Affichage MARC: :: Library Catalog

Enregistré dans:

Détails bibliographiques
Auteur principal:	Scretching, Daniel
Format:	Recurso digital
Langue:
Publié:	Zenodo 2026
Accès en ligne:	https://doi.org/10.5281/zenodo.19042483
Tags:	Ajouter un tag Pas de tags, Soyez le premier à ajouter un tag!

_version_	1866901197393231872
author	Scretching, Daniel
author_facet	Scretching, Daniel
contents	<p><em>The SQC Invariant in Hydrogenic Theory II: Exact Recovery of the Two Einstein-B Invariants from <span><span>B12B_{12}</span><span><span><span><span>B</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span></span></span></span> and <span><span>B21B_{21}</span><span><span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span></em> extends the Part I <span><span>A21A_{21}</span><span><span><span><span>A</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span>-sector result into the stimulated-transition sector by working in the standard Einstein convention defined with radiation energy density per unit ordinary frequency, <span><span>ρν\rho_\nu</span><span><span><span><span>ρ</span><span><span><span><span><span><span>ν</span></span></span><span></span></span></span></span></span></span></span></span>. In that convention, the spontaneous-emission relation</p> <p><span><span><span>A21=ISQC(g1g2)f12ν2A_{21}=I_{\mathrm{SQC}}\left(\frac{g_1}{g_2}\right)f_{12}\nu^2</span><span><span><span><span>A</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span></span><span></span></span></span></span></span><span><span><span>(</span></span><span><span><span><span><span><span>g</span><span><span><span>2</span></span><span></span></span><span>g</span><span><span><span>1</span></span><span></span></span></span><span></span></span></span></span></span><span><span>)</span></span></span><span><span>f</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span><span><span>ν</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span></span></span></span></span></p> <p>is combined with the standard Einstein relations</p> <p><span><span><span>A21=8πhν3c3B21,g1B12=g2B21.A_{21}=\frac{8\pi h\nu^3}{c^3}B_{21},\qquad g_1B_{12}=g_2B_{21}.</span><span><span><span><span>A</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>c</span><span><span><span><span>3</span></span></span></span>8<span>πh</span><span>ν</span><span><span><span><span>3</span></span></span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span><span>,</span><span><span>g</span><span><span><span><span><span><span>1</span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>g</span><span><span><span><span><span><span>2</span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span><span>.</span></span></span></span></span></p> <p>From these, the paper derives two exact recovery formulas for the same electromagnetic time invariant from the two stimulated coefficients separately:</p> <p><span><span><span>ISQC(B12)=8πhc3νB12f12,ISQC(B21)=8πhc3νB21(g1/g2)f12.I_{\mathrm{SQC}}^{(B_{12})}=\frac{8\pi h}{c^3}\frac{\nu B_{12}}{f_{12}}, \qquad I_{\mathrm{SQC}}^{(B_{21})}=\frac{8\pi h}{c^3}\frac{\nu B_{21}}{(g_1/g_2)f_{12}}.</span><span><span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span><span><span><span>(</span><span>B</span><span>12</span><span></span><span>)</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>c</span><span><span><span><span>3</span></span></span></span>8<span>πh</span></span><span></span></span></span></span></span><span><span><span><span><span><span>f</span><span><span><span>12</span></span><span></span></span><span>ν</span><span>B</span><span><span><span>12</span></span><span></span></span></span><span></span></span></span></span></span><span>,</span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span><span><span><span>(</span><span>B</span><span>21</span><span></span><span>)</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>c</span><span><span><span><span>3</span></span></span></span>8<span>πh</span></span><span></span></span></span></span></span><span><span><span><span><span><span>(</span><span>g</span><span><span><span>1</span></span><span></span></span>/<span>g</span><span><span><span>2</span></span><span></span></span><span>)</span><span>f</span><span><span><span>12</span></span><span></span></span><span>ν</span><span>B</span><span><span><span>21</span></span><span></span></span></span><span></span></span></span></span></span><span>.