線形ポテンシャルを持つDirac方程式の束縛解利用統計を見る

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著者

手塚洋一

雑誌名

東洋大学紀要. 自然科学篇 = Journal of Toyo

University. 東洋大学自然科学研究室編

号

59 ページ

97-154

発行年

2015-03

URL

http://id.nii.ac.jp/1060/00007024/

Creative Commons : 表示 - 非営利 - 改変禁止 http://creativecommons.org/licenses/by-nc-nd/3.0/deed.ja

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手塚洋一

Bound State Solutions of the Dirac Equation with Linear Potential

Hirokazu T

EZUKA

東洋大学紀要自然科学篇第 59 号抜刷

Reprinted from

Journal of Toyo University, Natural Science No. 59, pp.97 ∼ 154, March, 2015

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　The Dirac equation with a linear potential is discussed. The linear potential has often been used for the confinement potential of valence quarks. Numerical solutions for nonrelativistic bound-state problems have been obtained by many authors. Analytical solution of the Schrödinger equation with a linear potential is known as an Airy function for zero orbital angular momentum. Those for arbitrary orbital angular momenta have also been obtained with the aid of additional potentials.

　We investigate the Dirac equation of a particle with mass moving in a central linear potential. The Dirac equation with a vector linear potential has no bound state solution. The scalar linear potential gives bound state solutions to the Dirac equation. Since the potentials depend on only radial coordinate r, the equations of motion can be separeted into the radial part and the angular part. The radial wave function is assumed to be of an asymptotic form times a polynomial expression of the radial coordinate. Analytical solutions are shown to exist when there are some quantitative relations between the strength constant of the linear potential and the mass of the particle. We show analytical solutions up to the fifth-order expansion of the radial wave function.

Keywords: Dirac equation, linear potential, bound state

線形ポテンシャルを持つ Dirac 方程式の束縛解

Abstract

手塚洋一

*

Bound State Solutions of the Dirac Equation

with Linear Potential

Hirokazu T

EZUKA *

*）東洋大学自然科学研究室　112-8606　東京都文京区白山 5-28-20

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1 ͸͡Ίʹ

ڑ཭ r ʹൺྫ͢Δϙςϯγϟϧʢ ઢܗϙςϯγϟϧ ʣ͸͠͹͠͹ΫΥʔΫͷด͡ࠐΊϙςϯ γϟϧͱͯ͠૝ఆ͞ΕΔɻଟ͘ͷܭࢉ͕͜ͷλΠϓͷϙςϯγϟϧΛ࢖ͬͯͳ͞Ε͍ͯΔ͕ɺ ͦͷେ෦෼͸ඇ૬ର࿦తͳӡಈํఔࣜΛ਺஋తʹղ͘͜ͱʹΑͬͯͳ͞Ε͍ͯΔʢ Eichten,E. 1975ɺGunion,J.F.-Willey,R.S. 1975ɺKaushal,R.S. 1975 ʣɻԿਓ͔ͷஶऀʹΑΔ૬ର࿦త ޮՌΛߟྀͨ͠਺஋ܭࢉ΋ଘࡏ͢Δʢ Kang,J.S.-Schnitzer,H.J. 1975ɺGunion,J.F.-Li,L.F. 1975ʣɻৄ͍͠ϨϏϡʔ͸ʢ Quigg,C-Rosner,J.L. 1979ɺGrosse,H-Martin,A. 1980 ʣʹ͋Δɻ ઢܗϙςϯγϟϧΛ࣋ͭඇ૬ର࿦తͳ Schr¨odinger ํఔࣜͷղੳతͳղʹؔͯ͠͸͢Ͱ ʹٞ࿦͕͋Δʢ Tezuka,H. 1991 ʣɻ֯ӡಈྔʹґଘ͢Δ൒੔਺࣍਺ͷ෇Ճతͳϙςϯγϟϧ Λಋೖ͢Δ͜ͱʹΑͬͯઢܗϙςϯγϟϧ͸ղੳతͳղΛ࣋ͭ͜ͱ͕ূ໌͞Ε͍ͯΔɻ ͜͜Ͱ͸ΫΥʔΫ͕ै͏Ͱ͋Ζ͏૬ର࿦తͳӡಈํఔࣜͰ͋Δ Dirac ํఔࣜʹઢܗϙς ϯγϟϧΛՃ͑ͯͦͷղΛݕ౼͢ΔɻDirac ํఔࣜʹϙςϯγϟϧΛಋೖ͢Δʹ͸̎ͭͷλ Πϓ͕஌ΒΕ͍ͯΔɻ̍ͭ͸ӡಈྔ߲ʹ̐ݩϕΫτϧͱͯ͠ಋೖ͢ΔϕΫτϧϙςϯγϟ ϧͰ͋Γɺ΋͏̍ͭ͸࣭ྔ߲ʹՃ͑ΒΕΔεΧϥʔϙςϯγϟϧͰ͋Δɻ͜ͷ࿦จͰ͸த ৺ྗʢ ڑ཭ r ͷΈʹґଘ͢Δϙςϯγϟϧ ʣΛߟ͑ɺϕΫτϧϙςϯγϟϧͱͯ͠͸࣌ؒ ੒෼ͷΈΛߟ͕͑ͨɺ͜ͷλΠϓͷϙςϯγϟϧ͸ด͡ࠐΊϙςϯγϟϧͱͯ͠͸ػೳ͠ ͳ͍͜ͱ͕Θ͔ΔɻεΧϥʔϙςϯγϟϧ͸ด͡ࠐΊϙςϯγϟϧͱͯ͠ಇ͘͜ͱ͕Θ͔ ΔͷͰɺղੳతͳଋറঢ়ଶͷղͱͯ͠͸͜ͷλΠϓͷϙςϯγϟϧͷ৔߹Λݕ౼ͨ͠ɻ ઢܗϙςϯγϟϧΛ࣋ͭ Dirac ํఔࣜΛղ͘ࢼΈ͸͍͔ͭ͘ͳ͞Ε͍ͯΔɻAbe-Fujita ͸࣭ྔ 0 ͷཻࢠʹର͠ઢܗͷεΧϥʔϙςϯγϟϧΛ࣋ͭ Dirac ํఔࣜͷۙࣅతͳղੳղ ΛٻΊ͍ͯΔʢ Abe,S-Fujita,T 1987 ʣɻHofer-Stocker ͸Ұ࣍ݩ໰୊ͱͯ͠εΧϥʔϙςϯ γϟϧΛ࣋ͭ Dirac ํఔࣜΛղ͍͍ͯΔʢ Hofer,D.-Stocker,W. 1989 ʣɻ ͜ͷ࿦จͰ͸ઢܗͷεΧϥʔϙςϯγϟϧΛ࣋ͭ Dirac ํఔ͕ࣜࢦ਺ؔ਺తʹऩଋ͢Δ ઴ۙղͱ༗ݶͳଟ߲ࣜͷ૊Έ߹ΘͤͰղੳతʹղ͚Δ͜ͱΛࣔ͢ɻͨͩ͠ɺղੳతͳղ͕ ଘࡏ͢Δͷ͸ઢܗεΧϥʔϙςϯγϟϧͷେ͖͞ͱཻࢠͷ࣭ྔʹ͋Δؔ܎͕ଘࡏ͢Δ৔ ߹ʹݶΔɻղੳతͳղ͸ଟ߲ࣜͷ r5_{·ͰٻΊͨɻ͜͜·ͰͰɺنଇੑ͸͍͍ͩͨ૝૾͕} ͭ͘ɻ

2 த৺ྗϙςϯγϟϧΛ࣋ͭ

Dirac

ํఔࣜ

֯౓ํ޲ʹґଘͤͣɺڑ཭ r ͷΈʹґଘ͢Δத৺ྗϙςϯγϟϧΛ࣋ͭ૬ର࿦తͳӡಈ ํఔࣜͰ͋Δ Dirac ํఔࣜͷଋറঢ়ଶΛߟ͑Δɻϙςϯγϟϧͱͯ͠͸εΧϥʔܕ S(r) ͱϕΫτϧܕ Vµ_(r)_{͕૝ఆ͞ΕΔɻϕΫτϧϙςϯγϟϧ͸̐ݩӡಈྔʹର͠} pµ → pµ− Vµ(r) (2.1) ͷܗͰಋೖ͞ΕɺεΧϥʔϙςϯγϟϧ͸࣭ྔ߲ʹ m → m + S(r) (2.2) ͷܗͰಋೖ͞ΕΔɻϕΫτϧϙςϯγϟϧͱͯ͠͸࣌ؒ੒෼ͷΈΛߟྀ͢Δɻ͢ͳΘͪ Vµ(r) = V (r) δµ 0 (2.3) ͱ͢Δɻཻࢠ͸࣭ྔ m ͷϑΣϧϛཻࢠͰ͋ΔͱԾఆͯ͠ɺDirac ํఔࣜ [α· p + β{m + S(r)}] ψ(r) = {E − V (r)}ψ(r) (2.4) ʹै͏΋ͷͱ͢Δɻ α = ( 0 σ σ 0 ) β = ( I 0 0 −I ) (2.5) ͸ 4 ੒෼ͷ Dirac ߦྻͰ͋Δɻσ ͸ 2 ੒෼ͷ Pauli ߦྻͰ͋ΓɺIɺ0 ͸ 2 ੒෼ͷ୯Ґߦྻ ͱྵߦྻΛද͢ɻཻࢠͷݻ༗ΤωϧΪʔ͸ Eɺӡಈྔ͸ pɺۭؒ࠲ඪ r Λ΋ͭ೾ಈؔ਺ ͸ 4 ੒෼εϐϊʔϧͷ ψ(r) Ͱදݱ͞Ε͍ͯΔɻϙςϯγϟϧ V (r)ɺS(r) ͸ڑ཭͚ͩʹ ґଘ͢Δத৺ྗͱԾఆ͢Δɻ ϙςϯγϟϧ͸த৺ྗͰ֯౓ґଘੑΛ࣋ͨͳ͍ͱԾఆ͞Ε͍ͯΔ͔Βɺ4 ੒෼εϐϊʔ ϧ ψ(r) Λ 2 ੒෼ͷಈܘ෦෼ G(r)ɺF (r) ͱ֯౓෦෼ φl j,m(Ω)ʹ෼͚ ψ(r) =    iG(r) r φ l j,m(Ω) F (r) r σ· r r φ l j,m(Ω)    (2.6) ͱ͓͍ͯɺํఔࣜʢ 2.4 ʣΛม਺෼཭Ͱ͖Δɻ͢ͳΘͪ dG(r) dr =− κ rG(r) +{E − V (r) + m + S(r)} F (r) (2.7) dF (r) dr = κ rF (r)− {E − V (r) − m − S(r)} G(r) (2.8) ͱೋͭͷࣜʹ෼཭Ͱ͖Δɻͨͩ͠ j = l ±1₂ ʹରͯ͠ κ =∓(j +1₂) (2.9) ͱఆٛ͞Ε͍ͯΔɻ ࣜʢ 2.7 ʣ͔Β F (r) ΛٻΊΔͱ F (r) = 1 E− V (r) + m + S(r) { dG(r) dr + κ rG(r) } (2.10) ͱͳΓɺ͜ΕΛඍ෼ͯ͠ dF (r) dr = dV (r) dr − dS(r) dr {E − V (r) + m + S(r)}2 { dG(r) dr + κ rG(r) } + 1 E− V (r) + m + S(r) { d2G(r) dr2 − κ r2G(r) + κ r dG(r) dr } (2.11) ͕ٻ·Δɻ͜ΕΒΛʢ 2.8 ʣʹ୅ೖͯ͠ F (r) Λফڈ͢Δͱ dV (r) dr − dS(r) dr {E − V (r) + m + S(r)}2 { dG(r) dr + κ rG(r) }

