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Teaching Feynman’s sum-over-paths quantum theory
1.F. Dyson, in Some Strangeness in the Proportion, edited by H. Woolf (Addison–Wesley, Reading, MA, 1980), p. 376.
2.R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).
3.R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill, New York, 1965).
4.R. P. Feynman, QED, The Strange Theory of Light and Matter (Princeton University Press, Princeton, 1985).
5.R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. III.
6.See, for example, N. J. Dowrick, Eur. J. Phys. 18, 75 (1997).
6.Titles of articles on this subject in the Am. J. Phys. may be retrieved online at http://www.amherst.edu/∼ajp.
7.R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Plenum, New York, 1994). This text includes a nice introduction of the sum-over-paths theory and many applications, suitable for an upper undergraduate or graduate course.
8.For a description of the National Teachers Enhancement Network at Montana State University and a listing of current courses, see the Web site http://www.montana.edu//wwwxs.
9.Draft software written by Taylor in the computer language cT. For a description of this language, see the Web site http://cil.andrew.cum.edu/ct.html.
10.To conform to the “stopwatch” picture, rotation is taken to be clockwise, starting with the stopwatch hand straight up. We assume that later [for example, with Eq. (8) in step 18] this convention will be “professionalized” to the standard counterclockwise rotation, starting with initial orientation in the rightward direction. The choice of either convention, consistently applied, has no effect on probabilities calculated using the theory.
11.Feynman explains later in his popular QED book (page 104 of Ref. 4) that the photon stopwatch hand does not rotate while the photon is in transit. Rather, the little arrows summed at the detection event arise from a series of worldlines originating from a “rotating” source.
12.In Fig. 1, the computer simply adds up stopwatch-hand arrows for a sampling of alternative paths in two spatial dimensions. The resulting arrow at the detector is longer than the original arrow at the emitter. Yet the probability of detection (proportional to the square of the length of the arrow at the detector) cannot be greater than unity. Students do not seem to worry about this at the present stage in the argument.
13.R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. II, Chap. 19.
14.See, Ref. 2, Sec. 4, postulate II.
15.The classical principle of least action assumes fixed initial and final events. This is exactly what the sum-over-paths formulation of quantum mechanics needs also, with fixed events of emission and detection. The classical principle of least action is valid only when dissipative forces (such as friction) are absent. This condition is also satisfied by quantum mechanics, since there are no dissipative forces at the atomic level.
16.A naive reading of Eq. (2) seems to be inconsistent with the deBroglie relation when one makes the substitutions and and In Ref. 3, pp. 44–45, Feynman and Hibbs resolve this apparent inconsistency, which reflects the difference between group velocity and phase velocity of a wave.
17.See a similar figure in Ref. 3, Fig. 3-3, p. 48.
18.We have found three kinds of errors that result from representing a continuous wavefunction with a finite series of equally spaced arrows. (1) Representing a wavefunction of wide x extension with a narrower width of arrows along the x direction leads to propagation of edge effects into the body of the wavefunction. The region near the center changes a negligible amount if the elapsed time is sufficiently short. (2) The use of discrete arrows can result in a Cornu spiral that does not complete its inward scroll to the theoretically predicted point at each end. For example, in the Cornu spiral in the left-hand panel of Fig. 5, the use of discrete arrows leads to repeating small circles at each end, rather than convergence to a point. The overall resulting arrow (from the tail of the first little arrow to the head of the final arrow) can differ slightly in length from the length it would have if the scrolls at both ends wound to their centers. The fractional error is typically reduced by increasing the number of arrows, thereby increasing the ratio of resulting arrow length to the length of the little component arrows. (3) The formation of a smooth Cornu spiral at the detection event requires that the difference in rotation to a point on the final wavefunction be small between arrows that are adjacent in the original wavefunction. But for very short times between the initial and later wavefunctions, some of the connecting worldlines are nearly horizontal in spacetime diagrams similar to Figs. 3 and 4, corresponding to large values of kinetic energy KE, and therefore high rotation frequency Under such circumstances, the difference in rotation at an event on the final wavefunction can be great between arrows from adjacent points in the initial wavefunction. This may lead to distortion of the Cornu spiral or even its destruction. In summary, a finite series of equally spaced arrows can adequately represent a continuous wavefunction provided the number of arrows (for a given total extension) is large and the time after the initial wavefunction is neither too small nor too great. We have done a preliminary quantitative analysis of these effects showing that errors can be less than 2% for a total number of arrows easily handled by desktop computers. This accuracy is adequate for teaching purposes.
19.In Ref. 3, p. 42ff, Feynman and Hibbs carry out a complicated integration to find the propagator for a free electron. However, the normalization constant used in their integration is determined only later in their treatment (Ref. 3, p. 78) in the course of deriving the Schrödinger equation.
20.This is verified by the usual Schrödinger analysis. The initial free-particle wavefunction shown in Figs. 8910 has zero second derivative, so it will also have a zero time derivative.
21.We add a linear taper to each end of the initial wavefunctions used in constructing Figs. 891011 to suppress “high-frequency components” that otherwise appear along the entire length of a later wavefunction when a finite initial wavefunction has a sharp space termination. The tapered portions lie outside the views shown in these figures.
22.In Ref. 3, Eq. (3-42), p. 57.
23.In Ref. 3, Eq. (3-3), p. 42.
24.In Refs. 2 and 3; see also D. Derbes, Am. J. Phys. 64, 881 (1996).
25.In Ref. 3; L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
26.R. C. Smith and E. F. Taylor, Am. J. Phys. 63, 1090 (1995).
27.A complete tabulation of the spring 1996 questionnaire results is available from Taylor.
28.To obtain draft exercises and software, see the Web site http://cil.andrew.cmu.edu/people/edwin.taylor.html.
29.Feynman implies this connection in his popular presentation (Ref. 4).
30.For example, the principle of extremal aging can be used to derive expressions for energy and angular momentum of a satellite moving in the Schwarzschild metric. See, for example, E. F. Taylor and J. A. Wheeler, Scouting Black Holes, desktop published, Chap. 11. Available from Taylor (Website in Ref. 28).
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