Quantum mechanics of the 1/x 2 potential

American Journal of Physics

Volumn 74, Issue 2, 2006, Pages 109-117

  Essin, Andrew M ^a   Griffiths, David J ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b REED COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 32944456430     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.2165248     Document Type: Article

References (55)

1. 2 potential" Am. J. Phys. 70, 513-519 (2002).
2. 2 potential with α=1/4 [Eq. (3)] - see Exercise 1 in Ref. 30. For discussion of the two-dimensional delta function see L. R. Mead, and J. Godines, "An analytical example of renormalization in two-dimensional quantum mechanics," Am. J. Phys. 59, 935-937 (1991);
3. R. Jackiw, "Delta-function potentials in two- and three-dimensional quantum mechanics," in M. A. B. Bég Memorial Volume, edited by A. Ali and P. Hoodbhoy (World Scientific, Singapore, 1991), pp. 25-42.
4. β(x).
5. K. S. Gupta and S. G. Rajeev, "Renormalization in quantum mechanics," Phys. Rev. D 48, 5940-5945 (1993).
6. ig(κx), diverges for large x.
7. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, (1980), Eqs. (6.576.4) and (8.332.3). Incidentally, states with different K are not orthogonal.
8. 2v/(v+2) [see D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, Englewood Cliffs, NJ, 2004), 2nd ed., Problem 8.11], and the exponent is infinite when v=-2. The variational principle only confirms what we already knew-that the ground state is lower than every negative energy.
9. Reference 6, p. 962, Eqs. (3) and (4).
10. Some authors use a plus sign in Eq. (24), which adds π/2 to the phase shift. We prefer the minus sign, because it reduces to δ=0 when the potential is zero.
11. For a list of accessible references see C. V. Siclen, "The one-dimensional hydrogen atom," Am. J. Phys. 56, 9-10 (1988).
12. 2.
13. 4∈=0.043 089.
14. For the ground state this inequality would appear to require g≪3 (see Fig. 5), but in practice the approximation is good up to g=3. For the excited states κ is smaller, and the approximation is valid for even higher g.
15. Reference 5, Eqs. (11.112) and (11.118), and Ref. 6, Eqs. (8.331) and (8.332).
16. This holds for g < 3, as we can easily confirm by comparing the graph of Eq. (30) with Fig. 5. For larger values of g the approximation itself is invalid for the ground state. Incidentally, Eq. (29) has solutions for negative n, but these are spurious, because they violate the assumption κ∈ ≪ 1.
17. n=(2/γ∈)exp(- nπ/g), where γ≡(C) = 1.781 072. See Ref. 6, Eq. (8.321.1), and p. xxviii.
18. (1)(x)]* (valid for real x and real g). See Ref. 6, p. 969.
19. v, and Eq. (8.332.3) to eliminate T(1-ig).
20. Reference 6, Eqs. (8.331) and (8.332.1).
21. Reference 6, Eqs. (8.441.1) and (8.444.1), and p. xxviii.
22. 1 in Eq. (42) fixed. This is not the same as going straight to g=0 and then letting ∈ → 0 [Eq. (41)], which only reproduces the limiting value π/4.
23. 2 potential. Other regularizations have been proposed. See, for example, C. Schwartz, "Almost singular potentials," J. Math. Phys. 17, 863-867 (1976);
24. H. E. Camblong, L. N. Epele, H. Fanchiotti, and C. A. Garca Canal, "Dimensional transmutation and dimensional regularization in quantum mechanics, I. General theory," and "II. Rotational invariance," Ann. Phys. 287, 14-100 (2001). Our approach follows Ref. 4. It is important in principle to demonstrate that all regularizations lead to the same physical predictions. If they do not, the theory is non-renormalizable and there is very little that can be done with it.
25. 2 potential; renormalization offers a means for doing so. By the way, something very similar happens in quantum electrodynamics, where the theory, naively construed, yields an infinite mass for the electron. The introduction of a cutoff renders the mass finite but indeterminate. We take the observed mass of the electron as input and use it to eliminate any explicit reference to the cutoff. The resulting renormalized theory has been spectacularly successful, yielding by far the most precise (and precisely confirmed) predictions in all of physics.