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1 ͸͡Ίʹ

ڑ཭ r ʹൺྫ͢Δϙςϯγϟϧʢ ઢܗϙςϯγϟϧ ʣ͸͠͹͠͹ΫΥʔΫͷด͡ࠐΊϙςϯ γϟϧͱͯ͠૝ఆ͞ΕΔɻଟ͘ͷܭࢉ͕͜ͷλΠϓͷϙςϯγϟϧΛ࢖ͬͯͳ͞Ε͍ͯΔ͕ɺ ͦͷେ෦෼͸ඇ૬ର࿦తͳӡಈํఔࣜΛ਺஋తʹղ͘͜ͱʹΑͬͯͳ͞Ε͍ͯΔʢ Eichten,E. 1975ɺGunion,J.F.-Willey,R.S. 1975ɺKaushal,R.S. 1975 ʣɻԿਓ͔ͷஶऀʹΑΔ૬ର࿦త ޮՌΛߟྀͨ͠਺஋ܭࢉ΋ଘࡏ͢Δʢ Kang,J.S.-Schnitzer,H.J. 1975ɺGunion,J.F.-Li,L.F. 1975ʣɻৄ͍͠ϨϏϡʔ͸ʢ Quigg,C-Rosner,J.L. 1979ɺGrosse,H-Martin,A. 1980 ʣʹ͋Δɻ ઢܗϙςϯγϟϧΛ࣋ͭඇ૬ର࿦తͳ Schr¨odinger ํఔࣜͷղੳతͳղʹؔͯ͠͸͢Ͱ ʹٞ࿦͕͋Δʢ Tezuka,H. 1991 ʣɻ֯ӡಈྔʹґଘ͢Δ൒੔਺࣍਺ͷ෇Ճతͳϙςϯγϟϧ Λಋೖ͢Δ͜ͱʹΑͬͯઢܗϙςϯγϟϧ͸ղੳతͳղΛ࣋ͭ͜ͱ͕ূ໌͞Ε͍ͯΔɻ ͜͜Ͱ͸ΫΥʔΫ͕ै͏Ͱ͋Ζ͏૬ର࿦తͳӡಈํఔࣜͰ͋Δ Dirac ํఔࣜʹઢܗϙς ϯγϟϧΛՃ͑ͯͦͷղΛݕ౼͢ΔɻDirac ํఔࣜʹϙςϯγϟϧΛಋೖ͢Δʹ͸̎ͭͷλ Πϓ͕஌ΒΕ͍ͯΔɻ̍ͭ͸ӡಈྔ߲ʹ̐ݩϕΫτϧͱͯ͠ಋೖ͢ΔϕΫτϧϙςϯγϟ ϧͰ͋Γɺ΋͏̍ͭ͸࣭ྔ߲ʹՃ͑ΒΕΔεΧϥʔϙςϯγϟϧͰ͋Δɻ͜ͷ࿦จͰ͸த ৺ྗʢ ڑ཭ r ͷΈʹґଘ͢Δϙςϯγϟϧ ʣΛߟ͑ɺϕΫτϧϙςϯγϟϧͱͯ͠͸࣌ؒ ੒෼ͷΈΛߟ͕͑ͨɺ͜ͷλΠϓͷϙςϯγϟϧ͸ด͡ࠐΊϙςϯγϟϧͱͯ͠͸ػೳ͠ ͳ͍͜ͱ͕Θ͔ΔɻεΧϥʔϙςϯγϟϧ͸ด͡ࠐΊϙςϯγϟϧͱͯ͠ಇ͘͜ͱ͕Θ͔ ΔͷͰɺղੳతͳଋറঢ়ଶͷղͱͯ͠͸͜ͷλΠϓͷϙςϯγϟϧͷ৔߹Λݕ౼ͨ͠ɻ ઢܗϙςϯγϟϧΛ࣋ͭ Dirac ํఔࣜΛղ͘ࢼΈ͸͍͔ͭ͘ͳ͞Ε͍ͯΔɻAbe-Fujita ͸࣭ྔ 0 ͷཻࢠʹର͠ઢܗͷεΧϥʔϙςϯγϟϧΛ࣋ͭ Dirac ํఔࣜͷۙࣅతͳղੳղ ΛٻΊ͍ͯΔʢ Abe,S-Fujita,T 1987 ʣɻHofer-Stocker ͸Ұ࣍ݩ໰୊ͱͯ͠εΧϥʔϙςϯ γϟϧΛ࣋ͭ Dirac ํఔࣜΛղ͍͍ͯΔʢ Hofer,D.-Stocker,W. 1989 ʣɻ ͜ͷ࿦จͰ͸ઢܗͷεΧϥʔϙςϯγϟϧΛ࣋ͭ Dirac ํఔ͕ࣜࢦ਺ؔ਺తʹऩଋ͢Δ ઴ۙղͱ༗ݶͳଟ߲ࣜͷ૊Έ߹ΘͤͰղੳతʹղ͚Δ͜ͱΛࣔ͢ɻͨͩ͠ɺղੳతͳղ͕ ଘࡏ͢Δͷ͸ઢܗεΧϥʔϙςϯγϟϧͷେ͖͞ͱཻࢠͷ࣭ྔʹ͋Δؔ܎͕ଘࡏ͢Δ৔ ߹ʹݶΔɻղੳతͳղ͸ଟ߲ࣜͷ r5_{·ͰٻΊͨɻ͜͜·ͰͰɺنଇੑ͸͍͍ͩͨ૝૾͕} ͭ͘ɻ

2 த৺ྗϙςϯγϟϧΛ࣋ͭ

Dirac

ํఔࣜ

֯౓ํ޲ʹґଘͤͣɺڑ཭ r ͷΈʹґଘ͢Δத৺ྗϙςϯγϟϧΛ࣋ͭ૬ର࿦తͳӡಈ ํఔࣜͰ͋Δ Dirac ํఔࣜͷଋറঢ়ଶΛߟ͑Δɻϙςϯγϟϧͱͯ͠͸εΧϥʔܕ S(r) ͱϕΫτϧܕ Vµ_(r)_{͕૝ఆ͞ΕΔɻϕΫτϧϙςϯγϟϧ͸̐ݩӡಈྔʹର͠} pµ → pµ− Vµ(r) (2.1) ͷܗͰಋೖ͞ΕɺεΧϥʔϙςϯγϟϧ͸࣭ྔ߲ʹ m → m + S(r) (2.2) ͷܗͰಋೖ͞ΕΔɻϕΫτϧϙςϯγϟϧͱͯ͠͸࣌ؒ੒෼ͷΈΛߟྀ͢Δɻ͢ͳΘͪ Vµ(r) = V (r) δµ 0 (2.3) ͱ͢Δɻཻࢠ͸࣭ྔ m ͷϑΣϧϛཻࢠͰ͋ΔͱԾఆͯ͠ɺDirac ํఔࣜ [α· p + β{m + S(r)}] ψ(r) = {E − V (r)}ψ(r) (2.4) ʹै͏΋ͷͱ͢Δɻ α = ( 0 σ σ 0 ) β = ( I 0 0 −I ) (2.5) ͸ 4 ੒෼ͷ Dirac ߦྻͰ͋Δɻσ ͸ 2 ੒෼ͷ Pauli ߦྻͰ͋ΓɺIɺ0 ͸ 2 ੒෼ͷ୯Ґߦྻ ͱྵߦྻΛද͢ɻཻࢠͷݻ༗ΤωϧΪʔ͸ Eɺӡಈྔ͸ pɺۭؒ࠲ඪ r Λ΋ͭ೾ಈؔ਺ ͸ 4 ੒෼εϐϊʔϧͷ ψ(r) Ͱදݱ͞Ε͍ͯΔɻϙςϯγϟϧ V (r)ɺS(r) ͸ڑ཭͚ͩʹ ґଘ͢Δத৺ྗͱԾఆ͢Δɻ ϙςϯγϟϧ͸த৺ྗͰ֯౓ґଘੑΛ࣋ͨͳ͍ͱԾఆ͞Ε͍ͯΔ͔Βɺ4 ੒෼εϐϊʔ ϧ ψ(r) Λ 2 ੒෼ͷಈܘ෦෼ G(r)ɺF (r) ͱ֯౓෦෼ φl j,m(Ω)ʹ෼͚ ψ(r) =    iG(r) r φ l j,m(Ω) F (r) r σ· r r φ l j,m(Ω)    (2.6) ͱ͓͍ͯɺํఔࣜʢ 2.4 ʣΛม਺෼཭Ͱ͖Δɻ͢ͳΘͪ dG(r) dr =− κ rG(r) +{E − V (r) + m + S(r)} F (r) (2.7) dF (r) dr = κ rF (r)− {E − V (r) − m − S(r)} G(r) (2.8) ͱೋͭͷࣜʹ෼཭Ͱ͖Δɻͨͩ͠ j = l ± 1₂ ʹରͯ͠ κ =∓(j +1₂) (2.9) ͱఆٛ͞Ε͍ͯΔɻ ࣜʢ 2.7 ʣ͔Β F (r) ΛٻΊΔͱ F (r) = 1 E− V (r) + m + S(r) { dG(r) dr + κ rG(r) } (2.10) ͱͳΓɺ͜ΕΛඍ෼ͯ͠ dF (r) dr = dV (r) dr − dS(r) dr {E − V (r) + m + S(r)}2 { dG(r) dr + κ rG(r) } + 1 E− V (r) + m + S(r) { d2G(r) dr2 − κ r2G(r) + κ r dG(r) dr } (2.11) ͕ٻ·Δɻ͜ΕΒΛʢ 2.8 ʣʹ୅ೖͯ͠ F (r) Λফڈ͢Δͱ dV (r) dr − dS(r) dr {E − V (r) + m + S(r)}2 { dG(r) dr + κ rG(r) }

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+ 1 E− V (r) + m + S(r) { d2_G(r) dr2 − κ r2G(r) + κ r dG(r) dr } = κ r 1 E− V (r) + m + S(r) { dG(r) dr + κ rG(r) } − {E − V (r) − m − S(r)} G(r) (2.12) ͱͳΔɻ੔ཧ͢Δͱ {E − V (r) + m + S(r)}d 2_G(r) dr2 + { dV (r) dr − dS(r) dr } dG(r) dr ={E − V (r) + m + S(r)}κ(κ + 1)_r₂ G(r)− { dV (r) dr − dS(r) dr } κ rG(r) − {E − V (r) + m + S(r)}2{E − V (r) − m − S(r)} G(r) (2.13) ͱͳΔɻ

3 ઢܗεΧϥʔϙςϯγϟϧ

·ͣεΧϥʔϙςϯγϟϧͷΈͷ৔߹Λݕ౼͢Δɻϙςϯγϟϧ͸ઢܗͰ S(r) = ar (3.1) ͱ͢Δɻa > 0 ͷ৔߹ʹ͸ڑ཭ r ͱͱ΋ʹ༗ޮ࣭ྔ m∗_{= m + S(r)} _{͕େ͖͘ͳΔͨΊɺ} Ҿྗతͱߟ͑ΒΕΔɻଋറঢ়ଶΛߟ͑ΔͷͰɺa > 0 ͱԾఆ͢ΔɻV (r) = 0 ͱͯ͠ɺ͜Ε Λʢ 2.13 ʣʹ୅ೖͯ͠੔ཧ͢Δͱ (E + m + ar)d 2_G(r) dr2 − a dG(r) dr = (E + m + ar)κ(κ + 1) r2 G(r) + a κ rG(r) − (E + m + ar)2(E− m − ar) G(r) (3.2) ͕ٻ·Δɻ͜Ε͕ղ͘΂͖εΧϥʔઢܗϙςϯγϟϧΛ࣋ͭ Dirac ํఔࣜͰ͋Δɻ͜ͷํ ఔࣜͷղੳղʹؔͯ͠͸͢Ͱʹʢ Tezuka,H. 2013 ʣͷใࠂ͕͋Δ͕ɺ͜͜Ͱ͸গ͠ৄ͘͠ ܭࢉΛݟ͍ͯ͘ɻ ·ͣଋറঢ়ଶ͕ଘࡏ͢Δ͔Ͳ͏͔֬ೝ͢ΔͨΊɺ઴ۙղΛٻΊΔɻे෼ԕํ r → ∞ Ͱ ํఔࣜʢ 3.2 ʣͰ࢒Δ߲͸ ard 2_G(r) dr2 − a dG(r) dr = a 3_r3_G(r) _(3.3) ͱͳΔɻ೾ಈؔ਺ G(r) ͷଋറঢ়ଶͷ઴ۙղΛ e−αrδ ͱԾఆ͢Δɻଋറঢ়ଶͱͳΔΑ͏ʹ ن֨ԽՄೳͳͨΊʹ͸े෼ԕํͰ೾ಈؔ਺͸े෼খ͘͞ͳΒͳͯ͘͸ͳΒͳ͍͔Β α > 0 ͱͳΔ͸ͣͰ͋Δɻr Ͱඍ෼ͯ͠ dG(r) dr ∼ −αδr δ−1_e−αrδ (3.4) d2_G(r) dr2 ∼ { −αδ(δ − 1)rδ−2+ α2δ2r2(δ−1)} e−αrδ (3.5) ͱͳΔͷͰɺ઴ۙํఔࣜʢ 3.3 ʣʹ୅ೖ͢Δͱ r{−αδ(δ − 1)rδ−2+ α2δ2r2(δ−1)} e−αrδ+ αδrδ−1e−αrδ = a2r3e−αrδ (3.6) ͱͳΔɻࠨลͷ࠷ߴ࣍਺ͷ߲Λ࢒͢ͱ α2δ2r2δ−1= a2r3 (3.7) ͱͳΔɻ͜ΕΑΓ δ = 2 4α2= a2 (3.8) ͕ٻ·Δɻα > 0ɺa > 0 Ͱ͋Δ͔Β α = a 2 (3.9) ͱͳΔɻ ͜ͷ઴ۙܗΛ࢖ͬͯɺ೾ಈؔ਺Λଟ߲ࣜల։͠ G(r) = n ∑ k=0 akrke−αr 2_−βr (3.10) ͱԾఆ͢Δɻͨͩ͠ɺak ͸ఆ਺Ͱ͋Γɺ೾ಈؔ਺͕ऩଋ͠ɺن֨ԽͰ͖ΔͨΊʹ n ͸༗ ݶͰ͋ΔͱԾఆ͢Δɻ dG(r) dr = n ∑ k=0 ak{krk−1− (2αr + β)rk}e−αr 2_−βr = n ∑ k=0 ak(krk−1− 2αrk+1− βrk)e−αr 2_−βr (3.11) d2_G(r) dr2 = n ∑ k=0 ak{k(k − 1)rk−2− 2α(k + 1)rk− βkrk−1 − k(2αr + β)rk−1+ (2αr + β)2rk}e−αr2−βr = n ∑ k=0 ak[k(k− 1)rk−2− 2βkrk−1+{−2α(2k + 1) + β2}rk + 4αβrk+1+ 4α2rk+2] e−αr2−βr (3.12) ͱͳΔͷͰɺํఔࣜʢ 3.2 ʣ͸ n ∑ k=0 ak(E + m + ar)[ k(k− 1)rk−2− 2βkrk−1 +{−2α(2k + 1) + β2_}rk_{+ 4αβr}k+1_{+ 4α}2_rk+2_] − n ∑ k=0 aka(krk−1− βrk− 2αrk+1)

