THE BOUND STATE ENERGY LEVELS OF MONOPOLE PAIR AND OF CHARGED MONOPOLE-ELECTRON SYSTEM—SOME SINGULAR STATE PROBLEMS IN RELATIVISTIC QUANTUM MECHANICS

Dai Xian-xi; Huang Fa-yang; Ni Guang-jiong

Chinese Physics C> 1980, Vol. 4> Issue(3) : 313-321

THE BOUND STATE ENERGY LEVELS OF MONOPOLE PAIR AND OF CHARGED MONOPOLE-ELECTRON SYSTEM—SOME SINGULAR STATE PROBLEMS IN RELATIVISTIC QUANTUM MECHANICS

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In this paper four kinds of singular state problems in relativistic quantum me-chanics and its solutions are summarized. In the references [2,3,5,6,11], it hasbeen pointed out that in some quantum mechanial problems involving singular statesthere exists phase angle uncertainty. The principle eliminating this kind of nncer-tainty——the orthogonality-variation principle has been obtained in [11] The re-sults of scattering and bound state problems of a neutral monopole and a chargedDirac particle are consistent with those obtained by C. N. Yang, Kazama, and Gold-haber. In [5], [11] and this paper the Case-type equations determining the boundstates of monopole pair and exotic atoms consisting of a charged monopole and anelectron are obtained. In this paper these energy levels are calculated numericallyBecause in these equations the number of singular points is infinite and the rune-tions oscillate rapidly, usual calculation methods are not suitable in these cases. Afteranalysing these equations and determining the positions of singular points and theranges of energy levels, these energy levels are calculated by computer. It is pointed out that the number of these energy levels is infinite and the posi-tions of positive and negative energy levels are asymmetric. The negative energylevels do not appear untilεis very near -1. The values of the positive energy levelsare spread in the range 0.9998ε<1 They are near those of the hydrogen-like atomand similar to those of impurity in solid. This is a perturbation problem of singularstates to which the usual perturbation theory can not be applied. The energy levels of monopole pair are not similar to those of the hydrogen-likeatom. In the range 0<|ε|≤0.99 there are many energy levels with positive andnegative energy. It is also pointed out that for the hydrogen-like atoms, 119<z<137, somenegative energy bound state solutions satisfying square integrability condition arepossible. The ordinary standard condition does not exclude these states, but the or-thogonality criterion does exclude them. The conclusion is in agreement with thatreached by Professor C. N. Yang. The results shown suggest that the orthogonality-variation principle is reasona-ble.

References

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C. G. Darwin, Proc. Roya, Soc. London, Ser, A118(1928), 654.[2] K. M. Oase, Phys. Rev., 80(1950),797.[3] Von Neumann, Private Communication from Pauli.[4] I. Pomeranchuk et al., J. Phys., USSR 9(1945). 97.[5] 戴显熹,复旦学报,1977年第1期,100页.[6] 戴显熹,高能物理与核物理,2 (1977), 305;戴显熹、倪光炯,复且学报,1977年,第3期,第1页.[7] 戴显熹、沈纯理,复且学报,1978年,第3期,第89页.[8] Tai Tsun Wu and Chen Ning Yang, Nucle, Physics, B107(1976), 365.[9] Y. Fiazama et al., Phys. Rev., D15(1977), 2287,[10] Y. Kazama et al., Phys, Rev., D15(1977), 2300[11] 戴显熹、倪光炯,高能物理与核物理,2 (1978), 225.[12] A. S. Goldhaber, Phys. Rev. D16(1977), 1815.

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Dai Xian-xi, Huang Fa-yang and Ni Guang-jiong. THE BOUND STATE ENERGY LEVELS OF MONOPOLE PAIR AND OF CHARGED MONOPOLE-ELECTRON SYSTEM—SOME SINGULAR STATE PROBLEMS IN RELATIVISTIC QUANTUM MECHANICS[J]. Chinese Physics C, 1980, 4(3): 313-321.

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Received: 1979-03-10
Revised: 1900-01-01

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沈阳化工大学材料科学与工程学院沈阳 110142

THE BOUND STATE ENERGY LEVELS OF MONOPOLE PAIR AND OF CHARGED MONOPOLE-ELECTRON SYSTEM—SOME SINGULAR STATE PROBLEMS IN RELATIVISTIC QUANTUM MECHANICS

Received Date: 1979-03-10

Accepted Date: 1900-01-01

Available Online: 1980-06-05

Abstract: In this paper four kinds of singular state problems in relativistic quantum me-chanics and its solutions are summarized. In the references [2,3,5,6,11], it hasbeen pointed out that in some quantum mechanial problems involving singular statesthere exists phase angle uncertainty. The principle eliminating this kind of nncer-tainty——the orthogonality-variation principle has been obtained in [11] The re-sults of scattering and bound state problems of a neutral monopole and a chargedDirac particle are consistent with those obtained by C. N. Yang, Kazama, and Gold-haber. In [5], [11] and this paper the Case-type equations determining the boundstates of monopole pair and exotic atoms consisting of a charged monopole and anelectron are obtained. In this paper these energy levels are calculated numericallyBecause in these equations the number of singular points is infinite and the rune-tions oscillate rapidly, usual calculation methods are not suitable in these cases. Afteranalysing these equations and determining the positions of singular points and theranges of energy levels, these energy levels are calculated by computer. It is pointed out that the number of these energy levels is infinite and the posi-tions of positive and negative energy levels are asymmetric. The negative energylevels do not appear untilεis very near -1. The values of the positive energy levelsare spread in the range 0.9998ε<1 They are near those of the hydrogen-like atomand similar to those of impurity in solid. This is a perturbation problem of singularstates to which the usual perturbation theory can not be applied. The energy levels of monopole pair are not similar to those of the hydrogen-likeatom. In the range 0<|ε|≤0.99 there are many energy levels with positive andnegative energy. It is also pointed out that for the hydrogen-like atoms, 119<z<137, somenegative energy bound state solutions satisfying square integrability condition arepossible. The ordinary standard condition does not exclude these states, but the or-thogonality criterion does exclude them. The conclusion is in agreement with thatreached by Professor C. N. Yang. The results shown suggest that the orthogonality-variation principle is reasona-ble.

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THE BOUND STATE ENERGY LEVELS OF MONOPOLE PAIR AND OF CHARGED MONOPOLE-ELECTRON SYSTEM—SOME SINGULAR STATE PROBLEMS IN RELATIVISTIC QUANTUM MECHANICS

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