Bound states of a uniform spherical charge distribution - Revisited!

American Journal of Physics

Volumn 68, Issue 7, 2000, Pages 640-648

  Tiburzi, Brian C ^a,c   Holstein, Barry R ^b  

^a AMHERST COLLEGE   (United States)

^b UNIVERSITY OF MASSACHUSETTS   (United States)

^c UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034411927     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19502     Document Type: Article

References (14)

1. See, e.g., L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (McGraw-Hill, New York, 1935).
2. J. Hüfner, F. Scheck, and C. S. Wu, "Muonic Atoms," in Muon Physics, edited by V. W. Hughes and C. S. Wu (Academic, New York, 1977), Vol. I, pp. 202-304.
3. J. Zablotney, "Energy levels of a charged particle in the field of a spherically symmetric uniform charge distribution," Am. J. Phys. 43, 168-172 (1975).
4. E. T. Whittaker and G. N. Watson, Course of Modern Analysis (Macmillan, New York, 1944), Chap. 16.
5. For a more familiar exterior solution and its relation to the Whittaker function, see Appendix A.
6. Albert Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1961), p. 100.
7. Of course there is a limit to the success of the asymptotic expansion when z becomes very small. For example, considering the 5g state of muonic neon, we find ∈=5.741 40±0.000 05 keV corresponding to z=0.049. The actual 5g binding energy is 11.248 keV, the same as the Coulombic value. Thus we cannot determine the binding energies using the asymptotic series in Eq. (8) for very light atoms, states of high angular momentum, or states of small binding energy. Note: this means we cannot recover the Coulombic limit from use of the asymptotic series.
8. David Hartree, "The wave mechanics of an atom with a non-Coulomb central field. III," Proc. Cambridge Philos. Soc. 24, 426-433 (1928).
9. Terminating the infinite series also gives us spurious roots if we consider z too large where the approximation is no longer valid. This occurs when we try to find the largest binding energy for a given l since this corresponds to the largest value of z. Careful attention will expose the presence of a spurious root since ℱ(∈) will not have tangent-like behavior and/or the addition of more terms to the truncated series will continually boost the root to a higher energy. Such roots are manifestations of approximating the infinite series by a finite one.
10. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 13.
11. W. R. Johnson and Gerhard Soff, "The lamb shift in hydrogen-like atoms, 1 ≤Z≤ 110," At. Data Nucl. Data Tables 33, 405-446 (1985).
12. B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules (Longman, London, 1983), Appendix 7.
13. R. Engfer et al., "Charge-distribution parameters, isotope shifts, and magnetic hyperfine constants for muonic atoms," At. Data Nucl. Data Tables 14, 509-597 (1974).
14. T. L. Trueman, "Energy level shifts in atomic states of strongly-interacting particles," Nucl. Phys. 26, 57-67 (1961).