(7)

+ 1 E− V (r) + m + S(r) { d2_G(r) dr2 − κ r2G(r) + κ r dG(r) dr } = κ r 1 E− V (r) + m + S(r) { dG(r) dr + κ rG(r) } − {E − V (r) − m − S(r)} G(r) (2.12) ͱͳΔɻ੔ཧ͢Δͱ {E − V (r) + m + S(r)}d 2_G(r) dr2 + { dV (r) dr − dS(r) dr } dG(r) dr ={E − V (r) + m + S(r)}κ(κ + 1)_r₂ G(r)− { dV (r) dr − dS(r) dr } κ rG(r) − {E − V (r) + m + S(r)}2{E − V (r) − m − S(r)} G(r) (2.13) ͱͳΔɻ

3 ઢܗεΧϥʔϙςϯγϟϧ

·ͣεΧϥʔϙςϯγϟϧͷΈͷ৔߹Λݕ౼͢Δɻϙςϯγϟϧ͸ઢܗͰ S(r) = ar (3.1) ͱ͢Δɻa > 0 ͷ৔߹ʹ͸ڑ཭ r ͱͱ΋ʹ༗ޮ࣭ྔ m∗ _{= m + S(r)}_{͕େ͖͘ͳΔͨΊɺ} Ҿྗతͱߟ͑ΒΕΔɻଋറঢ়ଶΛߟ͑ΔͷͰɺa > 0 ͱԾఆ͢ΔɻV (r) = 0 ͱͯ͠ɺ͜Ε Λʢ 2.13 ʣʹ୅ೖͯ͠੔ཧ͢Δͱ (E + m + ar)d 2_G(r) dr2 − a dG(r) dr = (E + m + ar)κ(κ + 1) r2 G(r) + a κ rG(r) − (E + m + ar)2(E− m − ar) G(r) (3.2) ͕ٻ·Δɻ͜Ε͕ղ͘΂͖εΧϥʔઢܗϙςϯγϟϧΛ࣋ͭ Dirac ํఔࣜͰ͋Δɻ͜ͷํ ఔࣜͷղੳղʹؔͯ͠͸͢Ͱʹʢ Tezuka,H. 2013 ʣͷใࠂ͕͋Δ͕ɺ͜͜Ͱ͸গ͠ৄ͘͠ ܭࢉΛݟ͍ͯ͘ɻ ·ͣଋറঢ়ଶ͕ଘࡏ͢Δ͔Ͳ͏͔֬ೝ͢ΔͨΊɺ઴ۙղΛٻΊΔɻे෼ԕํ r → ∞ Ͱ ํఔࣜʢ 3.2 ʣͰ࢒Δ߲͸ ard 2_G(r) dr2 − a dG(r) dr = a 3_r3_G(r) _(3.3) ͱͳΔɻ೾ಈؔ਺ G(r) ͷଋറঢ়ଶͷ઴ۙղΛ e−αrδ ͱԾఆ͢Δɻଋറঢ়ଶͱͳΔΑ͏ʹ ن֨ԽՄೳͳͨΊʹ͸े෼ԕํͰ೾ಈؔ਺͸े෼খ͘͞ͳΒͳͯ͘͸ͳΒͳ͍͔Β α > 0 ͱͳΔ͸ͣͰ͋Δɻr Ͱඍ෼ͯ͠ dG(r) dr ∼ −αδr δ−1_e−αrδ (3.4) d2_G(r) dr2 ∼ { −αδ(δ − 1)rδ−2+ α2δ2r2(δ−1)} e−αrδ (3.5) ͱͳΔͷͰɺ઴ۙํఔࣜʢ 3.3 ʣʹ୅ೖ͢Δͱ r{−αδ(δ − 1)rδ−2+ α2δ2r2(δ−1)} e−αrδ+ αδrδ−1e−αrδ = a2r3e−αrδ (3.6) ͱͳΔɻࠨลͷ࠷ߴ࣍਺ͷ߲Λ࢒͢ͱ α2δ2r2δ−1= a2r3 (3.7) ͱͳΔɻ͜ΕΑΓ δ = 2 4α2= a2 (3.8) ͕ٻ·Δɻα > 0ɺa > 0 Ͱ͋Δ͔Β α = a 2 (3.9) ͱͳΔɻ ͜ͷ઴ۙܗΛ࢖ͬͯɺ೾ಈؔ਺Λଟ߲ࣜల։͠ G(r) = n ∑ k=0 akrke−αr 2_−βr (3.10) ͱԾఆ͢Δɻͨͩ͠ɺak ͸ఆ਺Ͱ͋Γɺ೾ಈؔ਺͕ऩଋ͠ɺن֨ԽͰ͖ΔͨΊʹ n ͸༗ ݶͰ͋ΔͱԾఆ͢Δɻ dG(r) dr = n ∑ k=0 ak{krk−1− (2αr + β)rk}e−αr 2_−βr = n ∑ k=0 ak(krk−1− 2αrk+1− βrk)e−αr 2_−βr (3.11) d2_G(r) dr2 = n ∑ k=0 ak{k(k − 1)rk−2− 2α(k + 1)rk− βkrk−1 − k(2αr + β)rk−1+ (2αr + β)2rk}e−αr2−βr = n ∑ k=0 ak[k(k− 1)rk−2− 2βkrk−1+{−2α(2k + 1) + β2}rk + 4αβrk+1+ 4α2rk+2] e−αr2−βr (3.12) ͱͳΔͷͰɺํఔࣜʢ 3.2 ʣ͸ n ∑ k=0 ak(E + m + ar)[ k(k− 1)rk−2− 2βkrk−1 +{−2α(2k + 1) + β2_}rk_{+ 4αβr}k+1_{+ 4α}2_rk+2_] − n ∑ k=0 aka(krk−1− βrk− 2αrk+1)

(8)

= n ∑ k=0 ak(E + m + ar)κ(κ + 1)rk−2+ n ∑ k=0 akaκrk−1 − n ∑ k=0 ak(E + m + ar)2(E− m − ar)rk (3.13) ͱͳΔɻ੔ཧ͢Δͱ n ∑ k=0 ak[{k(k − 1)(E + m) − κ(κ + 1)(E + m)}rk−2

+{ak(k − 1) − 2βk(E + m) − ak − aκ(κ + 1) − aκ}rk−1

+{−2βak − 4αk(E + m) − 2α(E + m) + β2(E + m) + aβ + (E + m)2(E− m)}rk

+{−4aαk − 2aα + aβ2+ 4αβ(E + m) + 2αa

− a(E + m)2+ 2a(E + m)(E− m)}rk+1

+{4αβa + 4α2(E + m) + a2(E− m) − 2a2(E + m)}rk+2 + (4α2a− a3_)rk+3_{] = 0} _(3.14) ͱͳΔ͕ɺα2₌ a2 4 Λ୅ೖ͢Δͱ࠷ޙͷ r k+3 _ͷ߲͸ফ͑ n ∑ k=0 ak[{k(k − 1) − κ(κ + 1)}(E + m)rk−2

+{ak(k − 2) − aκ(κ + 2) − 2βk(E + m)}rk−1

+{−aβ(2k − 1) − a(2k + 1)(E + m) + β2(E + m) + (E + m)2(E− m)}rk

+{−2a2k + aβ2+ 2aβ(E + m)− a(E + m)2+ 2a(E + m)(E− m)}rk+1

+{2a2β− 2a2m}rk+2= 0 (3.15) ͱͳΔɻ͜ͷࣜΛ r ͷ߃౳ࣜͱͯ͠ղ͚͹ଋറঢ়ଶͷղ͕ٻ·Δ͜ͱʹͳΔɻ࠷ޙͷ rk+2 ͷ߲͕࠷ߴ࣍਺ͱͳΔ͕ɺͦͷ܎਺͸ k ʹґଘ͠ͳ͍ͷͰҰൠతʹղ͚Δɻa ̸= 0 ͔ͩΒ 2β− 2m = 0 ∴ β = m (3.16) ͱͳΓɺࢦ਺ؔ਺ͷ෦෼͸͜ΕͰܾఆ͞ΕΔɻβ = m ͸ཻࢠͷ࣭ྔʹ૬౰͢ΔͷͰɺਖ਼· ͨ̌ͱߟ͑Δɻ͢ͳΘͪ β ≥ 0 ͱ͢Δɻɹ ࠷ऴతʹ੔ཧ͢Δͱ n ∑ k=0 ak[ (k + κ)(k− κ − 1)(E + m)rk−2 +{a(k + κ)(k − κ − 2) − 2mk(E + m)}rk−1

+{−am(2k − 1) − a(2k + 1)(E + m) + E2(E + m)}rk

+ (−2a2k + aE2)rk+1] = 0 (3.17) ͱͳΓɺ͜ΕΛ r ʹؔ͢Δ߃౳ࣜͱߟ͑ͯղ͖ɺκɺE ΛܾΊΔɻ ࣍ʹ࠷΋࣍਺ͷখ͞ͳ߲ (k = 0) Λߟ͑Δͱ r−2_ͷ߲ͱͳΓ a0{−κ(κ + 1)}(E + m) = 0 (3.18) ͱͳΔ͕ɺκ = ±1, ±2, ±3, · · · Ͱ͋Δ͜ͱΛߟྀ͢Δͱ a0= 0 ·ͨ͸ κ = −1 ·ͨ͸ E + m = 0 (3.19) ͱ͍͏৚͕݅ٻ·ΔɻE < 0 ͷ৔߹ʹ͸ɺཻࢠ͕ੜ੒͞Εͨํ͕ΤωϧΪʔతʹ༗རͱͳ Δෆ҆ఆղͱͳΔͷͰɺE ≥ 0 ͱԾఆ͢Δɻm = 0ɺE = 0 ͷղ΋ҙຯͷ͋Δղͱ͸ࢥ͑ ͳ͍ͷͰɺE + m > 0 ͱԾఆ͢Δɻ࣮ࡍʹ m = 0ɺE = 0 ͱ͢Δͱʢ 3.17 ʣ͸ n ∑ k=0 ak{a(k + κ)(k − κ − 2)rk−1− am(2k − 1)rk− 2a2krk+1} = 0 (3.20) ͱͳΓɺ࠷ߴ࣍਺ͷ rn+1_{ͷ߲͕ −2a}2_nrn+1 _{ͱͳΔͷͰɺ͜ͷ߲͕ 0 ͱͳΓ߃౳ࣜΛຬ} ͨ͢͜ͱ͸Ͱ͖ͳ͍ɻ E + m > 0ͱͯ͠ κ = −1 ·ͨ͸ a0= 0ͷͦΕͧΕʹରͯ͠ɺଟ߲ࣜల։͕༗ݶͷ n ͰऴΘΔ৔߹ʹղ͕ଘࡏ͢Δ͔Ͳ͏͔ௐ΂Δɻ

4 κ =

₋₁

ͷ৔߹

ղ͘΂͖ํఔࣜʢ 3.17 ʣ͸ κ = −1 ʹରͯ͠ n ∑ k=0

ak[ k(k− 1)(E + m)rk−2+{a(k − 1)2− 2mk(E + m)}rk−1

+{−am(2k − 1) − a(2k + 1)(E + m) + E2(E + m)}rk

+ (−2a2k + aE2)rk+1] = 0 (4.1) ͱͳΔɻҎԼɺଟ߲ࣜల։ͷ߲਺ͷগͳ͍ॱʹݕ౼͢Δɻ

4.1 n = 0

a

0

̸= 0

ͷ৔߹ ࠷ॳʹ೾ಈؔ਺͕࠷΋࣍਺ͷখ͞ͳ G(r) = a0e−αr 2_−βr (4.2) ͱͳΔղ͕ଘࡏ͢Δ͔Ͳ͏͔ݕ౼͢Δɻʢ 4.1 ʣ͸ a0[ ar−1+{am − a(E + m) + E2(E + m)}r0+ aE2r1] = 0 (4.3) ͱͳΔ͕ɺ͜ͷ͕ࣜ͢΂ͯͷ࣍਺ͷ r ʹରͯ͠߃౳ࣜͱͳΔͨΊʹ͸ɺr−1_{ͷ߲͔Β a = 0} ͱͳΔ͕ɺ͜Ε͸ઢܗϙςϯγϟϧ͕ଘࡏ͠ͳ͍͜ͱΛҙຯ͢ΔͷͰແҙຯͰ͋Δɻκ = −1 ʹର͠ɺʢ 4.2 ʣͷΑ͏ͳ೾ಈؔ਺Λ࣋ͭଋറঢ়ଶͷղ͸ଘࡏ͠ͳ͍ɻ

(9)

= n ∑ k=0 ak(E + m + ar)κ(κ + 1)rk−2+ n ∑ k=0 akaκrk−1 − n ∑ k=0 ak(E + m + ar)2(E− m − ar)rk (3.13) ͱͳΔɻ੔ཧ͢Δͱ n ∑ k=0 ak[{k(k − 1)(E + m) − κ(κ + 1)(E + m)}rk−2

+{ak(k − 1) − 2βk(E + m) − ak − aκ(κ + 1) − aκ}rk−1

+{−2βak − 4αk(E + m) − 2α(E + m) + β2(E + m) + aβ + (E + m)2(E− m)}rk

+{−4aαk − 2aα + aβ2+ 4αβ(E + m) + 2αa

− a(E + m)2+ 2a(E + m)(E− m)}rk+1

+{4αβa + 4α2(E + m) + a2(E− m) − 2a2(E + m)}rk+2 + (4α2a− a3_)rk+3_{] = 0} _(3.14) ͱͳΔ͕ɺα2₌ a2 4 Λ୅ೖ͢Δͱ࠷ޙͷ r k+3 _ͷ߲͸ফ͑ n ∑ k=0 ak[{k(k − 1) − κ(κ + 1)}(E + m)rk−2