26. 0 → γ + γ, which could not occur without the breaking of chiral symmetry.
27. See, for instance, E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics (Springer, New York, 1997), pp. 116-117.
28. A then Eq. (52) holds.
29. H. Weyl, "Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen," Math. Ann. 68, 220-269 (1910);
30. J. von Neumann, "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren," Math. Ann. ibid. 102, 49-131 (1929);
31. M. H. Stone, "On one-parameter unitary groups in Hilbert space," Ann. Math. 33, 643-648 (1932).
32. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (Dover, New York, 1993);
33. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, London, 1975);
34. G. Hellwig, Differential Operators of Mathematical Physics (Addison-Wesley, Reading, MA, 1964).
35. G. Bonneau, J. Faraut, and G. Valent, "Self-adjoint extensions of operators and the teaching of quantum mechanics," Am. J. Phys. 69, 322-331 (2001).
36. V. S. Araujo, F. A. B. Coutinho, and J. F. Perez, "Operator domains and self-adjoint operators," Am. J. Phys. 72, 203-213 (2004).
37. There exist pathological functions that are square-integrable and yet do not go to zero at infinity, but there is no penalty for excluding them here. See, for example, D. V. Widder, Advanced Calculus (Dover, New York, 1998), 2nd ed., p. 325.
38. + to obtain the same result.
39. H vanish if 0 ≤ x ≤ ∈ or x ≥ τ for arbitrarily small ∈ and arbitrarily large τ.
40. We follow the treatment in Ref. 30, where the special case α=1/4 is posed as an exercise.
41. H +.
42. Mathematicians usually take η=1, but this choice offends the physicist's concern for dimensional consistency. In any case, it combines with other arbitrary constants at the end.
43. 2 potential, but we shall not do so here.
44. 0 carries the dimensions of length, and hence the choice of a particular self-adjoint extension entails breaking the scale invariance that led to all the difficulties in Sec. II.
45. This case violates our assumption in Ref. 37, so it should be taken with a grain of salt. See Ref. 30, Example 1, for a more rigorous analysis.
46. This result agrees with Eq. (80) of Ref. 30, with r → x and φ → ψ/x.
47. The term "self-adjoint extension" is potentially misleading, because at first sight it appears to involve a contraction, not an expansion, of the domain. The point is that you must start out with a Hermitian operator, and H is not Hermitian with respect to the set of functions that satisfy the boundary condition ψ(0)=0. That is why we first had to restrict the domain (see Ref. 33), and the "extension" is with respect to that much more limited domain.
48. 2 potential, and if the remedy is necessarily radical, so be it.
49. C. Zhu and J. R. Klauder, "Classical symptoms of quantum illnesses," Am. J. Phys. 61, 605-611 (1993);
50. N. A. Wheeler, "Relativistic classical fields," unpublished notes, Reed College, 1973, pp. 246-251.
51. J.-M. Lévy-Leblond, "Electron capture by polar molecules," Phys. Rev. 153, 1-4 (1967).
52. M. H. Mittleman and V. P. Myerscough, "Minimum moment required to bind a charged particle to an extended dipole," Phys. Lett. 23, 545-546 (1966);
53. W. B. Brown and R. E. Roberts, "On the critical binding of an electron by an electric dipole," J. Chem. Phys. 46, 2006-2007 (1966).
54. 2 potentials: Electron exchange between Rydberg atoms and polar molecules," Phys. Rev. Lett. 73, 2436-2439 (1994);
55. Coon and Holstein (Ref. 1); many articles by H. E. Camblong and collaborators, especially "Quantum anomaly in molecular physics," Phys. Rev. Lett. 87, 220402-1-4 (2001).