Avoided crossings of diamagnetic hydrogen as functions of magnetic field strength and angular momentum

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 58, Issue 6, 1998, Pages 4668-4682

  Walkup, J R ^a   Dunn, M ^a   Watson, D K ^a   Germann, T C ^b  

^a UNIVERSITY OF OKLAHOMA   (United States)

^b LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001818863     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.58.4668     Document Type: Article

References (110)

1. R. H. Garstang, Rep. Prog. Phys. 40, 105 (1977), and references therein.RPPHAG
2. D. Delande and J. C. Gay, Phys. Rev. Lett. 66, 3237 (1991); PRLTAO
3. Phys. Rev. Lett.J. C. Gay, 66, 145 (1991);
4. W. M. Rösner, G. Wunner, H. Herold, and H. Ruder, J. Phys. B 17, 29 (1984); JPAMA4
5. D. Wintgen and H. Friedrich, Phys. Rev. A 36, 131 (1987).PLRAAN
6. D. Delande and J. C. Gay, Comments At. Mol. Phys. 19, 35 (1986).CAMPBS
7. C. Iu, G. R. Welch, M. M. Kash, D. Kleppner, D. Delande, and J. C. Gay, Phys. Rev. Lett. 66, 145 (1991).PRLTAO
8. H. Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989).PRPLCM
9. P. Fassbinder and W. Schweizer, Phys. Rev. A 53, 2135 (1996); PLRAAN
10. M. Robnik, J. Phys. A 14, 105 (1977); JPHAC5
11. J. L. Greenstein and J. B. Oke, Astrophys. J. 252, 285 (1982); ASJOAB
12. G. Wunner, H. Ruder, and H. Herold, Phys. Lett. 79A, 159 (1980); PYLAAG
13. Phys. Lett.G. WunnerH. RuderH. Herold247A, 374 (1981).
14. C. M. Dai and D. S. Chu, Physica B 172, 445 (1991); PHYBE3
15. A. K. Ramdas and S. Rodriguez, Rep. Prog. Phys. 44, 1297 (1981); RPPHAG
16. H. S. Brandi, Phys. Rev. A 11, 1835 (1975); PLRAAN
17. J. G. Mavroides, Optical Properties of Solids, edited by F. Abeles (North-Holland, Amsterdam, 1972);
18. G. E. Stillman, D. M. Larsen, C. M. Wolfe, and R. C. Brandt, Solid State Commun. 9, 2245 (1971); SSCOA4
19. H. Hasegawa, Physics of Solids in Intense Magnetic Fields (Plenum, New York, 1969).
20. J. C. Gay, Comments At. Mol. Phys. 25, 185 (1991); CAMPBS
21. J. Main and G. Wunner, J. Phys. B 27, 2835 (1994); JPAPEH
22. D. Delande, Chaos and Quantum Physics, 1989 Les Houches Lectures Session LII, edited by M. J. Giannoni, A. Voos, and J. Zinn-Justin (North-Holland, Amsterdam, 1991);
23. H. Friedrich, Theoretical Atomic Physics (Springer-Verlag, Berlin, 1991), Sec. 5.3.4;
24. H. Hasegawa, M. Robnik, and G. Wunner, Prog. Theor. Phys. Suppl. 98, 198 (1989).PTPSAL
25. S. Watanabe and H. Komine, Phys. Rev. Lett. 67, 3227 (1991); PRLTAO
26. H. Hasegawa, M. Robnik, and G. Wunner, Prog. Theor. Phys. Suppl. 98, 198 (1989); PTPSAL
27. J. Main, G. Wiebush, A. Holle, and K. H. Welge, Phys. Rev. Lett. 56, 2594 (1986); PRLTAO
28. J. C. Gay and D. Delande, Atomic Excitation and Recombination in External Fields, edited by M. H. Nayfeh and C. W. Clark (Gordon and Breach, New York, 1985);
29. D. Kleppner, M. G. Littman, and L. Zimmerman, Rydberg States of Atoms and Molecules, edited by R. F. Stebbings and F. B. Dunning (Cambridge University Press, Cambridge, 1983).
