An operator analysis for the Schrödinger, Klein–Gordon, and Dirac equations with a Coulomb potential
1.W. Pauli, Z. Phys. 36, 336 (1926)
1.[translated in Sources of Quantum Mechanics, edited by B. L. van der Waerden (Dover, New York, 1968), pp. 387–415].
2.See, for example, J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory (Springer, New York, 1982), Vol. 3, Chap. 4.
3.See, for example, A. Böhm, Quantum Mechanics (Springer, New York, 1979), Chap. 6.
4.O. L. de Lange and R. E. Raab, Phys. Rev. A 35, 951 (1987).
5.J. D. Newmarch and R. M. Golding, Am. J. Phys. 46, 658 (1978).
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8.L. Infeld and T. E. Hull, Rev. Mod. Phys. 23, 21 (1951).
9.P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw‐Hill, New York, 1953), Pt. 1, Chap. 6.
10.L. C. Biedenharn and P. J. Brussaard, Coulomb Excitation (Clarendon, Oxford, 1965), Chap. 3.
11.L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics (Addison‐Wesley, Reading, MA, 1981), p. 357.
12.L. C. Biedenharn, Found. Phys. 13, 13 (1983).
13.There is, however, a useful analogy: the radial Dirac equations can be formally identified with angular momentum shift operations to in the radial wave functions of a second‐order equation. This result yields the energy eigenvalues of the Dirac equation (see Ref. 8, Sec. 8.4). A similar method for calculating energy eigenvalues is discussed in H. Green, Matrix Mechanics (Noordhoff, Groningen, 1965), Chap. 7.
14.E. Schrödinger, Proc. R. Irish Acad. Sect. A 46, 183 (1941). An alternative treatment of Schrödinger’s recurrence relations, and their extension to the Dirac‐Coulomb equation, is given in Ref. 8, Secs. 8.2 and 8.3. In this method, quantum numbers in wave functions are replaced by variables and operators are defined that involve differentiation with respect to these variables. These “O operators” are different from the energy shift operators derived above: the former cannot be expressed in terms of the radial operators r and and they do not factorize any radial equation.
15.R. Musto, Phys. Rev. 148, 1274 (1966). Note that the scaling operators in this paper, and in Ref. 16, differ by constant factors from the nonrelativistic limit of our Eq. (33).
16.R. H. Pratt and T. F. Jordan, Phys. Rev. 148, 1276 (1966).
17.See, for example, B. G. Wybourne, Classical Groups for Physicists (Wiley, New York, 1974), Chap. 21, and the references therein.
18.An alternative algebraic method of calculating these eigenvalues is the group‐theoretical method based on the SO(2,l) spectrum‐generating algebra that is common to the radial Schrödinger, Klein‐Gordon, and Dirac equations with a Coulomb potential: see Ref. 17, Chap. 18, and the references therein.
19.See, for example, H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One‐ and Two‐Electron Atoms (Academic, New York, 1957), Sec. 14. The complexity of the Dirac‐Coulomb problem, and the desirability of alternative solutions to this problem, have been stressed by Biedenharn (Ref. 12).
20.P. R. Auvil and L. M. Brown, Am. J. Phys. 46, 679 (1978).
21.S. Waldenstro/m, Am. J. Phys. 47, 1098 (1979).
22.E. H. de Groot, Am. J. Phys. 50, 1141 (1982).
23.J. Y. Su, Phys. Rev. A 32, 3251 (1985). Our Eqs. (11) and (12) correspond to Eqs. (2.11) and (2.12) of this reference. The latter are expressed in the coordinate representation. This is an inessential feature, and it is straightforward to derive Eqs. (2.11) and (2.12) in terms of the radial operator
24.See, for example, G. Baym, Lectures on Quantum Mechanics (Benjamin, New York, 1969), Chap. 23.
25.P. C. Martin and R. J. Glauber, Phys. Rev. 109, 1307 (1958);
25.L. C. Biedenharn, Phys. Rev. 126, 845 (1962);
25.M. K. F. Wong and H. Y. Yeh, Phys. Rev. D 25, 3396 (1982).
26.J. O. Hirschfelder, J. Chem. Phys. 33, 1762 (1960).
27.J. H. Epstein and S. T. Epstein, Am. J. Phys. 30, 266 (1962) and references therein.
28.S. T. Epstein, Am. J. Phys. 44, 251 (1976). There is a misprint in Eq. (7) of this reference: the coefficient of should be
29.H. Hellmann, Acta Physicochim. URSS 1, 913 (1935);
29.H. Hellmann, 4, 225 (1936);
29.Einfuhrung in die Quantechemie (Deuticke, Leipzig, 1937);
29.R. P. Feynman, Phys. Rev. 56, 340 (1939). See also E. C. Kemble, Ref. 31, Chap. 13.
30.The shift operation that raises l in the nonrelativistic kets is See, for example, Refs. S and 6.
31.This is a standard application for the Schrödinger‐Coulomb equation: see, for example, E. C. Kemble, The Fundamental Principles of Quantum Mechanics (Dover, New York, 1958), Chap. 5. For the Dirac‐Coulomb equation, see Ref. 23.
32.E. C. Kemble, Phys. Rev. 48, 549 (1935). See also Ref. 31, Chap. 3.
33.R. E. Langer, Phys. Rev. 51, 669 (1937).
34.A similar coincidence exists between the Bohr‐Wilson‐Sommerfeld quantization rules of the old quantum theory applied to the relativistic Coulomb problem, and the exact solution to the Dirac‐Coulomb equation. This “Sommerfeld puzzle” has been discussed in detail by Biedenharn (Ref. 12).
35.M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 504.
36.A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw‐Hill, New York, 1953), Vol. 1, Chap. 3.
37.Equations (10) on p. 258 of Ref. 36 are helpful in obtaining these results.
38.B. H. Armstrong and E. A. Power, Am. J. Phys. 31, 262 (1963).
39.B. H. Armstrong, Phys. Rev. 130, 2506 (1963).
40. is the subspace in which the kinetic energy is finite. See, for example, R. L. Liboff, I. Nebenzahl, and H. H. Fleischmann, Am. J. Phys. 41, 976 (1973),
40.and H. A. Bethe and R. Jackiw, Intermediate Quantum Mechanics, (Benjamin, New York, 1968) 2nd. ed., Chap. 21.
41.T. Tietz, Sov. Phys. JETP 3, 777 (1956).
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