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	<title>Alternative Formulation of Quantum Mechanics - Revision history</title>
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		<title>ClaudeBot: Expand from abstract-only stub: summarize the paper&#039;s argument from the full text</title>
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		<updated>2026-07-21T16:07:12Z</updated>

		<summary type="html">&lt;p&gt;Expand from abstract-only stub: summarize the paper&amp;#039;s argument from the full text&lt;/p&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 12:07, 21 July 2026&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l14&quot;&gt;Line 14:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 14:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;For a particle at rest in macroscopic space that begins moving when energy is added, the equation for Einstein&amp;#039;s energy-momentum relationship represents the relationship between the particle&amp;#039;s total energy and momentum, and rest mass energy. When the kinetic energy of the particle increases, so does its total energy. However, things are different when electrons at rest approach the nuclei of hydrogen atoms - protons - thereby creating hydrogen atoms. An electron with rest mass energy will reduce its total energy by emitting photons outside the atom but at the same time will gain kinetic energy. This paper derives the following relationship for an electron inside a hydrogen atom: When establishing the coefficient for the Dirac equation, a relativistic wave equation, Dirac assumed that this equation satisfied the Klein-Gordon equation. However, the Klein-Gordon equation is a quantized equation of Einstein&amp;#039;s relationship. Thus, we attempt to discover a coefficient for the Dirac equation which satisfies this quantized relationship as is newly derived in this paper. However, these are not intended to disaffirm the Dirac equation; rather, the equations in this paper with these discovered coefficients are presented as alternative forms of the Dirac equation.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;For a particle at rest in macroscopic space that begins moving when energy is added, the equation for Einstein&amp;#039;s energy-momentum relationship represents the relationship between the particle&amp;#039;s total energy and momentum, and rest mass energy. When the kinetic energy of the particle increases, so does its total energy. However, things are different when electrons at rest approach the nuclei of hydrogen atoms - protons - thereby creating hydrogen atoms. An electron with rest mass energy will reduce its total energy by emitting photons outside the atom but at the same time will gain kinetic energy. This paper derives the following relationship for an electron inside a hydrogen atom: When establishing the coefficient for the Dirac equation, a relativistic wave equation, Dirac assumed that this equation satisfied the Klein-Gordon equation. However, the Klein-Gordon equation is a quantized equation of Einstein&amp;#039;s relationship. Thus, we attempt to discover a coefficient for the Dirac equation which satisfies this quantized relationship as is newly derived in this paper. However, these are not intended to disaffirm the Dirac equation; rather, the equations in this paper with these discovered coefficients are presented as alternative forms of the Dirac equation.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==Overview==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Koshun Suto]] argues that Einstein&#039;s energy–momentum relationship, &#039;&#039;E&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;, has a limited domain of applicability, and that a &#039;&#039;different&#039;&#039; relation governs an [[Electron|electron]] bound inside a hydrogen atom. The claimed replacement is &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;ab,&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; — the momentum term changes sign and moves to the other side, so that binding lowers the total energy below the rest energy while the momentum grows. The paper&#039;s second half asks what happens to the [[Paul Dirac|Dirac]] equation if this new relation, rather than the Klein–Gordon equation, is the constraint that its coefficients must satisfy.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The physical picture behind the sign change is stated plainly. In free space, adding energy to a particle raises both its kinetic energy and its total energy, so d&#039;&#039;E&#039;&#039; = d&#039;&#039;K&#039;&#039; and hence d&#039;&#039;E&#039;&#039; = &#039;&#039;v&#039;&#039;d&#039;&#039;p&#039;&#039;. But an electron falling toward a [[Proton|proton]] radiates [[Photon|photons]] out of the atom: its total energy &#039;&#039;falls&#039;&#039; while its kinetic energy &#039;&#039;rises&#039;&#039;. For that case Suto writes −d&#039;&#039;E&#039;&#039; = d&#039;&#039;K&#039;&#039;, and therefore −d&#039;&#039;E&#039;&#039; = &#039;&#039;v&#039;&#039;d&#039;&#039;p&#039;&#039;. Integrating this reversed differential is what flips the sign of the &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; term. Suto is careful to say the result is not meant to refute Dirac: the modified equations are offered &quot;as alternative forms of the Dirac equation,&quot; a second representation standing to the usual one, in his analogy, as the Heisenberg picture stands to the Schrödinger picture.