30. A. V. Sergeev (private communication).
31. P. A. Braun and V. I. Savichev, J. Phys. B 26, 3739 (1993); JPAPEH
32. J. Phys. BP. Cacciani, C. Delsart, E. Luckoenig, and J. Pinard, 25, 1991 (1992);
33. Q. Wang and C. Greene, Phys. Rev. A 44, 1874 (1991); PLRAAN
34. I. Sh. Averbukh and N. F. Perelman, Phys. Lett. A 139, 449 (1989); PYLAAG
35. G. R. Welch, M. M. Kash, C. Iu, L. Hsu, and D. Kleppner, Phys. Rev. Lett. 62, 893 (1989); PRLTAO
36. Phys. Rev. Lett.C. Iu, G. R. Welch, M. M. Kash, L. Hsu, and D. Kleppner, 63, 1133 (1989);
37. T. F. Gallagher, Rep. Prog. Phys. 51, 143 (1988); RPPHAG
38. J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 (1986).PRLTAO
39. L. Chen, M. Cheret, F. Roussel, and G. Spiess, J. Phys. B 26, L437 (1993); JPAPEH
40. D. Delande and J. C. Gay, Europhys. Lett. 5, 303 (1988); EULEEJ
41. J. Liang, M. Gross, P. Goy, and S. Haroche, Phys. Rev. A 33, 4437 (1986); PLRAAN
42. R. G. Hulet and D. Kleppner, Phys. Rev. Lett. 51, 1430 (1983).PRLTAO
43. H. Friedrich, Theoretical Atomic Physics (Springer-Verlag, Berlin, 1990).
44. For a general discussion of the square-root branch point of physical systems see J. Goldberg and W. Schweizer, J. Phys. A 24, 2785 (1991).JPHAC5
45. The chaotic region is much harder to describe since avoided crossings are so close together that assigning a well-defined character to each energy level becomes impossible.
46. It is technically incorrect to say that the character exchanges at the avoided crossing because the states close to the avoided crossing are actually made up of a superposition of the states that are used to label each energy level.
47. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977), Sec. 79.
48. M. Dunn, D. K. Watson, J. R. Walkup, and T. C. Germann, J. Chem. Phys. 104, 9870 (1996).JCPSA6
49. E. A. Solov’ev, Zh. Eksp. Teor. Fiz. 81, 1681 (1981) [Sov. Phys. JETP 54, 893 (1981)];SPHJAR
50. S. Yu. Ovchinnikov and E. A. Solov’ev, Comments At. Mol. Phys. 22, 69 (1988).CAMPBS
51. Dimensional Scaling in Chemical Physics, edited by D. R. Herschbach, J. Avery, and O. Goscinski (Kluwer, Dordrecht, 1992).
52. C. M. Bender, L. D. Mlodinow, and N. Papanicolaou, Phys. Rev. A 25, 1305 (1982).PLRAAN
53. Y. Y. Goldschmidt, Nucl. Phys. B 393, 507 (1993); NUPBBO
54. J. Rudnick and G. Gaspari, Science 237, 384 (1987); SCIEAS
55. R. A. Ferrell and D. J. Scalapino, Phys. Rev. A 9, 846 (1974); PLRAAN
56. A. J. Bray, J. Phys. A 7, 2144 (1974); JPHAC5
57. H. E. Stanley, Phys. Rev. 176, 718 (1968); PHRVAO
58. Phys. Rev.T. H. Berlin and M. Kac, 86, 821 (1952).
59. M. Dunn, T. C. Germann, D. Z. Goodson, C. A. Traynor, J. D. Morgan III, D. K. Watson, and D. R. Herschbach, J. Chem. Phys. 101, 5987 (1994).JCPSA6
