Wave Function of the Dirac Equation for an Electron in the Field of a Nucleus Expressed in Terms of an Eigenfunction of the Spin Projection Operator and a Wave Function of the Schrödinger Equation. Radiative Processes of a Hydrogen-Like Atom and Selection Rules
A solution of the Dirac equation for an electron in the field of a point nucleus (Ze), expressed in terms of an eigenfunction of the operator of the spin projection onto the third axis and the corresponding solution of the Schrödinger equation is derived. This solution is suitable for practical calc...
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| Veröffentlicht in: | Russian physics journal Jg. 59; H. 1; S. 48 - 64 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.05.2016
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| Abstract | A solution of the Dirac equation for an electron in the field of a point nucleus (Ze), expressed in terms of an eigenfunction of the operator of the spin projection onto the third axis and the corresponding solution of the Schrödinger equation is derived. This solution is suitable for practical calculations. On the basis of this solution, using ordinary methods of QED and field theory, general principles for the emission of photons, axions, and neutrinos
Z
e
*
→
Z
e
+
γ
,
a
,
v
v
¯
by a hydrogen-like atom are formulated which take into account the spin state of the electron and, in the case of photons, their polarization. This range of questions pertaining to a comparative characteristic of processes of emission of massless or almost massless particles has, to this day, not been discussed from this point of view in the literature. Selection rules for
γ
,
a
,
v
v
¯
emission processes are also obtained, where for axions and neutrinos they coincide with the existing selection rules in the literature ∆m = 0,±1; with ∆l = ±1 pertaining to photons, but for photon emission a few of them do in fact differ from them with the hypothesis of odd values of ∆l, not established by us and additional to the usual values ∆l = ±1 of variation of the azimuthal quantum number l due to the appearance of “new” integrals over the spherical angle
θ
for ∆m = ±1, where for ∆m = 0, as before, ∆l = ±1. Moreover, the dependence of the amplitude of the photon emission process on the quantum numbers is in principle different than in the previously adopted approach to the problem although the lifetime in the excited state for small values of the quantum numbers coincides in order of magnitude with the accepted value ~10
–9
s. |
|---|---|
| AbstractList | A solution of the Dirac equation for an electron in the field of a point nucleus (Ze), expressed in terms of an eigenfunction of the operator of the spin projection onto the third axis and the corresponding solution of the Schrodinger equation is derived. This solution is suitable for practical calculations. On the basis of this solution, using ordinary methods of QED and field theory, general principles for the emission of photons, axions, and neutrinos (Ze)* → (Ze) + γ, a, v[bar.v] by a hydrogen-like atom are formulated which take into account the spin state of the electron and, in the case of photons, their polarization. This range of questions pertaining to a comparative characteristic of processes of emission of massless or almost massless particles has, to this day, not been discussed from this point of view in the literature. Selection rules for y, a, v[bar.v] emission processes are also obtained, where for axions and neutrinos they coincide with the existing selection rules in the literature Δm = 0, [+ or -]1; with Δl = ±1 pertaining to photons, but for photon emission a few of them do in fact differ from them with the hypothesis of odd values of Al, not established by us and additional to the usual values Δl = ±1 of variation of the azimuthal quantum number l due to the appearance of "new" integrals over the spherical angle d for Δm = ±1, where for Δm = 0, as before, Δl = [+ or -]1. Moreover, the dependence of the amplitude of the photon emission process on the quantum numbers is in principle different than in the previously adopted approach to the problem although the lifetime in the excited state for small values of the quantum numbers coincides in order of magnitude with the accepted value ~[10.sup.