</span></span></span></span></span></p> <p>Both reduce identically to</p> <p><span><span><span>ISQC=2πe2ε0mec3=7.42165600145×10−22 s,I_{\mathrm{SQC}}=\frac{2\pi e^2}{\varepsilon_0 m_e c^3} =7.42165600145\times10^{-22}\ \text{s},</span><span><span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>ε</span><span><span><span>0</span></span><span></span></span><span>m</span><span><span><span>e</span></span><span></span></span><span>c</span><span><span><span><span>3</span></span></span></span>2<span>π</span><span>e</span><span><span><span><span>2</span></span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span>7.42165600145</span><span>×</span></span><span><span>1</span><span>0<span><span><span><span><span><span>−22</span></span></span></span></span></span></span><span> </span><span><span>s</span></span><span>,</span></span></span></span></span></p> <p>using CODATA 2022 values as published by NIST.</p> <p>The paper also recovers a second exact stimulated-sector cofactor,</p> <p><span><span><span>νB12f12=νB21(g1/g2)f12=e24ε0meh=c38πhISQC,\frac{\nu B_{12}}{f_{12}} = \frac{\nu B_{21}}{(g_1/g_2)f_{12}} = \frac{e^2}{4\varepsilon_0 m_e h} = \frac{c^3}{8\pi h}I_{\mathrm{SQC}},</span><span><span><span><span><span><span><span><span>f</span><span><span><span>12</span></span><span></span></span><span>ν</span><span>B</span><span><span><span>12</span></span><span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>(</span><span>g</span><span><span><span>1</span></span><span></span></span>/<span>g</span><span><span><span>2</span></span><span></span></span><span>)</span><span>f</span><span><span><span>12</span></span><span></span></span><span>ν</span><span>B</span><span><span><span>21</span></span><span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span>4<span>ε</span><span><span><span>0</span></span><span></span></span><span>m</span><span><span><span>e</span></span><span></span></span><span>h</span><span>e</span><span><span><span><span>2</span></span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span>8<span>πh</span><span>c</span><span><span><span><span>3</span></span></span></span></span><span></span></span></span></span></span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span></span><span></span></span></span></span></span><span>,</span></span></span></span></span></p> <p>showing that both <span><span>BB</span><span><span><span>B</span></span></span></span>-channels collapse onto the same invariant structure when written with the correct degeneracy weighting. A central interpretive point is that the bare identity <span><span>B12=B21B_{12}=B_{21}</span><span><span><span><span>B</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span> is not generally exact in the standard Einstein formalism; the exact relation is <span><span>g1B12=g2B21g_1B_{12}=g_2B_{21}</span><span><span><span><span>g</span><span><span><span><span><span><span>1</span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>g</span><span><span><span><span><span><span>2</span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span>. Equality of the two <span><span>BB</span><span><span><span>B</span></span></span></span> coefficients occurs only in the special case <span><span>g1=g2g_1=g_2</span><span><span><span><span>g</span><span><span><span><span><span><span>1</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>g</span><span><span><span><span><span><span>2</span></span></span><span></span></span></span></span></span></span></span></span>, or when one works with suitably degeneracy-weighted coefficients. Hilborn’s review is especially useful here because it carefully distinguishes conventions based on energy density per unit angular frequency, per unit ordinary frequency, and related alternative definitions, where factors of <span><span>2π2\pi</span><span><span><span>2</span><span>π</span></span></span></span> and <span><span>cc</span><span><span><span>c</span></span></span></span> can otherwise be confused.</p> <p>Overall, the paper places the stimulated Einstein coefficients into the same SQC-style invariant framework previously established for <span><span>A21A_{21}</span><span><span><span><span>A</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span>, showing that the stimulated and spontaneous sectors recover the same fixed electromagnetic constant while preserving the standard degeneracy structure of Einstein’s original formalism.</p>
format	Recurso digital
id	zenodo_https___doi_org_10_5281_zenodo_19042483
institution	Zenodo
language
publishDate	2026
publisher	Zenodo
record_format	zenodo
