first noted by L. Q. English and R. R. Winters, "Continued Fractions and the Harmonic Oscillator using Feynman's Path Integral," Am. J. Phys. 65, 390-393 (1997). Alternative means of evaluation were noted by K. Unnikrishnan, "Comment on 'Continued Fractions and the Harmonic Oscillator using Feynman's Path Integral," by L. Q. English and R. R. Winters [Am. J. Phys. 65, 390-393 (1997)]," ibid. 65, 1212 (1997) and by B. R. Holstein, "Answer to Question #4. Is there a physics application that is best analyzed in terms of Continued Fractions?" ibid. 65, 1133-1135 (1997).