+{ak(k − 2) − aκ(κ + 2) − 2βk(E + m)}rk−1

+{−aβ(2k − 1) − a(2k + 1)(E + m) + β2(E + m) + (E + m)2(E− m)}rk

+{−2a2k + aβ2+ 2aβ(E + m)− a(E + m)2+ 2a(E + m)(E− m)}rk+1

+{2a2β− 2a2m}rk+2= 0 (3.15) ͱͳΔɻ͜ͷࣜΛ r ͷ߃౳ࣜͱͯ͠ղ͚͹ଋറঢ়ଶͷղ͕ٻ·Δ͜ͱʹͳΔɻ࠷ޙͷ rk+2 ͷ߲͕࠷ߴ࣍਺ͱͳΔ͕ɺͦͷ܎਺͸ k ʹґଘ͠ͳ͍ͷͰҰൠతʹղ͚Δɻa ̸= 0 ͔ͩΒ 2β− 2m = 0 ∴ β = m (3.16) ͱͳΓɺࢦ਺ؔ਺ͷ෦෼͸͜ΕͰܾఆ͞ΕΔɻβ = m ͸ཻࢠͷ࣭ྔʹ૬౰͢ΔͷͰɺਖ਼· ͨ̌ͱߟ͑Δɻ͢ͳΘͪ β ≥ 0 ͱ͢Δɻɹ ࠷ऴతʹ੔ཧ͢Δͱ n ∑ k=0 ak[ (k + κ)(k− κ − 1)(E + m)rk−2 +{a(k + κ)(k − κ − 2) − 2mk(E + m)}rk−1

+{−am(2k − 1) − a(2k + 1)(E + m) + E2(E + m)}rk

+ (−2a2k + aE2)rk+1] = 0 (3.17) ͱͳΓɺ͜ΕΛ r ʹؔ͢Δ߃౳ࣜͱߟ͑ͯղ͖ɺκɺE ΛܾΊΔɻ ࣍ʹ࠷΋࣍਺ͷখ͞ͳ߲ (k = 0) Λߟ͑Δͱ r−2 _ͷ߲ͱͳΓ a0{−κ(κ + 1)}(E + m) = 0 (3.18) ͱͳΔ͕ɺκ = ±1, ±2, ±3, · · · Ͱ͋Δ͜ͱΛߟྀ͢Δͱ a0= 0 ·ͨ͸ κ = −1 ·ͨ͸ E + m = 0 (3.19) ͱ͍͏৚͕݅ٻ·ΔɻE < 0 ͷ৔߹ʹ͸ɺཻࢠ͕ੜ੒͞Εͨํ͕ΤωϧΪʔతʹ༗རͱͳ Δෆ҆ఆղͱͳΔͷͰɺE ≥ 0 ͱԾఆ͢Δɻm = 0ɺE = 0 ͷղ΋ҙຯͷ͋Δղͱ͸ࢥ͑ ͳ͍ͷͰɺE + m > 0 ͱԾఆ͢Δɻ࣮ࡍʹ m = 0ɺE = 0 ͱ͢Δͱʢ 3.17 ʣ͸ n ∑ k=0 ak{a(k + κ)(k − κ − 2)rk−1− am(2k − 1)rk− 2a2krk+1} = 0 (3.20) ͱͳΓɺ࠷ߴ࣍਺ͷ rn+1_{ͷ߲͕ −2a}2_nrn+1 _{ͱͳΔͷͰɺ͜ͷ߲͕ 0 ͱͳΓ߃౳ࣜΛຬ} ͨ͢͜ͱ͸Ͱ͖ͳ͍ɻ E + m > 0ͱͯ͠ κ = −1 ·ͨ͸ a0= 0ͷͦΕͧΕʹରͯ͠ɺଟ߲ࣜల։͕༗ݶͷ n ͰऴΘΔ৔߹ʹղ͕ଘࡏ͢Δ͔Ͳ͏͔ௐ΂Δɻ

4 κ =

₋₁

ͷ৔߹

ղ͘΂͖ํఔࣜʢ 3.17 ʣ͸ κ = −1 ʹରͯ͠ n ∑ k=0

ak[ k(k− 1)(E + m)rk−2+{a(k − 1)2− 2mk(E + m)}rk−1

+{−am(2k − 1) − a(2k + 1)(E + m) + E2(E + m)}rk

+ (−2a2k + aE2)rk+1] = 0 (4.1) ͱͳΔɻҎԼɺଟ߲ࣜల։ͷ߲਺ͷগͳ͍ॱʹݕ౼͢Δɻ

4.1 n = 0

a

0

̸= 0

ͷ৔߹ ࠷ॳʹ೾ಈؔ਺͕࠷΋࣍਺ͷখ͞ͳ G(r) = a0e−αr 2_−βr (4.2) ͱͳΔղ͕ଘࡏ͢Δ͔Ͳ͏͔ݕ౼͢Δɻʢ 4.1 ʣ͸ a0[ ar−1+{am − a(E + m) + E2(E + m)}r0+ aE2r1] = 0 (4.3) ͱͳΔ͕ɺ͜ͷ͕ࣜ͢΂ͯͷ࣍਺ͷ r ʹରͯ͠߃౳ࣜͱͳΔͨΊʹ͸ɺr−1_{ͷ߲͔Β a = 0} ͱͳΔ͕ɺ͜Ε͸ઢܗϙςϯγϟϧ͕ଘࡏ͠ͳ͍͜ͱΛҙຯ͢ΔͷͰແҙຯͰ͋Δɻκ = −1 ʹର͠ɺʢ 4.2 ʣͷΑ͏ͳ೾ಈؔ਺Λ࣋ͭଋറঢ়ଶͷղ͸ଘࡏ͠ͳ͍ɻ

(10)

4.2 n = 1

a

1

̸= 0

ͷ৔߹

͜ͷ৔߹ɺଟ߲ࣜ͸ k = 0 ͱ k = 1 ͕ڐ͞ΕΔͷͰ

a0[ ar−1+{am − a(E + m) + E2(E + m)}r0+ aE2r1]

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2+ aE2)r2] = 0 (4.4) ͱͳΔɻୈ߲̍ͷ r−1_{ͷ߲͸ a ̸= 0 Ͱ͋Δ͔Β a}₀_{= 0}_{ͱͳΔ͜ͱΛཁٻ͢Δɻ} a1 ͷ࠷ॳͷ߲ r0 ͷ߲͔Β m(E + m) = 0 (4.5) ͱͳΔ͕ɺԾఆ͔Β E + m ̸= 0 Ͱ͋Δ͔Βɺղ͕ଘࡏ͢Δͷ͸ m = 0 ͷ৔߹͚ͩͰ͋ Δɻr2_ͷ߲͸ −2a2+ aE2= a(−2a + E2) = 0 (4.6) Λ༩͑Δ͕ɺa ̸= 0 Ͱ͋Δ͔Β E2_{= 2a}_ͱͳΔɻr1_ͷ߲͸ −am − 3a(E + m) + E2(E + m) = 0 (4.7) ͱͳΔ͕ɺ m = 0ɺE2_{= 2a}_{Λ୅ೖ͢Δͱ}

−3aE + 2aE = −aE ̸= 0 (4.8)

ͱͳΓɺ͜ͷ৔߹ʹ΋ղ͕ଘࡏ͠ͳ͍͜ͱ͕Θ͔Δɻ

4.3 n = 2

a

2

̸= 0

ͷ৔߹

k = 0ɺ1ɺ2 ͕ڐ͞ΕΔ͔Βʢ 4.1 ʣ͸

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2_{+ aE}2_)r2_]

+ a2[ 2(E + m)r0+{a − 4m(E + m)}r1

+{−3am − 5a(E + m) + E2(E + m)}r2+ (−4a2+ aE2)r3]

= 0 (4.9) ͱͳΔɻ࠷ߴ࣍਺ͷ߲͸ r3_ͷ߲Ͱ a2(−4a2+ aE2) = 0 (4.10) ͱͳΔ͕ a2̸= 0ɺa ̸= 0 Ͱ͋Δ͔Β E2= 4a (4.11) ͕ٻ·Δɻa ̸= 0 Λߟྀ͢Δͱɺ࠷௿࣍਺ͷ r−1_{͔Β a}₀_{= 0}_͕ٻ·Δɻ r0 _ͷ߲͸ a1{−2m(E + m)} + a2{2(E + m)} = 0 (4.12) ͱͳΔ͕ɺ੔ཧ͢Δͱ (−2a1m + 2a2)(E + m) = 0 (4.13) Ͱ͋ΔɻE + m > 0 Λߟྀ͢Ε͹ a1m− a2= 0 (4.14) ͱͳΔɻ r1 _ͷ߲

a1{−am − 3a(E + m) + E2(E + m)} + a2{a − 4m(E + m)} = 0 (4.15)

ʹ a1m = a2Λ୅ೖͯ͠੔ཧ͢Δͱ (−3a + E2− 4m2)(E + m) = 0 (4.16) ͱͳΔɻE + m > 0 Λߟྀ͠ɺE2_{= 4a}_{Λ୅ೖ͢Δͱ} a = 4m2 (4.17) ͱ͍͏ؔ܎͕ࣜٻ·Δɻ r2 _ͷ߲

a1(−2a2+ aE2) + a2{−3am − 5a(E + m) + E2(E + m)} = 0 (4.18)

ʹ a2= ma1ɺE2= 4aΛ୅ೖͯ͠੔ཧ͢Δͱ 2a2− 3am2_{− am(E + m) = 0} _(4.19) ͱͳΔɻa ̸= 0 Λߟྀ͠ɺa = 4m2_{Λ୅ೖ͢Δͱ} 4m2= mE (4.20) ͱͳΔɻm = 0 ͷ৔߹ʹ͸ a = 4m2_{ΑΓɺa ΋ 0 ͱͳͬͯ͠·͏ͷͰɺm ̸= 0 ͱͳΒͳ} ͚Ε͹ͳΒͳ͍ɻ͜ΕΛߟྀ͢Δͱɺ্͔ࣜΒ E = 4m ͕ٻ·Δɻ ݁ہɺղ͕ଘࡏ͢Δͷ͸ κ =−1 a2= ma1 a0= 0 a = 4m2 (4.21) ͷ৔߹Ͱɺ೾ಈؔ਺͸ G(r) = a1(r + mr2)e−2m 2_r2_−mr (4.22) ͱͳΓɺΤωϧΪʔݻ༗஋͸ E = 4m =√4a (4.23) Ͱ͋Δɻ܎਺ a1͸ن֨Խ৚͔݅ΒٻΊΒΕΔɻ ͜ͷଋറղ͸ϙςϯγϟϧͷେ͖͞ a ͱཻࢠͷ࣭ྔ m ͷؒʹ a = 4m2 _ͷؔ܎͕͋Δ ͱ͖ʹଘࡏ͢ΔղͰ͋Δɻ

(11)

4.2 n = 1

a

1

̸= 0

ͷ৔߹

͜ͷ৔߹ɺଟ߲ࣜ͸ k = 0 ͱ k = 1 ͕ڐ͞ΕΔͷͰ

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2+ aE2)r2] = 0 (4.4) ͱͳΔɻୈ߲̍ͷ r−1_{ͷ߲͸ a ̸= 0 Ͱ͋Δ͔Β a}₀_{= 0}_{ͱͳΔ͜ͱΛཁٻ͢Δɻ} a1 ͷ࠷ॳͷ߲ r0 ͷ߲͔Β m(E + m) = 0 (4.5) ͱͳΔ͕ɺԾఆ͔Β E + m ̸= 0 Ͱ͋Δ͔Βɺղ͕ଘࡏ͢Δͷ͸ m = 0 ͷ৔߹͚ͩͰ͋ Δɻr2_ͷ߲͸ −2a2+ aE2= a(−2a + E2) = 0 (4.6) Λ༩͑Δ͕ɺa ̸= 0 Ͱ͋Δ͔Β E2_{= 2a}_ͱͳΔɻr1 _ͷ߲͸ −am − 3a(E + m) + E2(E + m) = 0 (4.7) ͱͳΔ͕ɺ m = 0ɺE2_{= 2a}_{Λ୅ೖ͢Δͱ}

−3aE + 2aE = −aE ̸= 0 (4.8)

ͱͳΓɺ͜ͷ৔߹ʹ΋ղ͕ଘࡏ͠ͳ͍͜ͱ͕Θ͔Δɻ

4.3 n = 2

a

2

̸= 0

ͷ৔߹

k = 0ɺ1ɺ2 ͕ڐ͞ΕΔ͔Βʢ 4.1 ʣ͸

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2_{+ aE}2_)r2_]

+ a2[ 2(E + m)r0+{a − 4m(E + m)}r1

+{−3am − 5a(E + m) + E2(E + m)}r2+ (−4a2+ aE2)r3]