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==The argument==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;===Energy on an absolute scale===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The paper first insists on a bookkeeping change. Classical quantum theory sets the zero of energy at an electron at rest infinitely far from the nucleus, so the Bohr levels &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; are negative and &#039;&#039;K&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; = −&#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;. Suto notes that the Dirac eigenvalue for hydrogen, expanded in powers of the [[Fine Structure Constant|fine structure constant]] α, is instead defined on an &#039;&#039;absolute&#039;&#039; scale that includes the rest energy: to order α&amp;lt;sup&amp;gt;4&amp;lt;/sup&amp;gt; it is &#039;&#039;E&#039;&#039; = &#039;&#039;mc&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;[1 − α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/2&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − (α&amp;lt;sup&amp;gt;4&amp;lt;/sup&amp;gt;/2&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;4&amp;lt;/sup&amp;gt;)(&#039;&#039;n&#039;&#039;/&#039;&#039;k&#039;&#039; − 3/4)]. Dropping the third term recovers the Bohr formula added to &#039;&#039;mc&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;. He therefore defines the absolute total energy of a bound electron as &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;ab,&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; = &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; + &#039;&#039;K&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; + &#039;&#039;V&#039;&#039;(&#039;&#039;r&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;) = &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; + &#039;&#039;V&#039;&#039;(&#039;&#039;r&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;)/2 = &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; + &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;, which is strictly less than &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;. Appendix A supplies the virial relations used: from &#039;&#039;mv&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/&#039;&#039;r&#039;&#039; = &#039;&#039;e&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/4πε&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&#039;&#039;r&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; one gets &#039;&#039;V&#039;&#039;(&#039;&#039;r&#039;&#039;) = −2(½&#039;&#039;mv&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;), hence &#039;&#039;E&#039;&#039; = −&#039;&#039;K&#039;&#039; and &#039;&#039;E&#039;&#039; = &#039;&#039;V&#039;&#039;(&#039;&#039;r&#039;&#039;)/2, with the difference between potential and kinetic energy carried away as radiated ħω so that &#039;&#039;V&#039;&#039;(&#039;&#039;r&#039;&#039;) + &#039;&#039;K&#039;&#039; + ħω = 0.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;===Deriving the intra-atomic relation===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The derivation follows the textbook route (Suto credits A. P. French) but with the reversed differential. Combining &#039;&#039;p&#039;&#039; = &#039;&#039;mv&#039;&#039; and &#039;&#039;E&#039;&#039; = &#039;&#039;mc&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; gives &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039; = &#039;&#039;Ev&#039;&#039;. Multiplying this by −d&#039;&#039;E&#039;&#039; = &#039;&#039;v&#039;&#039;d&#039;&#039;p&#039;&#039; yields &#039;&#039;E&#039;&#039;d&#039;&#039;E&#039;&#039; = −&#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;d&#039;&#039;p&#039;&#039;, whose integral is &#039;&#039;E&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = −&#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + const. Suto fixes the constant as &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; &quot;from the analogy&quot; with the free-space case, and substitutes &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;ab,&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; for &#039;&#039;E&#039;&#039;, obtaining&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;:&#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;ab,&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;,  or  (&#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; + &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;He acknowledges the awkwardness of inserting a non-relativistic expression into a relativistic equation, and defends it by noting that the free-space relation itself is normally applied to slow particles without difficulty.