spellingShingle	The SQC Invariant in Hydrogenic Theory II: Exact Recovery of the Two Einstein-B Invariants from B_12 and B_21 Scretching, Daniel <p><em>The SQC Invariant in Hydrogenic Theory II: Exact Recovery of the Two Einstein-B Invariants from <span><span>B12B_{12}</span><span><span><span><span>B</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span></span></span></span> and <span><span>B21B_{21}</span><span><span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span></em> extends the Part I <span><span>A21A_{21}</span><span><span><span><span>A</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span>-sector result into the stimulated-transition sector by working in the standard Einstein convention defined with radiation energy density per unit ordinary frequency, <span><span>ρν\rho_\nu</span><span><span><span><span>ρ</span><span><span><span><span><span><span>ν</span></span></span><span></span></span></span></span></span></span></span></span>. In that convention, the spontaneous-emission relation</p> <p><span><span><span>A21=ISQC(g1g2)f12ν2A_{21}=I_{\mathrm{SQC}}\left(\frac{g_1}{g_2}\right)f_{12}\nu^2</span><span><span><span><span>A</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span></span><span></span></span></span></span></span><span><span><span>(</span></span><span><span><span><span><span><span>g</span><span><span><span>2</span></span><span></span></span><span>g</span><span><span><span>1</span></span><span></span></span></span><span></span></span></span></span></span><span><span>)</span></span></span><span><span>f</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span><span><span>ν</span><span><span><span><span><span><span>2</span></span></span></span></span></span></span></span></span></span></span></p> <p>is combined with the standard Einstein relations</p> <p><span><span><span>A21=8πhν3c3B21,g1B12=g2B21.A_{21}=\frac{8\pi h\nu^3}{c^3}B_{21},\qquad g_1B_{12}=g_2B_{21}.</span><span><span><span><span>A</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>c</span><span><span><span><span>3</span></span></span></span>8<span>πh</span><span>ν</span><span><span><span><span>3</span></span></span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span><span>,</span><span><span>g</span><span><span><span><span><span><span>1</span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>g</span><span><span><span><span><span><span>2</span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span><span>.</span></span></span></span></span></p> <p>From these, the paper derives two exact recovery formulas for the same electromagnetic time invariant from the two stimulated coefficients separately:</p> <p><span><span><span>ISQC(B12)=8πhc3νB12f12,ISQC(B21)=8πhc3νB21(g1/g2)f12.I_{\mathrm{SQC}}^{(B_{12})}=\frac{8\pi h}{c^3}\frac{\nu B_{12}}{f_{12}}, \qquad I_{\mathrm{SQC}}^{(B_{21})}=\frac{8\pi h}{c^3}\frac{\nu B_{21}}{(g_1/g_2)f_{12}}.</span><span><span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span><span><span><span>(</span><span>B</span><span>12</span><span></span><span>)</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>c</span><span><span><span><span>3</span></span></span></span>8<span>πh</span></span><span></span></span></span></span></span><span><span><span><span><span><span>f</span><span><span><span>12</span></span><span></span></span><span>ν</span><span>B</span><span><span><span>12</span></span><span></span></span></span><span></span></span></span></span></span><span>,</span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span><span><span><span>(</span><span>B</span><span>21</span><span></span><span>)</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>c</span><span><span><span><span>3</span></span></span></span>8<span>πh</span></span><span></span></span></span></span></span><span><span><span><span><span><span>(</span><span>g</span><span><span><span>1</span></span><span></span></span>/<span>g</span><span><span><span>2</span></span><span></span></span><span>)</span><span>f</span><span><span><span>12</span></span><span></span></span><span>ν</span><span>B</span><span><span><span>21</span></span><span></span></span></span><span></span></span></span></span></span><span>.