= 0 (4.9) ͱͳΔɻ࠷ߴ࣍਺ͷ߲͸ r3 _ͷ߲Ͱ a2(−4a2+ aE2) = 0 (4.10) ͱͳΔ͕ a2̸= 0ɺa ̸= 0 Ͱ͋Δ͔Β E2= 4a (4.11) ͕ٻ·Δɻa ̸= 0 Λߟྀ͢Δͱɺ࠷௿࣍਺ͷ r−1_{͔Β a}₀_{= 0}_͕ٻ·Δɻ r0_ͷ߲͸ a1{−2m(E + m)} + a2{2(E + m)} = 0 (4.12) ͱͳΔ͕ɺ੔ཧ͢Δͱ (−2a1m + 2a2)(E + m) = 0 (4.13) Ͱ͋ΔɻE + m > 0 Λߟྀ͢Ε͹ a1m− a2= 0 (4.14) ͱͳΔɻ r1_ͷ߲

a1{−am − 3a(E + m) + E2(E + m)} + a2{a − 4m(E + m)} = 0 (4.15)

ʹ a1m = a2Λ୅ೖͯ͠੔ཧ͢Δͱ (−3a + E2− 4m2)(E + m) = 0 (4.16) ͱͳΔɻE + m > 0 Λߟྀ͠ɺE2_{= 4a}_{Λ୅ೖ͢Δͱ} a = 4m2 (4.17) ͱ͍͏ؔ܎͕ࣜٻ·Δɻ r2_ͷ߲

a1(−2a2+ aE2) + a2{−3am − 5a(E + m) + E2(E + m)} = 0 (4.18)

ʹ a2= ma1ɺE2= 4aΛ୅ೖͯ͠੔ཧ͢Δͱ 2a2− 3am2_{− am(E + m) = 0} _(4.19) ͱͳΔɻa ̸= 0 Λߟྀ͠ɺa = 4m2_{Λ୅ೖ͢Δͱ} 4m2= mE (4.20) ͱͳΔɻm = 0 ͷ৔߹ʹ͸ a = 4m2_{ΑΓɺa ΋ 0 ͱͳͬͯ͠·͏ͷͰɺm ̸= 0 ͱͳΒͳ} ͚Ε͹ͳΒͳ͍ɻ͜ΕΛߟྀ͢Δͱɺ্͔ࣜΒ E = 4m ͕ٻ·Δɻ ݁ہɺղ͕ଘࡏ͢Δͷ͸ κ =−1 a2= ma1 a0= 0 a = 4m2 (4.21) ͷ৔߹Ͱɺ೾ಈؔ਺͸ G(r) = a1(r + mr2)e−2m 2_r2_−mr (4.22) ͱͳΓɺΤωϧΪʔݻ༗஋͸ E = 4m =√4a (4.23) Ͱ͋Δɻ܎਺ a1͸ن֨Խ৚͔݅ΒٻΊΒΕΔɻ ͜ͷଋറղ͸ϙςϯγϟϧͷେ͖͞ a ͱཻࢠͷ࣭ྔ m ͷؒʹ a = 4m2 _ͷؔ܎͕͋Δ ͱ͖ʹଘࡏ͢ΔղͰ͋Δɻ

(12)

4.4 n = 3

a

3

̸= 0

ͷ৔߹

k = 0, 1, 2, 3͕ڐ͞ΕΔͷͰɺํఔࣜ͸

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2+ aE2)r2]

+ a2[ 2(E + m)r0+{a − 4m(E + m)}r1

+{−3am − 5a(E + m) + E2(E + m)}r2+ (−4a2+ aE2)r3] + a3[ 6(E + m)r1+{4a − 6m(E + m)}r2

+{−5am − 7a(E + m) + E2(E + m)}r3+ (−6a2+ aE2)r4]

= 0 (4.24) ͱͳΔɻ r−1ͷ߲͔Β a0= 0͕ٻ·Γɺr0 ͷ߲ʹؔͯ͠͸ 4.3 અͱಉ͘͡ a1{−2m(E + m)} + a2{2(E + m)} = 0 (4.25) ͱͳΔͷͰɺE + m > 0 Λߟྀͯ͠ a2= ma1 (4.26) ͕ٻ·Δɻ ࠷ߴ࣍਺ͷ߲ r4 _͸ −6a2+ aE2= 0 (4.27) ͱͳΔ͕ɺa ̸= 0 Ͱ͋Δ͔Β E2= 6a (4.28) ͕ٻ·Δɻ r1ͷ߲͸ a1{−am − 3a(E + m) + E2(E + m)} + a2{a − 4m(E + m)} + a3{6(E + m)} = 0 (4.29) ͱͳΔ͕ɺa2= ma1Ͱ͋Δ͔Β a1(E + m)(−3a + E2− 4m2) + 6a3(E + m) = 0 (4.30) ͱͳΓɺE + m ̸= 0 Λߟྀ͢Δͱ a3= 1 6(3a− E 2_{+ 4m}2_)a 1 (4.31) ͱͳΔɻ͞Βʹ E2_{= 6a}_Λ୅ೖͯ͠ a3= 1 6(−3a + 4m 2_)a 1 (4.32) ͕ٻ·Δɻ r2 _ͷ߲

a1(−2a2+ aE2) + a2{−3am − 5a(E + m) + E2(E + m)}

+ a3{4a − 6m(E + m)} = 0 (4.33) Λ a2= ma1ɺa3= 1 6(−3a + 4m 2_)a 1ɺE2= 6aΛ࢖ͬͯ੔ཧ͢Δͱ (E2− m2)a + 12(a− m2)m(E + m) = 0 (4.34) ͱͳΓɺE + m ̸= 0 Ͱ͋Δ͔Β aE = m(12m2− 11a) (4.35) ͕ٻ·Δɻ ಉ༷ʹ r3_ͷ߲

a2(−4a2+ aE2) + a3{−5am − 7a(E + m) + E2(E + m)} = 0 (4.36)

Λ੔ཧ͢Δͱ 9 2a 2_m₋ 10 3 am 3_{+ (}1 2a 2₋2 3am 2_{)(E + m) = 0} _(4.37) ͱͳΔ͕ɺa ̸= 0 Ͱ͋Δ͔Β (3a− 4m2)E =−30am + 24m3 (4.38) ͱͳΔɻ m = 0ͱ͢Δͱʢ 4.35 ʣ͔Β E = 0 ͱͳͬͯ͠·͏ͷͰɺm ̸= 0 ͱͯ͠ʢ 4.35 ʣͱʢ 4.38 ʣ Λ࿈ཱ͢Δͱ 3a2− 56am2_{+ 48m}4_{= 0} _(4.39) ͕ٻ·Δɻ͜ΕΛղ͘ͱ a = 28± 8 √ 10 3 m 2_{> 0} _(4.40) ͱͳΔɻ͞Βʹ͜ͷࣜΛʢ 4.35 ʣʹ୅ೖͯ͠ E Λղ͘ͱ E = (−4 ∓ 2√10)m (4.41) ͱͳΔɻE > 0 ͷղΛٻΊΔͳΒ E = (−4 + 2√10)mɺa = 28− 8 √ 10 3 m 2_ΛͱΔ΂͖Ͱ ͋Δɻ·ͨʢ 4.35 ʣΛߟ͑Ε͹ a < 12 11m 2 _{ͱͳΔ͜ͱ΋Θ͔Δɻ͞Βʹʢ 4.32 ʣΑΓ} a3= 1 6(4m 2_{− 3a)a} 1= (−4 +4 3 √ 10)m2a1 (4.42) ͕ٻ·Δɻ

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4.4 n = 3

a

3

̸= 0

ͷ৔߹

k = 0, 1, 2, 3͕ڐ͞ΕΔͷͰɺํఔࣜ͸

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2+ aE2)r2]

+ a2[ 2(E + m)r0+{a − 4m(E + m)}r1

+{−5am − 7a(E + m) + E2(E + m)}r3+ (−6a2+ aE2)r4]

= 0 (4.24) ͱͳΔɻ r−1ͷ߲͔Β a0= 0͕ٻ·Γɺr0ͷ߲ʹؔͯ͠͸ 4.3 અͱಉ͘͡ a1{−2m(E + m)} + a2{2(E + m)} = 0 (4.25) ͱͳΔͷͰɺE + m > 0 Λߟྀͯ͠ a2= ma1 (4.26) ͕ٻ·Δɻ ࠷ߴ࣍਺ͷ߲ r4_͸ −6a2+ aE2= 0 (4.27) ͱͳΔ͕ɺa ̸= 0 Ͱ͋Δ͔Β E2= 6a (4.28) ͕ٻ·Δɻ r1ͷ߲͸ a1{−am − 3a(E + m) + E2(E + m)} + a2{a − 4m(E + m)} + a3{6(E + m)} = 0 (4.29) ͱͳΔ͕ɺa2= ma1Ͱ͋Δ͔Β a1(E + m)(−3a + E2− 4m2) + 6a3(E + m) = 0 (4.30) ͱͳΓɺE + m ̸= 0 Λߟྀ͢Δͱ a3= 1 6(3a− E 2_{+ 4m}2_)a 1 (4.31) ͱͳΔɻ͞Βʹ E2_{= 6a}_Λ୅ೖͯ͠ a3= 1 6(−3a + 4m 2_)a 1 (4.32) ͕ٻ·Δɻ r2_ͷ߲

a1(−2a2+ aE2) + a2{−3am − 5a(E + m) + E2(E + m)}

+ a3{4a − 6m(E + m)} = 0 (4.33) Λ a2= ma1ɺa3= 1 6(−3a + 4m 2_)a 1ɺE2= 6aΛ࢖ͬͯ੔ཧ͢Δͱ (E2− m2)a + 12(a− m2)m(E + m) = 0 (4.34) ͱͳΓɺE + m ̸= 0 Ͱ͋Δ͔Β aE = m(12m2− 11a) (4.35) ͕ٻ·Δɻ ಉ༷ʹ r3_ͷ߲

a2(−4a2+ aE2) + a3{−5am − 7a(E + m) + E2(E + m)} = 0 (4.36)

Λ੔ཧ͢Δͱ 9 2a 2_m₋ 10 3 am 3_{+ (}1 2a 2₋2 3am 2_{)(E + m) = 0} _(4.37) ͱͳΔ͕ɺa ̸= 0 Ͱ͋Δ͔Β (3a− 4m2)E =−30am + 24m3 (4.38) ͱͳΔɻ m = 0ͱ͢Δͱʢ 4.35 ʣ͔Β E = 0 ͱͳͬͯ͠·͏ͷͰɺm ̸= 0 ͱͯ͠ʢ 4.35 ʣͱʢ 4.38 ʣ Λ࿈ཱ͢Δͱ 3a2− 56am2_{+ 48m}4_{= 0} _(4.39) ͕ٻ·Δɻ͜ΕΛղ͘ͱ a = 28± 8 √ 10 3 m 2_{> 0} _(4.40) ͱͳΔɻ͞Βʹ͜ͷࣜΛʢ 4.35 ʣʹ୅ೖͯ͠ E Λղ͘ͱ E = (−4 ∓ 2√10)m (4.41) ͱͳΔɻE > 0 ͷղΛٻΊΔͳΒ E = (−4 + 2√10)mɺa =28− 8 √ 10 3 m 2_ΛͱΔ΂͖Ͱ ͋Δɻ·ͨʢ 4.35 ʣΛߟ͑Ε͹ a < 12 11m 2_{ͱͳΔ͜ͱ΋Θ͔Δɻ͞Βʹʢ 4.32 ʣΑΓ} a3= 1 6(4m 2_{− 3a)a} 1= (−4 +4 3 √ 10)m2a1 (4.42) ͕ٻ·Δɻ

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n = 3ʹରͯ͠͸ κ =−1 a3= (−4 +4 3 √ 10)m2a1 a2= ma1 a0= 0 a = 28− 8 √ 10 3 m 2_{∼ 0.90m}2 _(4.43) ͷ৔߹ʹղ͕ଘࡏ͠ɺ೾ಈؔ਺͸ G(r) = a1{r + mr2+ 1 6(4m 2_{− 3a)r}3_}e−ar2_/2_−mr = a1{r + mr2+ (−4 +4 3 √ 10)m2r3}e−ar2/2−mr (4.44) ΤωϧΪʔݻ༗஋͸ E =√6a = (−4 + 2√10)m∼ 2.32m (4.45) ͱͳΔɻ

4.5 n = 4

a

4

̸= 0

ͷ৔߹ k = 0, 1, 2, 3, 4͕ڐ͞ΕΔɻղ͘΂͖߃౳ࣜ͸ a0[ ar−1+{am − a(E + m) + E2(E + m)}r0+ aE2r1]

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2+ aE2)r2]

+ a2[ 2(E + m)r0+{a − 4m(E + m)}r1

+{−5am − 7a(E + m) + E2_{(E + m)}_}r3_{+ (}_−6a2_{+ aE}2_)r4_]

+ a4[ 12(E + m)r2+{9a − 8m(E + m)}r3

+{−7am − 9a(E + m) + E2(E + m)}r4+ (−8a2+ aE2)r5]

= 0 (4.46) Ͱ͋Δɻ r−1ɺr0_ɺr1_{ͷ߲ʹؔͯ͠͸ 4.4 અͱಉ͡Ͱ͋Γ} a0= 0 a2= ma1 (4.47) a3= 1 6(3a− E 2_{+ 4m}2_)a 1 (4.48) ͕ٻ·Δɻ ࠷ߴ࣍਺ͷ r5 _ͷ߲͸ −8a2+ aE2= 0 (4.49) ͱͳΓɺa ̸= 0 Λߟྀ͢Δͱ E2= 8a (4.50) ͱͳΔɻ͜ΕΛ࢖͏ͱ a3= (−5 6a + 2 3m 2_)a 1 (4.51) ͱͳΔɻ r2 _ͷ߲

a1(−2a2+ aE2) + a2{−3am − 5a(E + m) + E2(E + m)}

+ a3{4a − 6m(E + m)} + a4{12(E + m)} = 0 (4.52)

