Ehrenfest's theorem and the particle-in-a-box

American Journal of Physics

Volumn 64, Issue 11, 1996, Pages 1416-1418

  Rokhsar, D S ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030500601     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18367     Document Type: Article

References (11)

1. P. Ehrenfest, "Bemerkung uber die angenaherte Gultigkeit der klassichen Mechanik innerhalb der Quantenmechanik," Z. Phys. 45, 455-457 (1927).
2. Periodic boundary conditions also cause problems for Ehrenfest's theorem. See R. N. Hill, "A paradox involving the quantum mechanical Ehrenfest's theorem," Am. J. Phys. 41 (5), 736-738 (1973).
3. The quantization of the modes of a vibrating string was so well-known to the founders of quantum mechanics that the "particle-in-a-box" was used without much fanfare, even before the appearance of the Schrödinger equation, particularly in the statistical mechanics of ideal gases. The earliest use of piecewise constant potentials in quantum mechanics seems to be L. Nordheim, "Zur theorie der thermischen emission und der reflexion von elektronen an metallen," Z. Phys. 46, 833-855 (1928). I would be interested in any earlier references than this.
4. 2n, see L. S. Salter, "Quantum mechanical study of particles in 'softened' potential boxes and wells," Am. J. Phys. 58 (10), 961-967 (1990).
5. See, e.g., A. Messiah, Quantum Mechanics (Wiley, New York, 1958), pp. 86-96.
6. An early discussion of the restrictions on acceptable wave functions can be found in R. M. Langer and N. Rosen, "What requirements must the Schrödinger ψ-function satisfy?" Phys. Rev. 37, 658 (1931): "As a rough working rule we may demand of the function that it be integrable in the square and that it be finite and continuous wherever the potential energy is finite. If we introduce singularities ... in the potential energy, we must be prepared to put up with singularities in the wavefunction at those points. When, in the future, we get a deep enough insight into the nature of the physical problems so that we can replace the singular values of the potential with their true finite values, the singularities of the wave functions will also be removed. But until then we must be content with results which differ from the facts to the same degree as do the assumptions from which we start." For a detailed discussion of mathematical subtleties at singular points of the potential, see D. Branson, "Continuity conditions on Schrödinger wave functions at discontinuities of the potential," Am. J. Phys. 47, 1000-1005 (1979).
7. An early discussion of the restrictions on acceptable wave functions can be found in R. M. Langer and N. Rosen, "What requirements must the Schrödinger ψ-function satisfy?" Phys. Rev. 37, 658 (1931): "As a rough working rule we may demand of the function that it be integrable in the square and that it be finite and continuous wherever the potential energy is finite. If we introduce singularities ... in the potential energy, we must be prepared to put up with singularities in the wavefunction at those points. When, in the future, we get a deep enough insight into the nature of the physical problems so that we can replace the singular values of the potential with their true finite values, the singularities of the wave functions will also be removed. But until then we must be content with results which differ from the facts to the same degree as do the assumptions from which we start." For a detailed discussion of mathematical subtleties at singular points of the potential, see D. Branson, "Continuity conditions on Schrödinger wave functions at discontinuities of the potential," Am. J. Phys. 47, 1000-1005 (1979).
8. The notion of an effective particle-in-a-box width for a square well was introduced heuristically by S. Garrett in "Bound state energies of a particle in a finite square well: A simple approximation," Am. J. Phys. 47, (2), 195-196 (1979). See also B. I. Barker, G. H. Rayborn, J. W. Ioup, and G. E. Ioup, "Approximating the finite square well with an infinite well," ibid. 59 (11), 1038-1042 (1991).
9. The notion of an effective particle-in-a-box width for a square well was introduced heuristically by S. Garrett in "Bound state energies of a particle in a finite square well: A simple approximation," Am. J. Phys. 47, (2), 195-196 (1979). See also B. I. Barker, G. H. Rayborn, J. W. Ioup, and G. E. Ioup, "Approximating the finite square well with an infinite well," ibid. 59 (11), 1038-1042 (1991).
10. In the limit λ/L →∞, this is the same as saying that the amplitude at L/2 of the deep square well of width L is the same as that of an infinite well of width L + 2λ, in keeping with Eq. (20).
11. The matrix elements 〈n\X̂\m〉, 〈n\P̂\m〉, 〈n\F̂\m〉 all vanish by symmetry when n and m have the same parity, and are, respectively, real, imaginary, and real when n and m have opposite parity.