</span></span></span></span></span></p> <p>Both reduce identically to</p> <p><span><span><span>ISQC=2πe2ε0mec3=7.42165600145×10−22 s,I_{\mathrm{SQC}}=\frac{2\pi e^2}{\varepsilon_0 m_e c^3} =7.42165600145\times10^{-22}\ \text{s},</span><span><span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>ε</span><span><span><span>0</span></span><span></span></span><span>m</span><span><span><span>e</span></span><span></span></span><span>c</span><span><span><span><span>3</span></span></span></span>2<span>π</span><span>e</span><span><span><span><span>2</span></span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span>7.42165600145</span><span>×</span></span><span><span>1</span><span>0<span><span><span><span><span><span>−22</span></span></span></span></span></span></span><span> </span><span><span>s</span></span><span>,</span></span></span></span></span></p> <p>using CODATA 2022 values as published by NIST.</p> <p>The paper also recovers a second exact stimulated-sector cofactor,</p> <p><span><span><span>νB12f12=νB21(g1/g2)f12=e24ε0meh=c38πhISQC,\frac{\nu B_{12}}{f_{12}} = \frac{\nu B_{21}}{(g_1/g_2)f_{12}} = \frac{e^2}{4\varepsilon_0 m_e h} = \frac{c^3}{8\pi h}I_{\mathrm{SQC}},</span><span><span><span><span><span><span><span><span>f</span><span><span><span>12</span></span><span></span></span><span>ν</span><span>B</span><span><span><span>12</span></span><span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span><span>(</span><span>g</span><span><span><span>1</span></span><span></span></span>/<span>g</span><span><span><span>2</span></span><span></span></span><span>)</span><span>f</span><span><span><span>12</span></span><span></span></span><span>ν</span><span>B</span><span><span><span>21</span></span><span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span>4<span>ε</span><span><span><span>0</span></span><span></span></span><span>m</span><span><span><span>e</span></span><span></span></span><span>h</span><span>e</span><span><span><span><span>2</span></span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span><span><span><span>8<span>πh</span><span>c</span><span><span><span><span>3</span></span></span></span></span><span></span></span></span></span></span><span><span>I</span><span><span><span><span><span><span><span>SQC</span></span></span></span><span></span></span></span></span></span><span>,</span></span></span></span></span></p> <p>showing that both <span><span>BB</span><span><span><span>B</span></span></span></span>-channels collapse onto the same invariant structure when written with the correct degeneracy weighting. A central interpretive point is that the bare identity <span><span>B12=B21B_{12}=B_{21}</span><span><span><span><span>B</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span> is not generally exact in the standard Einstein formalism; the exact relation is <span><span>g1B12=g2B21g_1B_{12}=g_2B_{21}</span><span><span><span><span>g</span><span><span><span><span><span><span>1</span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>12</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>g</span><span><span><span><span><span><span>2</span></span></span><span></span></span></span></span></span><span><span>B</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span>. Equality of the two <span><span>BB</span><span><span><span>B</span></span></span></span> coefficients occurs only in the special case <span><span>g1=g2g_1=g_2</span><span><span><span><span>g</span><span><span><span><span><span><span>1</span></span></span><span></span></span></span></span></span><span>=</span></span><span><span><span>g</span><span><span><span><span><span><span>2</span></span></span><span></span></span></span></span></span></span></span></span>, or when one works with suitably degeneracy-weighted coefficients. Hilborn’s review is especially useful here because it carefully distinguishes conventions based on energy density per unit angular frequency, per unit ordinary frequency, and related alternative definitions, where factors of <span><span>2π2\pi</span><span><span><span>2</span><span>π</span></span></span></span> and <span><span>cc</span><span><span><span>c</span></span></span></span> can otherwise be confused.</p> <p>Overall, the paper places the stimulated Einstein coefficients into the same SQC-style invariant framework previously established for <span><span>A21A_{21}</span><span><span><span><span>A</span><span><span><span><span><span><span>21</span></span></span><span></span></span></span></span></span></span></span></span>, showing that the stimulated and spontaneous sectors recover the same fixed electromagnetic constant while preserving the standard degeneracy structure of Einstein’s original formalism.</p>
title	The SQC Invariant in Hydrogenic Theory II: Exact Recovery of the Two Einstein-B Invariants from B_12 and B_21
url	https://doi.org/10.5281/zenodo.19042483

Documents similaires