Λɺa2= ma1ɺa3= (− 5 6a + 2 3m 2_)a 1ɺE2= 8aΛ࢖ͬͯ੔ཧ͢Δͱ a1{8 3a 2₋1 3am

2_{+ (8am}_{− 4m}3_{)(E + m)}_{} + 12(E + m)a}

4= 0 (4.53) ͱͳΔɻE + m ̸= 0 Ͱ͋Δ͔Β a4= a1(−23 36am− 1 36aE + 1 3m 3₎ _(4.54) ͕ٻ·Δɻ r3 _ͷ߲͸

a2(−4a2+ aE2) + a3{−5am − 7a(E + m) + E2(E + m)}

+ a4{9a − 8m(E + m)} = 0 (4.55) ͱͳΔ͕ a2= ma1 a3= (− 5 6a + 2 3m 2_)a 1 a4= (−23 36am− 1 36aE + 1 3m 3_)a 1 E2= 8a Λ࢖ͬͯ੔ཧ͢Δͱ 1 3(E 2_{− m}2_)am₋ 1 4a 2_{(E + m)} + (−5₆a2+52 9 am 2₊2 9amE− 8 3m 4_{)(E + m) = 0} _(4.56) ͱͳΔɻ͞Βʹ E + m ̸= 0 Λߟྀ͢Ε͹ 1 3(E− m)am − 1 4a 2₋5 6a 2₊ 52 9 am 2₊ 2 9amE− 8 3m 4_{= 0} _(4.57) ͱͳΓ

20amE = 39a2− 196am2+ 96m4 (4.58) ͕ٻ·Δɻ

ಉ༷ʹ r4_ͷ߲

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n = 3ʹରͯ͠͸ κ =−1 a3= (−4 +4 3 √ 10)m2a1 a2= ma1 a0= 0 a = 28− 8 √ 10 3 m 2_{∼ 0.90m}2 _(4.43) ͷ৔߹ʹղ͕ଘࡏ͠ɺ೾ಈؔ਺͸ G(r) = a1{r + mr2+ 1 6(4m 2_{− 3a)r}3_}e−ar2_/2_−mr = a1{r + mr2+ (−4 +4 3 √ 10)m2r3}e−ar2/2−mr (4.44) ΤωϧΪʔݻ༗஋͸ E =√6a = (−4 + 2√10)m∼ 2.32m (4.45) ͱͳΔɻ

4.5 n = 4

a

4

̸= 0

ͷ৔߹ k = 0, 1, 2, 3, 4͕ڐ͞ΕΔɻղ͘΂͖߃౳ࣜ͸ a0[ ar−1+{am − a(E + m) + E2(E + m)}r0+ aE2r1]

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2+ aE2)r2]

+ a2[ 2(E + m)r0+{a − 4m(E + m)}r1

+{−5am − 7a(E + m) + E2_{(E + m)}_}r3_{+ (}_−6a2_{+ aE}2_)r4_]

+ a4[ 12(E + m)r2+{9a − 8m(E + m)}r3

+{−7am − 9a(E + m) + E2(E + m)}r4+ (−8a2+ aE2)r5]

= 0 (4.46) Ͱ͋Δɻ r−1ɺr0_ɺr1_{ͷ߲ʹؔͯ͠͸ 4.4 અͱಉ͡Ͱ͋Γ} a0= 0 a2= ma1 (4.47) a3= 1 6(3a− E 2_{+ 4m}2_)a 1 (4.48) ͕ٻ·Δɻ ࠷ߴ࣍਺ͷ r5 _ͷ߲͸ −8a2+ aE2= 0 (4.49) ͱͳΓɺa ̸= 0 Λߟྀ͢Δͱ E2= 8a (4.50) ͱͳΔɻ͜ΕΛ࢖͏ͱ a3= (−5 6a + 2 3m 2_)a 1 (4.51) ͱͳΔɻ r2_ͷ߲

a1(−2a2+ aE2) + a2{−3am − 5a(E + m) + E2(E + m)}

+ a3{4a − 6m(E + m)} + a4{12(E + m)} = 0 (4.52)

Λɺa2= ma1ɺa3= (− 5 6a + 2 3m 2_)a 1ɺE2= 8aΛ࢖ͬͯ੔ཧ͢Δͱ a1{8 3a 2₋1 3am

2_{+ (8am}_{− 4m}3_{)(E + m)}_{} + 12(E + m)a}

4= 0 (4.53) ͱͳΔɻE + m ̸= 0 Ͱ͋Δ͔Β a4= a1(−23 36am− 1 36aE + 1 3m 3₎ _(4.54) ͕ٻ·Δɻ r3_ͷ߲͸

a2(−4a2+ aE2) + a3{−5am − 7a(E + m) + E2(E + m)}

+ a4{9a − 8m(E + m)} = 0 (4.55) ͱͳΔ͕ a2= ma1 a3= (− 5 6a + 2 3m 2_)a 1 a4= (−23 36am− 1 36aE + 1 3m 3_)a 1 E2= 8a Λ࢖ͬͯ੔ཧ͢Δͱ 1 3(E 2_{− m}2_)am₋ 1 4a 2_{(E + m)} + (−5₆a2+52 9 am 2₊ 2 9amE− 8 3m 4_{)(E + m) = 0} _(4.56) ͱͳΔɻ͞Βʹ E + m ̸= 0 Λߟྀ͢Ε͹ 1 3(E− m)am − 1 4a 2₋5 6a 2₊52 9am 2₊2 9amE− 8 3m 4_{= 0} _(4.57) ͱͳΓ

20amE = 39a2− 196am2+ 96m4 (4.58) ͕ٻ·Δɻ

ಉ༷ʹ r4_ͷ߲

(16)

͸ (−5 6a + 2 3m 2₎₍_−6a2_{+ 8a}2_{) + (}₋23 36am− 1 36aE + 1 3m 3₎

× {−7am − 9a(E + m) + 8a(E + m)} = 0 (4.60)

ͱͳΔ͕ɺa ̸= 0 Ͱ͋Δ͔Βɺ੔ཧͯ͠

(31am− 12m3)E = 52a2− 232am2+ 96m4 (4.61) ͱͳΔɻ ղ͕ଘࡏ͢Δͷ͸ a2= ma1 a3= (−5 6a + 2 3m 2_)a 1 a4= (−23 36am− 1 36aE + 1 3m 3_)a 1 E2= 8a Λຬͨ͠ɺʢ 4.58 ʣɺʢ 4.61 ʣཱ͕྆͢Δ৔߹Ͱ͋Δɻ ʢ 4.58 ʣͱʢ 4.61 ʣΛ࿈ཱ͠੔ཧ͢Δͱ

169a3− 1904a2m2+ 3408am4− 1152m6= 0 (4.62)

ͱͳΔɻ͜ͷ̏࣍ํఔࣜΛຬͨ͢Α͏ͳ a ͱ m2 _{ͷؔ܎͕ଘࡏ͢Δ৔߹ʹͷΈղ͕ଘࡏ} ͢Δɻ ΤωϧΪʔݻ༗஋͸ʢ 4.58 ʣΑΓ E = 39 20 a m− 49 5m + 24 5 m3 a (4.63) ͱͳΔɻ κ =−1 Ͱ ೾ಈؔ਺͕ r4_{·ͰͰٻ·Δղ͸} a0= 0 a2= ma1 a3= (−5 6a + 2 3m 2_)a 1 a4= (−11 30am + 1 5m 3₋ 13 240 a2 m)a1 Ͱɺํఔࣜ

169a3− 1904a2m2+ 3408am4− 1152m6= 0 (4.64)

Λຬͨ͢৔߹Ͱ͋Δɻ͜ͷ̏࣍ํఔࣜͷ਺஋ղΛٻΊͯΈΔͱ a = 0.44m2, 1.68m2, 9.14m2 (4.65) ͱͳΔ͜ͱ͕Θ͔Δɻ͜ΕʹରԠ͢ΔΤωϧΪʔ͸ E = 1.88m, −3.67m, 8.55m (4.66) ͱͳΔɻ2 ͭ໨ͷղ͸ E < 0 Ͱ͋Γɺଋറঢ়ଶͷղͱ͸ߟ͑ΒΕͳ͍ɻ ݁ہɺ೾ಈؔ਺͸ G(r) =a1{r + mr2+ (−5 6a + 2 3m 2_)r3 + (−11 30am + 1 5m 3₋ 13 240 a2 m)r 4_}e−ar2_/2−mr (4.67) Ͱ͋ΓɺΤωϧΪʔݻ༗஋͸ E =√8a = { 1.88m for a = 0.44m2 8.55m for a = 9.14m2 (4.68) ͱͳΔɻ

4.6 n = 5

a

5

̸= 0

ͷ৔߹ ଟ߲ࣜ͸ k = 0, 1, 2, 3, 4, 5 ͕ڐ͞ΕΔͷͰɺํఔࣜ͸ a0[ ar−1+{am − a(E + m) + E2(E + m)}r0+ aE2r1]

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2+ aE2)r2]

+ a2[ 2(E + m)r0+{a − 4m(E + m)}r1

+{−3am − 5a(E + m) + E2_{(E + m)}_}r2_{+ (}_−4a2_{+ aE}2_)r3_]

+ a3[ 6(E + m)r1+{4a − 6m(E + m)}r2

+{−9am − 11a(E + m) + E2(E + m)}r5+ (−10a2+ aE2)r6]

= 0 (4.69) ͱͳΔɻ r−1ɺr0_ɺr1 _{ͷ߲ʹؔͯ͠͸ 4.5 અ ͱಉ͡Ͱɺٻ·Δ৚݅͸} a0= 0 a2= ma1 a3= 1 6(3a− E 2_{+ 4m}2_)a 1 Ͱ͋Γɺ࠷ߴ࣍਺ͷ߲͸ r6 _Ͱ −10a2+ aE2= 0 (4.70) ͱͳΓɺa ̸= 0 Λߟྀͯ͠ E2= 10a (4.71) ͕ٻ·Δɻ͜ΕΛ࢖͑͹ a3= (− 7 6a + 2 3m 2_)a 1 (4.72)

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͸ (−5 6a + 2 3m 2₎₍_−6a2_{+ 8a}2_{) + (}₋23 36am− 1 36aE + 1 3m 3₎

× {−7am − 9a(E + m) + 8a(E + m)} = 0 (4.60)

ͱͳΔ͕ɺa ̸= 0 Ͱ͋Δ͔Βɺ੔ཧͯ͠

(31am− 12m3)E = 52a2− 232am2+ 96m4 (4.61) ͱͳΔɻ ղ͕ଘࡏ͢Δͷ͸ a2= ma1 a3= (−5 6a + 2 3m 2_)a 1 a4= (−23 36am− 1 36aE + 1 3m 3_)a 1 E2= 8a Λຬͨ͠ɺʢ 4.58 ʣɺʢ 4.61 ʣཱ͕྆͢Δ৔߹Ͱ͋Δɻ ʢ 4.58 ʣͱʢ 4.61 ʣΛ࿈ཱ͠੔ཧ͢Δͱ

169a3− 1904a2m2+ 3408am4− 1152m6= 0 (4.62)

ͱͳΔɻ͜ͷ̏࣍ํఔࣜΛຬͨ͢Α͏ͳ a ͱ m2 _{ͷؔ܎͕ଘࡏ͢Δ৔߹ʹͷΈղ͕ଘࡏ} ͢Δɻ ΤωϧΪʔݻ༗஋͸ʢ 4.58 ʣΑΓ E = 39 20 a m− 49 5m + 24 5 m3 a (4.63) ͱͳΔɻ κ =−1 Ͱ ೾ಈؔ਺͕ r4 _{·ͰͰٻ·Δղ͸} a0= 0 a2= ma1 a3= (−5 6a + 2 3m 2_)a 1 a4= (−11 30am + 1 5m 3₋ 13 240 a2 m)a1 Ͱɺํఔࣜ

169a3− 1904a2m2+ 3408am4− 1152m6= 0 (4.64)

Λຬͨ͢৔߹Ͱ͋Δɻ͜ͷ̏࣍ํఔࣜͷ਺஋ղΛٻΊͯΈΔͱ a = 0.44m2, 1.68m2, 9.14m2 (4.65) ͱͳΔ͜ͱ͕Θ͔Δɻ͜ΕʹରԠ͢ΔΤωϧΪʔ͸ E = 1.88m, −3.67m, 8.55m (4.66) ͱͳΔɻ2 ͭ໨ͷղ͸ E < 0 Ͱ͋Γɺଋറঢ়ଶͷղͱ͸ߟ͑ΒΕͳ͍ɻ ݁ہɺ೾ಈؔ਺͸ G(r) =a1{r + mr2+ (−5 6a + 2 3m 2_)r3 + (−11 30am + 1 5m 3₋ 13 240 a2 m)r 4_}e−ar2_/2−mr (4.67) Ͱ͋ΓɺΤωϧΪʔݻ༗஋͸ E =√8a = { 1.88m for a = 0.44m2 8.55m for a = 9.14m2 (4.68) ͱͳΔɻ

4.6 n = 5

a

5

̸= 0

ͷ৔߹ ଟ߲ࣜ͸ k = 0, 1, 2, 3, 4, 5 ͕ڐ͞ΕΔͷͰɺํఔࣜ͸ a0[ ar−1+{am − a(E + m) + E2(E + m)}r0+ aE2r1]

