THE LEVINSON THEOREM AND ITS GENERALIZATION IN RELATIVISTIC QUANTUM MECHANICS

Ni Guang-jiong

Chinese Physics C> 1979, Vol. 3> Issue(4) : 432-449

THE LEVINSON THEOREM AND ITS GENERALIZATION IN RELATIVISTIC QUANTUM MECHANICS

Ni Guang-jiong

Fudan University,Shanghai

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Abstract：
The Levinson Theorem in non-relativistic quantum mechanics is derived by Green function method which leads to the following expression:n_l=1/π[δ_l（0）—δ_l（∞）]—（（—1）^l）/2 sin²δ_l（0）Then its generalization in Dirae equation is fuond as:n_k^（+）—n_k^-=1/π[δ_k（m）—δ_k（∞）+δ_k（—∞）—δ_k（—m）]—（k/（|k|））（（—1）^x/2）[sin²δ_k（m）+sin²δ_k（—m）].对于 Klein-Gordon There are two expressions for Klein-Gordon equation:n_l⁽⁺⁾±n_l^-=1/π{δ_l（m）—δ_l（∞）±[δ_l（—m）—δ_l（—∞）]}—（（—1）^l/2）[sin²δ_l（m）±sin²δ_l（—m）].The implication of these theorems and the range of their validity with relevantproblems are discussed.An example of S state ease in square well potential is trea-ted for testing these formulas.

References

[1]

N. Levinson, Danske Videnskab Selskab. Mat-fys. Medd., 25 (1949), No. 9.[2] L. Schiff. Quantum Mechanics; M. Goldberger. K. Watson, Collision Theory.[3] J. M. Jauch. Helv. Phys. Acta., 30 (1957), 143.[4] A. Martin, Nuovo Cimento, 7 (1958), 607.[5] P. Roman, Advanced quantum Theory, p. 384.[6] 见[5]，p. 300.[7] L. I. Schiff, H. Snyder, J. Weinberg, Phys. Rev., 57 (1940), 315.[8] 见[5], p.173. 但书上有误.[9] R. Jackiw, G. Woo, Phys. Rev., D12 (1975), 1643.[10] J. Fridel. Nuovo Cimento. Suppl. 7 (1958), 287; Phi. Mag.,43(1952), 153. R. J. Sokel, A. J. Phys., 45 (1977), 676.[11] 戴显熹、倪光炯，复旦学报，(自然科学版) 3 (1977), 1.[12] J. Reinhardt, W. Greiner, Reports on Progrees in Phys, 40 (1977). No. 3[13] J. Rafelski. B. Müller, W. Greiner, Nucl. Phys., B68 (1974), 585.[14] 戴显熹，复且学报(自然科学版)，1 (1977), 100.[15] S.Flüyde, Practical Quantum Mechanica, 2(1974), 210.[16] B, C. IIonos,.Snyder, J.}c}}}二 , 59(1970), 965.[17] H. Snyder, J. Weinberg, Phys. Rev., 57 (1940), 307.[18] B. 3enanosHV. B. C. IIonos, Y}H.. 105 (1971), 403.[19] A. Klein, J. Rafelski, Phys. Rev., D11 (195), 300; D12 (1975), 1194.[20] B. Müller, ARNS, 26 (1976), 351.[21] W. Pieper, W. Z. Physik., 218 (1969), 327.[22] A.b.I}oe. B. C. IIonoe,万中.，14 (1971), 874.[23J A. B. N，123 (1977), 369.[24] T. A. Osborn, D. Bolle, J. Math. Phys., 18 (1977), 432.[25] R. G. Newton. J. Math. Phys., 18 (1977), 1348; 1582.[26] R. Berg et al, Phys. Rev., D17 (1978), 1172.[27] J. Rafelski, L. P. Fulcher, A. Klein, Physics Reports, 38C (1978), Vo. 5.

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Ni Guang-jiong. THE LEVINSON THEOREM AND ITS GENERALIZATION IN RELATIVISTIC QUANTUM MECHANICS[J]. Chinese Physics C, 1979, 3(4): 432-449.

Ni Guang-jiong. THE LEVINSON THEOREM AND ITS GENERALIZATION IN RELATIVISTIC QUANTUM MECHANICS[J]. Chinese Physics C, 1979, 3(4): 432-449. shu

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Received: 1978-08-08
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 LEVINSON THEOREM AND ITS GENERALIZATION IN RELATIVISTIC QUANTUM MECHANICS

Fudan University,Shanghai

Received Date: 1978-08-08

Accepted Date: 1900-01-01

Available Online: 1979-08-05

Abstract: The Levinson Theorem in non-relativistic quantum mechanics is derived by Green function method which leads to the following expression:n_l=1/π[δ_l（0）—δ_l（∞）]—（（—1）^l）/2 sin²δ_l（0）Then its generalization in Dirae equation is fuond as:n_k^（+）—n_k^-=1/π[δ_k（m）—δ_k（∞）+δ_k（—∞）—δ_k（—m）]—（k/（|k|））（（—1）^x/2）[sin²δ_k（m）+sin²δ_k（—m）].对于 Klein-Gordon There are two expressions for Klein-Gordon equation:n_l⁽⁺⁾±n_l^-=1/π{δ_l（m）—δ_l（∞）±[δ_l（—m）—δ_l（—∞）]}—（（—1）^l/2）[sin²δ_l（m）±sin²δ_l（—m）].The implication of these theorems and the range of their validity with relevantproblems are discussed.An example of S state ease in square well potential is trea-ted for testing these formulas.

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