60. V. S. Popov, V. D. Mur, A. V. Sergeev, and V. M. Weinberg, Phys. Lett. A 149, 418 (1990); PYLAAG
61. Phys. Lett. AV. S. Popov, V. D. Mur, and A. V. Sergeev, 149, 425 (1990);
62. Phys. Lett. AV. S. Popov, V. D. Mur, A. V. Shcheblykin, and V. M. Weinberg, 124, 77 (1987).
63. A. Chatterjee, Phys. Rep. 186, 249 (1990).PRPLCM
64. A. A. Belov, Yu. E. Lozovik, and V. A. Mandelshtam, Zh. Eksp. Teor. Fiz. 98, 25 (1990) [Sov. Phys. JETP 71, 12 (1990)].SPHJAR
65. V. S. Popov and A. V. Sergeev, Phys. Lett. A 193, 165 (1994).PYLAAG
66. V. M. Vaĭnberg, V. S. Popov, and A. V. Sergeev, Zh. Eksp. Teor. Fiz. 98, 847 (1990) [Sov. Phys. JETP 71, 470 (1990)].SPHJAR
67. New Methods in Quantum Theory, edited by C. A. Tsipis, V. S. Popov, D. R. Herschbach and J. Avery, NATO Conference Book Vol. 8 (Kluwer Academic, Dordrecht, 1996).
68. T. C. Germann and S. Kais, J. Chem. Phys. 99, 7739 (1993).JCPSA6
69. L. J. Boya and R. Murray, Phys. Rev. A 50, 4397 (1994); PLRAAN
70. S. Kais and G. Beltrame, J. Phys. Chem. 97, 2453 (1993); JPCHAX
71. U. P. Sukhatme, B. M. Lauer, and T. D. Imbo, Phys. Rev. D 33, 1166 (1986); PRVDAQ
72. Phys. Rev. DR. S. Gangyopadhyay, R. Dutt, and Y. P. Varshni, 32, 3312 (1985);
73. D. Bollé and F. Gesztesy, Phys. Rev. A 30, 1279 (1984); PLRAAN
74. M. Sinha-Roy, R. S. Gangyopadhyay, and B. Dutta-Roy, J. Phys. A 17, L687 (1984).JPHAC5
75. S. M. Valone, Int. J. Quantum Chem. 49, 591 (1994); IJQCB2
76. S. Kais, D. R. Herschbach, N. C. Handy, C. W. Marray, and G. J. Laming, J. Phys. Chem. 99, 417 (1993); JPCHAX
77. G. F. Kventsel and J. Katriel, Phys. Rev. A 24, 2299 (1981); PLRAAN
78. Phys. Rev. AN. H. March, 30, 2936 (1984);
79. Phys. Rev. AN. H. March34, 5106 (1986);
80. N. H. MarchJ. Math. Phys. 26, 554 (1985).
81. A. A. Suvernev and D. Z. Goodson, Chem. Phys. Lett. 269, 177 (1997); CHPLBC
82. A. A. SuvernevD. Z. GoodsonJ. Chem. Phys. 107, 4099 (1997).
83. P. Serra and S. Kais, Phys. Rev. A 55, 238 (1997); PLRAAN
84. P. SerraS. KaisJ. Phys. A 30, 1483 (1997);
85. P. SerraS. KaisChem. Phys. Lett. 275, 211 (1997);
86. P. SerraS. KaisPhys. Rev. Lett. 77, 466 (1996);
87. Phys. Rev. Lett.J. P. Neirotti, P. Serra, and S. Kais, 79, 3142 (1997);
88. J. P. NeirottiP. SerraS. KaisChem. Phys. Lett. 260, 302 (1996).
89. T. C. Germann, D. R. Herschbach, and B. M. Boghosian, Comput. Phys. 8, 712 (1994).CPHYE2
90. J. R. Walkup et al(unpublished).
91. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 8.
92. L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932); PHZSAL
93. C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).PRLAAZ
94. G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, 1985), pp. 377 and 378.
95. These branch points are not only the values of the parameter (Formula presented) where (Formula presented) but the energy functions (Formula presented) and (Formula presented) actually share this point. This may be shown by following a path that takes (Formula presented) and encloses one of the branch points. From Eq. (8) this path takes (Formula presented) into (Formula presented) Thus, by letting the distance of the path to the branch point tend to zero while ensuring that the path continues to enclose the branch point we see that the branch point is common to both (Formula presented) and (Formula presented) (For a more detailed discussion see, for example, Ref. 18.)