+ a1[−2m(E + m)r0+{−am − 3a(E + m) + E2(E + m)}r1

+ (−2a2+ aE2)r2]

+ a2[ 2(E + m)r0+{a − 4m(E + m)}r1

+{−3am − 5a(E + m) + E2_{(E + m)}_}r2_{+ (}_−4a2_{+ aE}2_)r3_]

+ a3[ 6(E + m)r1+{4a − 6m(E + m)}r2

+{−9am − 11a(E + m) + E2(E + m)}r5+ (−10a2+ aE2)r6]

= 0 (4.69) ͱͳΔɻ r−1ɺr0_ɺr1_{ͷ߲ʹؔͯ͠͸ 4.5 અ ͱಉ͡Ͱɺٻ·Δ৚݅͸} a0= 0 a2= ma1 a3= 1 6(3a− E 2_{+ 4m}2_)a 1 Ͱ͋Γɺ࠷ߴ࣍਺ͷ߲͸ r6 _Ͱ −10a2+ aE2= 0 (4.70) ͱͳΓɺa ̸= 0 Λߟྀͯ͠ E2= 10a (4.71) ͕ٻ·Δɻ͜ΕΛ࢖͑͹ a3= (− 7 6a + 2 3m 2_)a 1 (4.72)

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ͱॻ͚Δɻ

r2_ͷ߲

a1(−2a2+ aE2) + a2{−3am − 5a(E + m) + E2(E + m)}

+ a3{4a − 6m(E + m)} + a4{12(E + m)} = 0 (4.73)

͸ a2= ma1 a3= (−7 6a + 2 3m 2_)a 1 E2= 10a Λ࢖ͬͯ੔ཧ͢Ε͹ a1{1 3a(E

2_{− m}2_{) + (12am}_{− 4m}3_{)(E + m)}_{} + 12(E + m)a}

4= 0 (4.74) ͱͳΔ͕ E + m ̸= 0 Λߟྀ͢Ε͹ a4= a1(−35 36am− 1 36aE + 1 3m 3₎ _(4.75) ͕ٻ·Δɻ r3ͷ߲

a2(−4a2+ aE2) + a3{−5am − 7a(E + m) + E2(E + m)}

+ a4{9a − 8m(E + m)} + a5{20(E + m)} = 0 (4.76)

ʹର͠ a2= ma1 a3= (−7 6a + 2 3m 2_)a 1 a4= (− 35 36am− 1 36aE + 1 3m 3_)a 1 E2= 10a Λ࢖ͬͯɺ੔ཧ͢Δͱ a1{1 3(E 2_{− m}2_)am₋1 4a 2_{(E + m) + (}₋7 2a 2₊ 88 9 am 2₊ 2 9amE− 8 3m 4₎ × (E + m)} + 20a5(E + m) = 0 (4.77) ͱͳΓɺE + m ̸= 0 Λߟྀͯ͠ a5= ( 3 16a 2₋ 1 36amE− 17 36am 2₊ 2 15m 4_)a 1 (4.78) ͕ٻ·Δɻ ಉ༷ʹ r4_ͷ߲

a3(−6a2+ aE2) + a4{−7am − 9a(E + m) + E2(E + m)}

+ a5{16a − 10m(E + m)} = 0 (4.79) Λ a3= (−7 6a + 2 3m 2_)a 1 a4= (−35 36am− 1 36aE + 1 3m 3_)a 1 a5= (3 16a 2₋ 1 36amE− 17 36am 2₊ 2 15m 4_)a 1 E2= 10a Λ࢖ͬͯ੔ཧ͢Δͱ − ₃₆7 a2(E2− m2) + 43 90am 2_(E2_{− m}2₎ + (−25 8a 2_{m +}16 3am 3₋4 3m 5_{)(E + m) = 0} _(4.80) ͱͳΔɻE + m ̸= 0 Λߟྀ͢Ε͹

(70a2− 172am2)E =−1055a2m + 1748am3− 480m5 (4.81) ͕ٻ·Δɻ

r5 ͷ߲͸

a4(−8a2+ aE2) + a5{−9am − 11a(E + m) + E2(E + m)} = 0 (4.82)

Ͱ͋Δ͕ɺ੔ཧ͢Δͱ

(175a2− 540am2+ 96m4)E =−2550a2m + 3880am3− 960m5 (4.83) ͕ٻ·Δɻ Ҏ্ΑΓɺղ͕ଘࡏ͢Δͷ͸ a0= 0 a2= ma1 a3= (−7 6a + 2 3m 2_)a 1 a4= (−35 36am− 1 36aE + 1 3m 3_)a 1 a5= ( 3 16a 2₋ 1 36amE− 17 36am 2₊ 2 15m 4_)a 1 E2= 10a Λຬͨ͠ɺ͞Βʹʢ 4.81 ʣͱʢ 4.83 ʣཱ͕྆͢Δ৔߹Ͱ͋Δɻ͜ͷ̎ͭͷ৚݅ࣜʢ 4.81 ʣͱ ʢ 4.83 ʣΛ࿈ཱ͢Δͱ

(70a2− 172am2)(−2550a2m + 3880am3− 960m5)

= (−1055a2m + 1748am3− 480m5)(175a2− 540am2+ 96m4) (4.84) ͱͳΔ͕ɺm ̸= 0 Λ࢖ͬͯ

(19)

ͱॻ͚Δɻ

r2_ͷ߲

a1(−2a2+ aE2) + a2{−3am − 5a(E + m) + E2(E + m)}

+ a3{4a − 6m(E + m)} + a4{12(E + m)} = 0 (4.73)

͸ a2= ma1 a3= (−7 6a + 2 3m 2_)a 1 E2= 10a Λ࢖ͬͯ੔ཧ͢Ε͹ a1{1 3a(E

2_{− m}2_{) + (12am}_{− 4m}3_{)(E + m)}_{} + 12(E + m)a}

4= 0 (4.74) ͱͳΔ͕ E + m ̸= 0 Λߟྀ͢Ε͹ a4= a1(−35 36am− 1 36aE + 1 3m 3₎ _(4.75) ͕ٻ·Δɻ r3ͷ߲

a2(−4a2+ aE2) + a3{−5am − 7a(E + m) + E2(E + m)}

+ a4{9a − 8m(E + m)} + a5{20(E + m)} = 0 (4.76)

ʹର͠ a2= ma1 a3= (−7 6a + 2 3m 2_)a 1 a4= (− 35 36am− 1 36aE + 1 3m 3_)a 1 E2= 10a Λ࢖ͬͯɺ੔ཧ͢Δͱ a1{1 3(E 2_{− m}2_)am₋1 4a 2_{(E + m) + (}₋7 2a 2₊ 88 9 am 2₊ 2 9amE− 8 3m 4₎ × (E + m)} + 20a5(E + m) = 0 (4.77) ͱͳΓɺE + m ̸= 0 Λߟྀͯ͠ a5= (3 16a 2₋ 1 36amE− 17 36am 2₊ 2 15m 4_)a 1 (4.78) ͕ٻ·Δɻ ಉ༷ʹ r4_ͷ߲

a3(−6a2+ aE2) + a4{−7am − 9a(E + m) + E2(E + m)}

+ a5{16a − 10m(E + m)} = 0 (4.79) Λ a3= (−7 6a + 2 3m 2_)a 1 a4= (−35 36am− 1 36aE + 1 3m 3_)a 1 a5= (3 16a 2₋ 1 36amE− 17 36am 2₊ 2 15m 4_)a 1 E2= 10a Λ࢖ͬͯ੔ཧ͢Δͱ −₃₆7a2(E2− m2) +43 90am 2_(E2_{− m}2₎ + (−25 8a 2_{m +}16 3am 3₋4 3m 5_{)(E + m) = 0} _(4.80) ͱͳΔɻE + m ̸= 0 Λߟྀ͢Ε͹

(70a2− 172am2)E =−1055a2m + 1748am3− 480m5 (4.81) ͕ٻ·Δɻ

r5ͷ߲͸

a4(−8a2+ aE2) + a5{−9am − 11a(E + m) + E2(E + m)} = 0 (4.82)

Ͱ͋Δ͕ɺ੔ཧ͢Δͱ

(175a2− 540am2+ 96m4)E =−2550a2m + 3880am3− 960m5 (4.83) ͕ٻ·Δɻ Ҏ্ΑΓɺղ͕ଘࡏ͢Δͷ͸ a0= 0 a2= ma1 a3= (−7 6a + 2 3m 2_)a 1 a4= (−35 36am− 1 36aE + 1 3m 3_)a 1 a5= ( 3 16a 2₋ 1 36amE− 17 36am 2₊ 2 15m 4_)a 1 E2= 10a Λຬͨ͠ɺ͞Βʹʢ 4.81 ʣͱʢ 4.83 ʣཱ͕྆͢Δ৔߹Ͱ͋Δɻ͜ͷ̎ͭͷ৚݅ࣜʢ 4.81 ʣͱ ʢ 4.83 ʣΛ࿈ཱ͢Δͱ

(70a2− 172am2)(−2550a2m + 3880am3− 960m5)

= (−1055a2m + 1748am3− 480m5)(175a2− 540am2+ 96m4) (4.84) ͱͳΔ͕ɺm ̸= 0 Λ࢖ͬͯ

(20)

ͱ·ͱΊΒΕΔɻ͜ͷํఔࣜ͸Ҽ਺෼ղͰ͖ͯ

(5a− 8m2)(1225a3− 31120a2m2+ 29136am4− 5760m6) = 0 (4.86) ͱͳΔɻ1 ͭͷղ͸໌Β͔ʹ a = 8 5m 2_{Ͱ͋Δ͕ɺ࢒Γͷ 3 ࣍ํఔࣜΛ਺஋తʹղ͘ͱ} a = 0.28m2, 0.68m2, 24.44m2 (4.87) ͕ٻ·Δɻ ্ʹٻΊͨ a ͷ஋Λʢ 4.81 ʣʹ୅ೖͯ͠ΤωϧΪʔΛٻΊΔͱ a = 8 5m 2 _{ʹରͯ͠͸ E = 4m} a = 0.28m2 ʹରͯ͠͸ E = 1.68m a = 0.68m2 ʹରͯ͠͸ E = −2.63m, a = 24.44m2 ʹରͯ͠͸ E = −15.65m ͱͳΔɻa = 0.68m2_ɺ24.44m2 _{ͷղ͸ E < 0 ͱͳΔͷͰෆదͰ͋Δɻ} ʢ 4.81 ʣͱʢ 4.83 ʣΛ࿈ཱͯ͠ٻΊͨΤωϧΪʔ E =−245 488 a2 m3 + 3007 244 a m− 5227 305m + 264 61 m3 a (4.88) Λ࢖͏ͱɺ೾ಈؔ਺ͷ܎਺͸ a4= ( 245 17568 a3 m3 − 3007 8784 a2 m − 1362 2745am + 13 61m 3_)a 1 (4.89) a5= ( 245 17568 a3 m2 − 85 549a 2₊ 7 1830am 2₊ 4 305m 4_)a 1 (4.90) ͱॻ͚Δɻ ݁࿦ͱͯ͠ κ = −1 Ͱ r5_{·Ͱͷଟ߲ࣜͰٻ·Δଋറղ͸} a0= 0 a2= ma1 a3= (−7 6a + 2 3m 2_)a 1 a4= ( 245 17568 a3 m3 − 3007 8784 a2 m − 1362 2745am + 13 61m 3_)a 1 a5= ( 245 17568 a3 m2 − 85 549a 2₊ 7 1830am 2₊ 4 305m 4_)a 1 Λຬͨ͠ɺa ͱͯ͠ a = 8 5m 2 _(4.91) ·ͨ͸ 3 ࣍ํఔࣜ

1225a3− 31120a2m2+ 29136am4− 5760m6= 0 (4.92) Λຬ଍͢Δ৔߹Ͱ͋Δɻ3 ࣍ํఔࣜͷ਺஋ղͷ͏ͪɺଋറղͱΈͳͤΔͷ͸ a = 0.28m2 ͚ͩͰ͋Δɻ ೾ಈؔ਺͸ G(r) =a1{r + mr2+ (−7 6a + 2 3m 2_)r3 + ( 245 17568 a3 m3 − 3007 8784 a2 m − 1362 2745am + 13 61m 3_)r4 + ( 245 17568 a3 m2 − 85 549a 2₊ 7 1830am 2₊ 4 305m 4_)r5_}e−ar2_/2_−mr (4.93) ͱͳΓɺΤωϧΪʔݻ༗஋͸ E =√10a =    1.68m for a = 0.28m2 4m for a =8 5m 2_{= 1.6m}2 (4.94) ͱͳΔɻ ͜ΕΑΓ࣍਺ͷେ͖ͳଟ߲ࣜʹ͍ͭͯ΋ಉ༷ͳٞ࿦͕Ͱ͖ΔɻҰൠʹɺκ = −1 ͱͳΔ ղ͸ E =√4a E =√6a E =√8a · · · · ͢ͳΘͪ E =√2na ( n = 2, 3, 4,· · · ) ͱͳΔ͜ͱ͕༧૝͞ΕΔɻ͜͜Ͱ n ͸ଟ߲ࣜల։ͷ࠷େ࣍਺Λද͢ɻ