-9] s. Keywords: hydrogen-like atom, radiation, selection rules. A solution of the Dirac equation for an electron in the field of a point nucleus (Ze), expressed in terms of an eigenfunction of the operator of the spin projection onto the third axis and the corresponding solution of the Schrödinger equation is derived. This solution is suitable for practical calculations. On the basis of this solution, using ordinary methods of QED and field theory, general principles for the emission of photons, axions, and neutrinos Z e * → Z e + γ , a , v v ¯ by a hydrogen-like atom are formulated which take into account the spin state of the electron and, in the case of photons, their polarization. This range of questions pertaining to a comparative characteristic of processes of emission of massless or almost massless particles has, to this day, not been discussed from this point of view in the literature. Selection rules for γ , a , v v ¯ emission processes are also obtained, where for axions and neutrinos they coincide with the existing selection rules in the literature ∆m = 0,±1; with ∆l = ±1 pertaining to photons, but for photon emission a few of them do in fact differ from them with the hypothesis of odd values of ∆l, not established by us and additional to the usual values ∆l = ±1 of variation of the azimuthal quantum number l due to the appearance of “new” integrals over the spherical angle θ for ∆m = ±1, where for ∆m = 0, as before, ∆l = ±1. Moreover, the dependence of the amplitude of the photon emission process on the quantum numbers is in principle different than in the previously adopted approach to the problem although the lifetime in the excited state for small values of the quantum numbers coincides in order of magnitude with the accepted value ~10 –9 s. |
| Audience | Academic |
| Author | Skobelev, V. V. |
| Author_xml | – sequence: 1 givenname: V. V. surname: Skobelev fullname: Skobelev, V. V. email: v.skobelev@inbox.ru organization: Moscow State University of Mechanical Engineering (MAMI) |
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| Cites_doi | 10.1016/0370-1573(87)90017-2 10.1007/s11182-015-0477-x 10.1103/PhysRevLett.38.1440 10.1016/j.physletb.2008.07.018 10.4213/tmf6420 |
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| References | AmslerCPhys. Lett.2008B66712008PhLB..667....1A10.1016/j.physletb.2008.07.018 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Cambridge, Massachusetts (2007). PecceiRDQuinnHRPhys. Rev. Lett.19773814401977PhRvL..38.1440P10.1103/PhysRevLett.38.1440 SkobelevVVRuss. Phys. J.2015582163341887210.1007/s11182-015-0477-x L. D. Landau and E M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Vol. 3, Pergamon Press, Oxford (1977). SkobelevVVTeor. Mat. Fiz.2009161174266488510.4213/tmf6420 RaffeltGGPhys. Rev1988D3713561988PhRvD..37.1356R J. E. Kim, Phys. Rep., 150, 1 (1987); H. Y. Cheng, ibid., 158, 1 (1988). V. V. Skobelev, Zh. Eksp. Teor. Fiz., 137, 241 (2010). V. B. Berestetskii, E. M. Lifshitz, and L P. Pitaevskii, Relativistic Quantum Theory, Vol. 4, Butterworth-Heinemann, London (1982). A. A. Sokolov, Yu. M. Loskutov, and I. M. Ternov, Quantum Mechanics [in Russian], Uchpedgiz, Moscow (1962). 737_CR11 737_CR6 C Amsler (737_CR10) 2008; B667 VV Skobelev (737_CR1) 2009; 161 VV Skobelev (737_CR3) 2015; 58 RD Peccei (737_CR5) 1977; 38 737_CR8 737_CR7 737_CR9 GG Raffelt (737_CR4) 1988; D37 737_CR2 |
| References_xml | – reference: V. V. Skobelev, Zh. Eksp. Teor. Fiz., 137, 241 (2010). – reference: RaffeltGGPhys. Rev1988D3713561988PhRvD..37.1356R – reference: L. D. Landau and E M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Vol. 3, Pergamon Press, Oxford (1977). – reference: AmslerCPhys. Lett.2008B66712008PhLB..667....1A10.1016/j.physletb.2008.07.018 – reference: PecceiRDQuinnHRPhys. Rev. Lett.19773814401977PhRvL..38.1440P10.1103/PhysRevLett.38.1440 – reference: I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Cambridge, Massachusetts (2007). – reference: A. A. Sokolov, Yu. M. Loskutov, and I. M. Ternov, Quantum Mechanics [in Russian], Uchpedgiz, Moscow (1962). – reference: V. B. Berestetskii, E. M. Lifshitz, and L P. Pitaevskii, Relativistic Quantum Theory, Vol. 4, Butterworth-Heinemann, London (1982). – reference: SkobelevVVTeor. Mat. Fiz.2009161174266488510.4213/tmf6420 – reference: J. E. Kim, Phys. Rep., 150, 1 (1987); H. Y. Cheng, ibid., 158, 1 (1988). – reference: SkobelevVVRuss. Phys. J.2015582163341887210.1007/s11182-015-0477-x – ident: 737_CR7 – ident: 737_CR9 doi: 10.1016/0370-1573(87)90017-2 – ident: 737_CR11 – ident: 737_CR8 – ident: 737_CR6 – volume: 58 start-page: 163 issue: 2 year: 2015 ident: 737_CR3 publication-title: Russ. Phys. J. doi: 10.1007/s11182-015-0477-x – ident: 737_CR2 – volume: 38 start-page: 1440 year: 1977 ident: 737_CR5 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.38.1440 – volume: B667 start-page: 1 year: 2008 ident: 737_CR10 publication-title: Phys. Lett. doi: 10.1016/j.physletb.2008.07.018 – volume: D37 start-page: 1356 year: 1988 ident: 737_CR4 publication-title: Phys. Rev – volume: 161 start-page: 74 issue: 1 year: 2009 ident: 737_CR1 publication-title: Teor. Mat. Fiz. doi: 10.4213/tmf6420 |
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| SubjectTerms | Condensed Matter Physics Hadrons Heavy Ions Hydrogen Lasers Mathematical and Computational Physics Nuclear Physics Optical Devices Optics Photonics Physics Physics and Astronomy Theoretical |
| Title | Wave Function of the Dirac Equation for an Electron in the Field of a Nucleus Expressed in Terms of an Eigenfunction of the Spin Projection Operator and a Wave Function of the Schrödinger Equation. Radiative Processes of a Hydrogen-Like Atom and Selection Rules |
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