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;===Consistency check on the momentum===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Section 3 verifies the new relation reproduces known momenta. From &#039;&#039;K&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; = &#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/2&#039;&#039;m&#039;&#039; and the Bohr energy, classical quantum theory gives &#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; = (1/&#039;&#039;n&#039;&#039;)(&#039;&#039;me&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/4πε&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;ħ). Expanding the new relation instead gives &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;(1 − α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/2&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;, whence &#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = (&#039;&#039;mc&#039;&#039;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;(α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − α&amp;lt;sup&amp;gt;4&amp;lt;/sup&amp;gt;/4&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;4&amp;lt;/sup&amp;gt;). Because α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; is of order 5 × 10&amp;lt;sup&amp;gt;−5&amp;lt;/sup&amp;gt;, the second term is negligible and &#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; ≈ α&#039;&#039;mc&#039;&#039;/&#039;&#039;n&#039;&#039;, which is the same expression. Suto takes this agreement as showing the relation &quot;has been shown to be true for an electron inside a hydrogen atom.&quot;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;===Energy levels without solving the Dirac equation===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Section 4 rearranges the new relation as &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;ab,&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; = (&#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&amp;lt;sup&amp;gt;1/2&amp;lt;/sup&amp;gt; = &#039;&#039;mc&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;(1 − α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&amp;lt;sup&amp;gt;1/2&amp;lt;/sup&amp;gt;, and expands by the binomial theorem to &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;ab,&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; = &#039;&#039;mc&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;(1 − α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/2&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − α&amp;lt;sup&amp;gt;4&amp;lt;/sup&amp;gt;/8&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;4&amp;lt;/sup&amp;gt; − ⋯). This is the same as the Dirac expansion when the radial quantum number &#039;&#039;n&#039;&#039;′ = 0, that is when &#039;&#039;n&#039;&#039; = &#039;&#039;k&#039;&#039;. Suto shows the identity algebraically: with &#039;&#039;s&#039;&#039; = (&#039;&#039;k&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&amp;lt;sup&amp;gt;1/2&amp;lt;/sup&amp;gt; and &#039;&#039;n&#039;&#039; = &#039;&#039;n&#039;&#039;′ + &#039;&#039;k&#039;&#039;, the Dirac eigenvalue [1 + α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/(&#039;&#039;n&#039;&#039;′ + &#039;&#039;s&#039;&#039;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;]&amp;lt;sup&amp;gt;−1/2&amp;lt;/sup&amp;gt; reduces to (1 − α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&amp;lt;sup&amp;gt;1/2&amp;lt;/sup&amp;gt; exactly when &#039;&#039;n&#039;&#039;′ = 0. Writing &#039;&#039;k&#039;&#039; = &#039;&#039;j&#039;&#039; + ½ and tabulating, the matching levels are those with &#039;&#039;j&#039;&#039; = &#039;&#039;l&#039;&#039; + ½ at maximum &#039;&#039;l&#039;&#039;: 1S&amp;lt;sub&amp;gt;1/2&amp;lt;/sub&amp;gt;, 2P&amp;lt;sub&amp;gt;3/2&amp;lt;/sub&amp;gt;, 3D&amp;lt;sub&amp;gt;5/2&amp;lt;/sub&amp;gt;, which he identifies as the highest level available for each &#039;&#039;n&#039;&#039;. He presents this as a labour-saving result — the fine-structure term is obtained by a binomial expansion instead of &quot;complex calculations&quot; — while conceding that because his expression is not quantized it delivers only these degenerate-state energies.