96. W. D. Heiss and A. L. Sannino [J. Phys. A 23, 1167 (1990)] have related the distribution of square-root branch points in matrix Hamiltonian problems to the appearance of quantum chaos in these systems.JPHAC5
97. Although the two-level system we chose in our example was real valued, we note that complex-valued two-level systems also require that real-valued branch points exist at the same location as long as the Hamiltonian is Hermitian.
98. T. C. Germann, D. R. Herschbach, M. Dunn, and D. K. Watson, Phys. Rev. Lett. 74, 658 (1995).PRLTAO
99. The algorithm of choice for calculating circular Rydberg states using dimensional perturbation theory is the matrix method, discussed in Ref. 23. The matrix method algorithm was used in this research. Twentieth-order calculations using the matrix method take only about 3 min to compute on a SparcStation 20 workstation.
100. E. Fermi, Z. Phys. 71, 250 (1931); ZEPYAA
101. G. Herzberg, Molecular Spectra and Molecular Structure V. II (Van Nostrand Reinhold, New York, 1945), p. 215.
102. Because we log scale both axes, the avoided crossing center is slightly displaced from where the narrowest approach of energy levels appears in the figures.
103. In this research, converged energy levels are found by Padé summing Eq. (17).
104. The avoided crossings that appear in the (Formula presented)-versus-(Formula presented) spectrum are harder to find than those appearing in the (Formula presented)-versus-(Formula presented) spectrum. One reason for this is the leading (Formula presented) in Eq. (17), which does not affect the dynamics of the problem (that is, which state is associated with each energy level), but because of its (Formula presented) dependence tends to mask the appearance of the avoided crossing. Therefore, subtracting this term from the energy series does not affect where an avoided crossing appears, but makes the avoided crossing more visible. The subsequent energy series has a leading δ that is also a function of (Formula presented) but, as in (Formula presented) does not affect the dynamics of the problem. Therefore, we factor this term out of the series as well. The result is a rescaling of the series as (Formula presented) whenever we calculate the energy with respect to changes in (Formula presented) Also, for these cases we use total energies, not binding energies. Therefore, the vertical axes on such plots change accordingly.
105. We used quadratic Padé summation to find the square-root branch points shown in the figures. See Ref. 18 and references therein.
106. In this research the quantum numbers (Formula presented) and (Formula presented) denoting the characters of the states, are related to the low-field quantum numbers (Formula presented) and (Formula presented) (see Ref. 5 and references therein) by (Formula presented) where (Formula presented) orders the states with respect to energy in each nth hydrogenic manifold in the low-(Formula presented)-field limit. (The higher the value of (Formula presented) the lower the energy.) Wintgen and Friedrich 554 find that only states with (Formula presented) (Formula presented) quanta that differ by 2 can exhibit a broad avoided crossing. In this case the Fermi resonance condition associated with these avoided crossings implies that, for the (Formula presented) Fermi resonance, the ratio (Formula presented) must be an integer. Since (Formula presented) must be larger than (Formula presented) ((Formula presented) must be larger than (Formula presented)) the only allowed Fermi resonance (other than the hydrogenic 1:1 Fermi resonance that occurs at (Formula presented) not relevant to this discussion) that satisfies the (Formula presented) condition is the 2:1 Fermi resonance. Note from Table I that not even all 2:1 Fermi resonances satisfy this condition and thus are expected to be broad. Furthermore, only avoided crossings involving two vibrational states (large (Formula presented)) are broad, further reducing the number of broad avoided crossings of the 2:1 Fermi resonance.
107. A. A. Kotze and W. D. Heiss [J. Phys. A 27, 3059 (1994)] have related the distribution of square-root branch points of diamagnetic hydrogen in the complex-(Formula presented) plane to the appearance of quantum chaos in these systems.JPHAC5
108. W. D. Heiss and W.-H. Steeb, J. Math. Phys. 32, 3003 (1991).JMAPAQ
109. A. Bohm, Quantum Mechanics: Foundations and Applications (Springer-Verlag, New York, 1986), Sec. I.7 and Chap. XXI.
110. D. Wintgen and H. Friedrich, J. Phys. B 19, 1261 (1986). JPAMA4