Bohmian mechanics as a heuristic device: Wave packets in the harmonic oscillator

American Journal of Physics

Volumn 70, Issue 3, 2002, Pages 313-318

  Bowman, Gary E ^a  

^a NORTHERN ARIZONA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23044531856     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1447539     Document Type: Article

References (36)

1. P. R. Holland, The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993).
2. D. Bohm and B. J. Hiley, The Undivided Universe (Routledge, London, 1993); J. T. Cushing, Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony (University of Chicago Press, Chicago, IL, 1994); Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing, A. Fine, and S. Goldstein (Kluwer, Dordrecht, 1996); S. Goldstein, "Quantum theory without observers-part two," Phys. Today 51,38-42 (1998).
3. D. F. Styer, "The motion of wave packets through their expectation values and uncertainties," Am. J. Phys. 58, 742-744 (1990).
4. D. S. Saxon, Elementary Quantum Mechanics (Holden-Day, San Francisco, 1968), pp. 144-147.
5. A. S. de Castro and N. C. da Cruz, "A pulsating Gaussian wave packet," Eur. J. Phys. 20, L19-L20 (1999).
6. J. Arnaud, "Pulsating Gaussian wavepackets and complex trajectories," Eur. J. Phys. 21, L15-L16 (2000).
7. M. Andrews, "Wave packets bouncing off walls," Am. J. Phys. 66, 252-254 (1998).
8. M. Andrews, "Invariant operators for quadratic Hamiltonians," Am. J. Phys. 67, 336-343 (1999).
9. R. Bluhm, V. A. Kostelecký, and J. A. Porter, "The evolution and revival structure of localized wave packets," Am. J. Phys. 64, 944-953 (1996).
10. Here "collapse" refers to the flattening of the packet due to time evolution, rather than to wave function collapse to an eigenstate due to measurement of an observable.
11. R. W. Robinett, "Visualizing the collapse and revival of wave packets in the infinite square well using expectation values," Am. J. Phys. 68, 410-420 (2000).
12. R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples (Oxford University Press, New York, 1997).
13. R. W. Henry and S. C. Glotzer, "A squeezed-state primer," Am. J. Phys. 56, 318-328 (1988).
14. M. M. Nieto, "Displaced and squeezed number states," Phys. Lett. A 229, 135-143 (1997).
15. M. M. Nieto, "What are squeezed states really like?," in Frontiers of Nonequilibrium Statistical Physics, edited by G. T. Moore and M. O. Scully (Plenum, New York, 1986), pp. 287-307.
16. E. Schrödinger, "Der stetige Übergang von der Mikro-zur Makro-mechanik," Naturwissenschaften 14, 664-666 (1926). Reprinted in English as "On the continuous transition from micro- to macro-mechanics," in Collected Papers on Wave Mechanics, edited by J. F. Shearer and W. M. Deans (Blackie and Son, London, 1928), pp. 41-44. Since this work predated Max Born's probability interpretation, Schrödinger examined the real part of the state, rather than its (complex) square. The features of interest are, however, unaltered-the packet envelope is a Gaussian of constant width, whose peak follows the classical trajectory.
17. See M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974), Sec. 2.2; M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), pp. 281-283.
18. D. Bohm, "A suggested interpretation of the quantum theory in terms of 'hidden' variables, I and II," Phys. Rev. 85, 166-179, 180-193 (1952); reprinted in Quantum Theory and Measurement, edited by J. Wheeler and W. Zurek (Princeton University Press, Princeton, NJ, 1983), pp. 369-396.
19. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980), 2nd ed. Chap. 10; Ref. 1, Chap. 2.
20. Measurements other than position are discussed in Ref. 1.
21. By treating Q as an additional potential energy, we may write a modified Hamiltonian as H=p2/2m + V+Q. Then we may use ∂H/∂p=ẋ (one of Hamilton's equations) to obtain the explicit form of p. If V and Q are independent of p, we obtain p=mẋ=mv.
22. Differences between classical potentials and Q are discussed in Ref. 1, Sec. 3.4.
23. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987). Bell discusses Bohmian mechanics in many of the papers in this collection of reprints; see in particular p. 115.
24. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), p. 75.
25. See Ref. 1, Sec. 4.9, for an alternative method.
26. The state itself also possesses this periodicity. Any SHO state may be expanded in energy eigenfunctions, the frequencies of which are integer multiples of the classical SHO frequency. Thus, any SHO state must be periodic with the classical period.
27. D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, NJ, 1951), Sec. 13.15.
28. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1964), p. 491.
29. Such SHO states are often called "squeezed states." See Ref. 12, p. 289 for other names.
30. Here, as in Eq. (8) for a TGS, we assume the phase vanishes at t=0. In Bohmian mechanics this means all the (Bohmian) trajectories are initially stationary.
31. This may also be seen from the analytical proof in G. Bowman, Ph.D. dissertation, University of Notre Dame, 2000, Sec. A1.3.
32. It may seem puzzling that our oscillator state now does not oscillate-it simply remains at the origin. But here we are interested only in the spreading/narrowing forces on the TGS trajectories, and (as just argued) these depend on λ and Δ.x, but not on the oscillator's overall motion.
33. -xe-β2x2 dx, the total probability to the respective 1/e points remains constant as the Gaussian's width is changed. Because Bohmian trajectories cannot cross, this constant integrated probability defines a unique Bohmian trajectory, which must always be located at the 1/e point.
34. Q behaves nonclassically, its dynamical effect is the same as that of a classical force: to accelerate (Bohmian) particles. Therefore, just as forces and trajectories are well-defined in Bohmian mechanics, so is the work-energy theorem.
35. C. Leubner, M. Alber, and N. Schupfer, "Critique and correction of the textbook comparison between classical and quantum harmonic oscillator probability densities," Am. J. Phys. 56, 1123-1129 (1988).
36. M. Born, The Bom-Einstein Utters (Walker and Co., New York, 1971), pp. 205-217; see also the related correspondence between Born and Wolf-gang Pauli, pp. 217-228.