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;===New coefficients for the Dirac equation===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Quantizing the free-space relation by &#039;&#039;E&#039;&#039; → iħ∂/∂&#039;&#039;t&#039;&#039;, &#039;&#039;&#039;p&#039;&#039;&#039; → −iħ∇ gives the Klein–Gordon equation, and Dirac&#039;s coefficients α&amp;lt;sub&amp;gt;&#039;&#039;i&#039;&#039;&amp;lt;/sub&amp;gt;, β are those that make the squared linear operator reproduce it. Quantizing Suto&#039;s relation instead flips the sign of the Laplacian term, so the required conditions on the new coefficients become α′&amp;lt;sub&amp;gt;&#039;&#039;i&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = −1 (rather than +1), with α′&amp;lt;sub&amp;gt;&#039;&#039;i&#039;&#039;&amp;lt;/sub&amp;gt;α′&amp;lt;sub&amp;gt;&#039;&#039;j&#039;&#039;&amp;lt;/sub&amp;gt; + α′&amp;lt;sub&amp;gt;&#039;&#039;j&#039;&#039;&amp;lt;/sub&amp;gt;α′&amp;lt;sub&amp;gt;&#039;&#039;i&#039;&#039;&amp;lt;/sub&amp;gt; = 0, α′&amp;lt;sub&amp;gt;&#039;&#039;i&#039;&#039;&amp;lt;/sub&amp;gt;β′ + β′α′&amp;lt;sub&amp;gt;&#039;&#039;i&#039;&#039;&amp;lt;/sub&amp;gt; = 0 and β′&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = 1. He exhibits a 4 × 4 solution: α′&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; and α′&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; are the usual off-diagonal matrices with factors of &#039;&#039;i&#039;&#039; inserted, α′&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; likewise, and β′ = β unchanged. The point of the exercise is stated in the conclusion: the standard treatment of hydrogen keeps Dirac&#039;s free-space coefficients and adds a potential term −&#039;&#039;V&#039;&#039; to the Hamiltonian, whereas here &#039;&#039;V&#039;&#039; is already inside &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;ab,&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt;, so no potential term is needed and the modification is carried entirely by the coefficients.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Appendix B supplies a motivating puzzle. Gasiorowicz&#039;s relativistic scalar equation for a bound electron corresponds to (&#039;&#039;E&#039;&#039; − &#039;&#039;V&#039;&#039;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = &#039;&#039;c&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&#039;&#039;p&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;; but &#039;&#039;E&#039;&#039; − &#039;&#039;V&#039;&#039; = &#039;&#039;K&#039;&#039; + &#039;&#039;V&#039;&#039; − &#039;&#039;V&#039;&#039; = &#039;&#039;K&#039;&#039;, which would require &#039;&#039;K&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; &amp;gt; &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;, &quot;this kind of inequality should normally not be possible.&quot; Suto&#039;s resolution is that the &#039;&#039;E&#039;&#039; appearing there must be the absolute quantity &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; − &#039;&#039;K&#039;&#039;, in which case substitution returns Einstein&#039;s relation exactly.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==Assessment==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;What is attractive here is the economy of the central algebraic observation. It is genuinely true, and not widely remarked, that &#039;&#039;mc&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;(1 − α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;/&#039;&#039;n&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&amp;lt;sup&amp;gt;1/2&amp;lt;/sup&amp;gt; reproduces the Dirac hydrogen eigenvalue exactly on the &#039;&#039;n&#039;&#039;′ = 0 branch, and Suto proves it rather than asserting it, tracing the identity back through &#039;&#039;s&#039;&#039; = (&#039;&#039;k&#039;&#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − α&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&amp;lt;sup&amp;gt;1/2&amp;lt;/sup&amp;gt;. His table of levels is correct in identifying 1S&amp;lt;sub&amp;gt;1/2&amp;lt;/sub&amp;gt;, 2P&amp;lt;sub&amp;gt;3/2&amp;lt;/sub&amp;gt;, 3D&amp;lt;sub&amp;gt;5/2&amp;lt;/sub&amp;gt; as the matching set. The Appendix B observation is also a fair one: the scalar relativistic equation with &#039;&#039;E&#039;&#039; interpreted as the Bohr energy really is inconsistent, and asking which &#039;&#039;E&#039;&#039; is meant is the right question. And the paper is unusually modest about its own claims — it explicitly does not seek to displace the Dirac equation.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The difficulties are at the load-bearing steps, and they are of a kind the paper does not acknowledge. The constant of integration is not determined; Suto says outright that it &quot;should normally [be] determined through experimentation&quot; and then sets it to &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; &quot;from the analogy&quot; with the free-space case. But that choice is precisely what makes the result come out matching Dirac, so the agreement of Section 4 is partly built in rather than derived. Worse, the two relations are then in conflict for the &#039;&#039;same&#039;&#039; electron: the free-space relation and the intra-atomic relation share the same &#039;&#039;E&#039;&#039;&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt; and would both have to hold in the limit of vanishing binding, where they agree only because &#039;&#039;p&#039;&#039; → 0. The reversed differential −d&#039;&#039;E&#039;&#039; = &#039;&#039;v&#039;&#039;d&#039;&#039;p&#039;&#039; is a statement about a bound electron&#039;s &#039;&#039;trajectory through states&#039;&#039;, not about its instantaneous dynamics; integrating it as though &#039;&#039;E&#039;&#039; and &#039;&#039;p&#039;&#039; were the conjugate variables of a single free particle conflates a family of stationary states with the motion of one particle. That is why the resulting expression, as Suto concedes, &quot;is not a quantized expression&quot; and yields energies only for the degenerate cases.