5 a

0

= 0

ͷ৔߹

ଋറղ͕ٻ·Δ৚݅ʢ 3.19 ʣͱͯ͠ κ = −1 ·ͨ͸ a0= 0͕੒Γཱͭ͜ͱ͕ඞཁͰ͋ͬ ͕ͨɺ͢Ͱʹݕ౼ͨ͠ κ = −1 ͷ৔߹ʹ΋࠷௿࣍਺ͷ߲͕߃౳ࣜͱͯ͠੒Γཱͭ৚͔݅Β a0= 0Ͱͳ͚Ε͹ͳΒͳ͔ͬͨɻ࣍ʹ͸ κ ̸= −1 Ͱ a0= 0ͷ৔߹Λݕ౼͢Δɻ a0= 0ͱ͢Δͱɺํఔࣜʢ 3.17 ʣ͸ n ∑ k=1 ak[{k(k − 1) − κ(κ + 1)}(E + m)rk−2

+{ak(k − 2) − aκ(κ + 2) − 2mk(E + m)}rk−1

+{−am(2k − 1) − a(2k + 1)(E + m) + E2(E + m)}rk

+ (−2a2k + aE2)rk+1] = 0 (5.1) ͱͳΔɻ࠷΋࣍਺ͷখ͞ͳ߲͸ r−1 _ͷ߲Ͱ a1{−κ(κ + 1)}(E + m) = 0 (5.2) ͱͳΔ͕ɺE + m ̸= 0 Ͱ κ = ±1, ±2, ±3, · · · Ͱ͋Δ͜ͱΛߟྀ͢Δͱ a1= 0 ·ͨ͸ κ = −1 (5.3) ͱͳΒͳ͚Ε͹ͳΒͳ͍ɻκ = −1 ͷ৔߹͸લষͰ͢Ͱʹղ͍͔ͨΒɺa1= 0ͷ৔߹Λݕ ౼͢Δɻ

(21)

ͱ·ͱΊΒΕΔɻ͜ͷํఔࣜ͸Ҽ਺෼ղͰ͖ͯ

(5a− 8m2)(1225a3− 31120a2m2+ 29136am4− 5760m6) = 0 (4.86) ͱͳΔɻ1 ͭͷղ͸໌Β͔ʹ a = 8 5m 2_{Ͱ͋Δ͕ɺ࢒Γͷ 3 ࣍ํఔࣜΛ਺஋తʹղ͘ͱ} a = 0.28m2, 0.68m2, 24.44m2 (4.87) ͕ٻ·Δɻ ্ʹٻΊͨ a ͷ஋Λʢ 4.81 ʣʹ୅ೖͯ͠ΤωϧΪʔΛٻΊΔͱ a = 8 5m 2 _{ʹରͯ͠͸ E = 4m} a = 0.28m2 ʹରͯ͠͸ E = 1.68m a = 0.68m2 ʹରͯ͠͸ E = −2.63m, a = 24.44m2 ʹରͯ͠͸ E = −15.65m ͱͳΔɻa = 0.68m2_ɺ24.44m2_{ͷղ͸ E < 0 ͱͳΔͷͰෆదͰ͋Δɻ} ʢ 4.81 ʣͱʢ 4.83 ʣΛ࿈ཱͯ͠ٻΊͨΤωϧΪʔ E =−245 488 a2 m3+ 3007 244 a m− 5227 305m + 264 61 m3 a (4.88) Λ࢖͏ͱɺ೾ಈؔ਺ͷ܎਺͸ a4= ( 245 17568 a3 m3− 3007 8784 a2 m − 1362 2745am + 13 61m 3_)a 1 (4.89) a5= ( 245 17568 a3 m2− 85 549a 2₊ 7 1830am 2₊ 4 305m 4_)a 1 (4.90) ͱॻ͚Δɻ ݁࿦ͱͯ͠ κ = −1 Ͱ r5 _{·Ͱͷଟ߲ࣜͰٻ·Δଋറղ͸} a0= 0 a2= ma1 a3= (−7 6a + 2 3m 2_)a 1 a4= ( 245 17568 a3 m3− 3007 8784 a2 m − 1362 2745am + 13 61m 3_)a 1 a5= ( 245 17568 a3 m2− 85 549a 2₊ 7 1830am 2₊ 4 305m 4_)a 1 Λຬͨ͠ɺa ͱͯ͠ a = 8 5m 2 _(4.91) ·ͨ͸ 3 ࣍ํఔࣜ

1225a3− 31120a2m2+ 29136am4− 5760m6= 0 (4.92) Λຬ଍͢Δ৔߹Ͱ͋Δɻ3 ࣍ํఔࣜͷ਺஋ղͷ͏ͪɺଋറղͱΈͳͤΔͷ͸ a = 0.28m2 ͚ͩͰ͋Δɻ ೾ಈؔ਺͸ G(r) =a1{r + mr2+ (−7 6a + 2 3m 2_)r3 + ( 245 17568 a3 m3− 3007 8784 a2 m − 1362 2745am + 13 61m 3_)r4 + ( 245 17568 a3 m2− 85 549a 2₊ 7 1830am 2₊ 4 305m 4_)r5_}e−ar2_/2_−mr (4.93) ͱͳΓɺΤωϧΪʔݻ༗஋͸ E =√10a =    1.68m for a = 0.28m2 4m for a = 8 5m 2_{= 1.6m}2 (4.94) ͱͳΔɻ ͜ΕΑΓ࣍਺ͷେ͖ͳଟ߲ࣜʹ͍ͭͯ΋ಉ༷ͳٞ࿦͕Ͱ͖ΔɻҰൠʹɺκ = −1 ͱͳΔ ղ͸ E =√4a E =√6a E =√8a · · · · ͢ͳΘͪ E =√2na ( n = 2, 3, 4,· · · ) ͱͳΔ͜ͱ͕༧૝͞ΕΔɻ͜͜Ͱ n ͸ଟ߲ࣜల։ͷ࠷େ࣍਺Λද͢ɻ

5 a

0

= 0

ͷ৔߹

ଋറղ͕ٻ·Δ৚݅ʢ 3.19 ʣͱͯ͠ κ = −1 ·ͨ͸ a0= 0͕੒Γཱͭ͜ͱ͕ඞཁͰ͋ͬ ͕ͨɺ͢Ͱʹݕ౼ͨ͠ κ = −1 ͷ৔߹ʹ΋࠷௿࣍਺ͷ߲͕߃౳ࣜͱͯ͠੒Γཱͭ৚͔݅Β a0= 0Ͱͳ͚Ε͹ͳΒͳ͔ͬͨɻ࣍ʹ͸ κ ̸= −1 Ͱ a0= 0ͷ৔߹Λݕ౼͢Δɻ a0= 0ͱ͢Δͱɺํఔࣜʢ 3.17 ʣ͸ n ∑ k=1 ak[{k(k − 1) − κ(κ + 1)}(E + m)rk−2

+{ak(k − 2) − aκ(κ + 2) − 2mk(E + m)}rk−1

+{−am(2k − 1) − a(2k + 1)(E + m) + E2(E + m)}rk

+ (−2a2k + aE2)rk+1] = 0 (5.1) ͱͳΔɻ࠷΋࣍਺ͷখ͞ͳ߲͸ r−1_ͷ߲Ͱ a1{−κ(κ + 1)}(E + m) = 0 (5.2) ͱͳΔ͕ɺE + m ̸= 0 Ͱ κ = ±1, ±2, ±3, · · · Ͱ͋Δ͜ͱΛߟྀ͢Δͱ a1= 0 ·ͨ͸ κ = −1 (5.3) ͱͳΒͳ͚Ε͹ͳΒͳ͍ɻκ = −1 ͷ৔߹͸લষͰ͢Ͱʹղ͍͔ͨΒɺa1= 0ͷ৔߹Λݕ ౼͢Δɻ

(22)

a1= 0ͷ৔߹ɺํఔࣜʢ 5.1 ʣ͸

n

∑

k=2

ak[{k(k − 1) − κ(κ + 1)}(E + m)rk−2

+{ak(k − 2) − aκ(κ + 2) − 2mk(E + m)}rk−1

+{−am(2k − 1) − a(2k + 1)(E + m) + E2_{(E + m)}_}rk

+ (−2a2_{k + aE}2_)rk+1_{] = 0} _(5.4)

ͱͳΓɺ࠷΋࣍਺ͷখ͞ͳ߲͸ r0_ͷ߲Ͱ

a2{2 − κ(κ + 1)}(E + m) = −a2(κ + 2)(κ− 1)(E + m) = 0 (5.5)

ͱͳΔɻE + m ̸= 0 Λߟྀ͢Δͱ ɹɹ a2= 0 ·ͨ͸ κ = −2 ·ͨ͸ κ = +1 (5.6) ͱͳΒͳ͚Ε͹ͳΒͳ͍ɻ

6 κ =

₋₂

ͷ৔߹

·ͣ࠷ॳʹ a0= a1= 0Ͱ κ = −2 ͷ৔߹Λݕ౼͢Δɻʢ 5.4 ʣࣜ͸ n ∑ k=2 ak[{k(k − 1) − 2}(E + m)rk−2 +{ak(k − 2) − 2mk(E + m)}rk−1

+{−am(2k − 1) − a(2k + 1)(E + m) + E2(E + m)}rk

+ (−2a2_{k + aE}2_)rk+1_{] = 0} _(6.1) ͱͳΔɻ

6.1 n = 2

a

2

̸= 0

ͷ৔߹ k = 2ͷΈڐ͞ΕΔ͔Βɺղ͘΂͖߃౳ࣜ͸ {−4m(E + m)}r1 +{−3am − 5a(E + m) + E2(E + m)}r2 + (−4a2+ aE2)r3= 0 (6.2) Ͱ͋Δɻr1_ͷ߲ −4m(E + m) = 0 (6.3) ͔ΒɺE + m ̸= 0 Ͱ͋Δ͔Β m = 0 ͕ٻ·Δɻm = 0 ͳΒ͹ r2 _ͷ߲͸ −5aE + E3= 0 (6.4) ͱͳΓɺE2_{= 5a}_{͕ٻ·Δɻͱ͜Ζ͕ɺr}3_{ͷ߲͸ a ̸= 0 ͱͯ͠} E2= 4a (6.5) Λཁٻ͢ΔͷͰɺཱ྆ͤͣɺ͜ͷ৔߹ʹ͸ղ͸ଘࡏ͠ͳ͍ɻ

6.2 n = 3

a

3

̸= 0

ͷ৔߹ ଟ߲ࣜ͸ k = 2, 3 ͕ڐ͞ΕΔ͔Β κ = −2 ͱͯ͠ a2[{−4m(E + m)}r1 +{−3am − 5a(E + m) + E2(E + m)}r2 + (−4a2_{+ aE}2_)r3_]

+ a3[ 4(E + m)r1+{3a − 6m(E + m)}r2

+{−5am − 7a(E + m) + E2(E + m)}r3 + (−6a2+ aE2)r4] = 0 (6.6) ͱͳΔɻ ࠷΋࣍਺ͷେ͖ͳ߲͸ r4_Ͱ −6a2_{+ aE}2_{= 0} _(6.7) ͱͳΔ͕ɺa ̸= 0 ΑΓ E2= 6a (6.8) ͕ٻ·Δɻ r1 ͷ߲ a2{−4m(E + m)} + 4a3(E + m) = 0 (6.9) ʹର͠ E + m ̸= 0 Λߟྀ͢Ε͹ a3= ma2 (6.10) ͕ٻ·Δɻ r2 _ͷ߲

a2{−3am − 5a(E + m) + E2(E + m)} + a3{3a − 6m(E + m)} = 0 (6.11)

ʹ a3= ma2ͱ E2= 6aΛ୅ೖͯ͠ a2(a− 6m2)(E + m) = 0 (6.12) ͱͳΔɻa2̸= 0ɺE + m ̸= 0 Λߟྀ͢Ε͹ a = 6m2 (6.13) ͱ͍͏৚͕݅ٻ·Δɻ r3 ͷ߲

a2(−4a2+ aE2) + a3{−5am − 7a(E + m) + E2(E + m)} = 0 (6.14)

ʹ a3= ma2ͱ E2= 6aΛ୅ೖ͢Δͱ

線形ポテンシャルを持つDirac方程式の束縛解 利用統計を見る

著者

手塚 洋一

雑誌名

東洋大学紀要. 自然科学篇 = Journal of Toyo

University. 東洋大学自然科学研究室 編

号

59

ページ

97-154

発行年

2015-03

URL

http://id.nii.ac.jp/1060/00007024/

手 塚 洋 一

Bound State Solutions of the Dirac Equation with Linear Potential

Hirokazu T

線形ポテンシャルを持つ Dirac 方程式の束縛解

手塚洋一

Bound State Solutions of the Dirac Equation

with Linear Potential

Hirokazu T

1

͸͡Ίʹ

2

த৺ྗϙςϯγϟϧΛ࣋ͭ

Dirac

ํఔࣜ

1

͸͡Ίʹ

2

த৺ྗϙςϯγϟϧΛ࣋ͭ

Dirac

ํఔࣜ

3

ઢܗεΧϥʔϙςϯγϟϧ

3

ઢܗεΧϥʔϙςϯγϟϧ

4

κ =

−1

ͷ৔߹

4.1

n = 0

a

̸= 0

4

κ =

−1

ͷ৔߹

4.1

n = 0

a

̸= 0

4.2

n = 1

a

̸= 0

4.3

n = 2

a

̸= 0

4.2

n = 1

a

̸= 0

4.3

n = 2

a

̸= 0

4.4

n = 3

a

̸= 0

4.4

n = 3

a

̸= 0

4.5

n = 4

線形ポテンシャルを持つDirac方程式の束縛解利用統計を見る

手塚洋一

University. 東洋大学自然科学研究室編

手塚洋一

₋₁

₋₁

₋₂