The Dirac equation with a confining potential

American Journal of Physics

Volumn 71, Issue 9, 2003, Pages 950-951

  Nogami, Y ^a   Toyama, F M ^b   Van Dijk, W ^a,c  

^a MCMASTER UNIVERSITY   (Canada)

^b KYOTO SANGYO UNIVERSITY   (Japan)

^c REDEEMER UNIVERSITY COLLEGE   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0043263397     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1555891     Document Type: Article

References (15)

1. R. S. Bhalerao and Budh Ram, "Fun and frustration with quarkonium in a 1 + 1 dimension," Am. J. Phys. 69, 817-818 (2001).
2. Antonio S. de Castro, "Comment on 'Fun and frustration with quarkonium in a 1 + 1 dimension,' by R. S. Bhalerao and B. Ram [Am. J. Phys. 69 (7), 817-818 (2001)]," Am. J. Phys. 70, 450-451 (2002).
3. M. Cavalcanti, "Comment on 'Fun and frustration with quarkonium in a 1 + 1 dimension,' by R. S. Bhalerao and B. Ram [Am. J. Phys. 69 (7), 817-818 (2001)]," Am. J. Phys. 70, 451-452 (2002).
4. John R. Killer, "Solution of the one-dimensional Dirac equation with a linear scalar potential," Am. J. Phys. 70, 522-524 (2002).
5. ν. The index ν was apparently assumed to be a positive integer in Ref. 1, which is why only one bound state was found. But ν is in general a noninteger as shown in Refs. 2-4.
6. C. L. Critchfield, "Scalar binding of quarks," Phys. Rev. D 12, 923-925 (1975); "Scalar potentials in the Dirac equation," J. Math. Phys. 17, 261-266 (1976).
7. C. L. Critchfield, "Scalar binding of quarks," Phys. Rev. D 12, 923-925 (1975); "Scalar potentials in the Dirac equation," J. Math. Phys. 17, 261-266 (1976).
8. Budh Ram, "Quark confining potential in relativistic Dequtions," Am. J. Phys. 50, 549-551 (1982).
9. The terminology such as "Lorentz scalar" in the present context is not strictly legitimate. For example, the "scalar potential" S(x)=g|x| is not a Lorentz scalar because |x| is not invariant under the Lorentz transformations. However, we adopt this commonly used terminology.
10. F. Cooper, A. Khare, R. Musto, and A. Wipf, "Supersymmetry and the Dirac equation," Ann. Phys. (N.Y.) 187, 1-28 (1987).
11. Y. Nogami and F. M. Toyama, "Supersymmetry aspects of the Dirac equation in one dimension with a Lorentz scalar potential," Phys. Rev. A 47, 1708-1714 (1993).
12. -1/2. This different choice of χ, which we do not use in this note, results in the same physics, that is, the same wave functions (in terms of u and v), and the same energy eigenvalues. In this connection, note that Eq. (2) is invariant under the simultaneous substitutions of u→v, v→u, E→-E, and V→-V. If V=0, there is a symmetry between positive and negative energy eigenvalues.
13. (n) of Eq. (9).
14. F. A. B. Coutinho and Y. Nogami, "Zero-range potential for the Dirac equation in two and three space dimensions: Elementary proof of Svenson's theorem," Phys. Rev. A 42, 5716-5719 (1990)
15. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), 3rd ed., Chap. 13.