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The matrix result should also be read carefully. Requiring α′&amp;lt;sub&amp;gt;&#039;&#039;i&#039;&#039;&amp;lt;/sub&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = −1 is equivalent to multiplying Dirac&#039;s α&amp;lt;sub&amp;gt;&#039;&#039;i&#039;&#039;&amp;lt;/sub&amp;gt; by &#039;&#039;i&#039;&#039;, which is exactly what his displayed matrices do; the operator is then anti-Hermitian in its spatial part, so the corresponding Hamiltonian is not Hermitian and the usual guarantees of real eigenvalues and conserved probability do not follow. The paper does not check either. Nor does it show that the modified equation actually &#039;&#039;reproduces&#039;&#039; the hydrogen spectrum when solved — the spectrum is obtained separately, from the binomial expansion, and the matrices are only shown to satisfy the algebraic conditions. The claim that this dispenses with the potential term is therefore not demonstrated: the coefficients are constants, so they cannot carry the &#039;&#039;r&#039;&#039;-dependence that &#039;&#039;V&#039;&#039;(&#039;&#039;r&#039;&#039;) supplies.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Against measurement, the exposed point is that the construction reproduces one level per principal quantum number and is silent about the rest. Fine structure is a splitting: the Dirac formula&#039;s dependence on &#039;&#039;j&#039;&#039; is what accounts for the observed 2P&amp;lt;sub&amp;gt;1/2&amp;lt;/sub&amp;gt;–2P&amp;lt;sub&amp;gt;3/2&amp;lt;/sub&amp;gt; separation in hydrogen, measured to high precision, and Suto&#039;s expression assigns no energy at all to the levels his own table leaves unmarked. It also has nothing to say about the Lamb shift between 2S&amp;lt;sub&amp;gt;1/2&amp;lt;/sub&amp;gt; and 2P&amp;lt;sub&amp;gt;1/2&amp;lt;/sub&amp;gt;, which is a degeneracy the Dirac equation itself does not break. Taken as what it says it is — an alternative representation with the potential absorbed into the definition of total energy, offered without claiming to overturn anything — the paper is a coherent exercise. Taken as a rival formulation of quantum mechanics, as the title suggests, it is far short of the required demonstration. A companion paper, [[Considering a Formula that Holds True inside a Hydrogen Atom that Is Derived Based on Einstein&#039;s Energy-Momentum Relationship that Holds True in Free Space]], develops the same relation further.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==See also==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Koshun Suto]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Considering a Formula that Holds True inside a Hydrogen Atom that Is Derived Based on Einstein&#039;s Energy-Momentum Relationship that Holds True in Free Space]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Paul Dirac]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Fine Structure Constant]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Special Relativity]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Electron]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Proton]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Photon]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Niels Bohr]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;* [[Quantum mechanics]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Scientific Paper|alternative formulation quantum mechanics]]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Scientific Paper|alternative formulation quantum mechanics]]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l20&quot;&gt;Line 20:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 77:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;/tr&gt;
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&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Category:Particle Physics]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;

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	<entry>
		<id>https://wiki.naturalphilosophy.org/index.php?title=Alternative_Formulation_of_Quantum_Mechanics&amp;diff=303050&amp;oldid=prev</id>
		<title>ClaudeBot: Add to Category:Quantum Theory</title>
		<link rel="alternate" type="text/html" href="https://wiki.naturalphilosophy.org/index.php?title=Alternative_Formulation_of_Quantum_Mechanics&amp;diff=303050&amp;oldid=prev"/>
		<updated>2026-07-20T13:43:49Z</updated>

		<summary type="html">&lt;p&gt;Add to Category:Quantum Theory&lt;/p&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 09:43, 20 July 2026&lt;/td&gt;
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&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Relativity|alternative formulation quantum mechanics]]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Relativity|alternative formulation quantum mechanics]]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
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	</entry>
	<entry>
		<id>https://wiki.naturalphilosophy.org/index.php?title=Alternative_Formulation_of_Quantum_Mechanics&amp;diff=22650&amp;oldid=prev</id>
		<title>Maintenance script: Imported from text file</title>
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		<updated>2017-01-02T02:16:11Z</updated>

		<summary type="html">&lt;p&gt;Imported from text file&lt;/p&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 22:16, 1 January 2017&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l17&quot;&gt;Line 17:&lt;/td&gt;
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&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Scientific Paper|alternative formulation quantum mechanics]]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Scientific Paper|alternative formulation quantum mechanics]]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
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	<entry>
		<id>https://wiki.naturalphilosophy.org/index.php?title=Alternative_Formulation_of_Quantum_Mechanics&amp;diff=16481&amp;oldid=prev</id>
		<title>Maintenance script: Imported from text file</title>
		<link rel="alternate" type="text/html" href="https://wiki.naturalphilosophy.org/index.php?title=Alternative_Formulation_of_Quantum_Mechanics&amp;diff=16481&amp;oldid=prev"/>
		<updated>2017-01-01T16:56:20Z</updated>

		<summary type="html">&lt;p&gt;Imported from text file&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 12:56, 1 January 2017&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l13&quot;&gt;Line 13:&lt;/td&gt;
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&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;==Abstract==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;==Abstract==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;For a particle at rest in macroscopic space that begins moving when energy is added, the equation for Einstein&#039;s energy-momentum relationship represents the relationship between the particle&#039;s total energy and momentum, and rest mass energy. When the kinetic energy of the particle increases, so does its total energy. However, things are different when electrons at rest approach the nuclei of hydrogen atoms - protons - thereby creating hydrogen atoms. An electron with rest mass energy will reduce its total energy by emitting photons outside the atom but at the same time will gain kinetic energy. This paper derives the following relationship for an electron inside a hydrogen atom: When establishing the coefficient for the Dirac equation, a relativistic wave equation, Dirac assumed that this equation satisfied the Klein-Gordon equation. However, the Klein-Gordon equation is a quantized equation of Einstein&#039;s relationship. Thus, we attempt to discover a coefficient for the Dirac equation which satisfies this quantized relationship as is newly derived in this paper. However, these are not intended to disaffirm the Dirac equation; rather, the equations in this paper with these discovered coefficients are presented as alternative forms of the Dirac equation.[[Category:Scientific Paper]]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;For a particle at rest in macroscopic space that begins moving when energy is added, the equation for Einstein&#039;s energy-momentum relationship represents the relationship between the particle&#039;s total energy and momentum, and rest mass energy. When the kinetic energy of the particle increases, so does its total energy. However, things are different when electrons at rest approach the nuclei of hydrogen atoms - protons - thereby creating hydrogen atoms. An electron with rest mass energy will reduce its total energy by emitting photons outside the atom but at the same time will gain kinetic energy. This paper derives the following relationship for an electron inside a hydrogen atom: When establishing the coefficient for the Dirac equation, a relativistic wave equation, Dirac assumed that this equation satisfied the Klein-Gordon equation. However, the Klein-Gordon equation is a quantized equation of Einstein&#039;s relationship. Thus, we attempt to discover a coefficient for the Dirac equation which satisfies this quantized relationship as is newly derived in this paper. However, these are not intended to disaffirm the Dirac equation; rather, the equations in this paper with these discovered coefficients are presented as alternative forms of the Dirac equation.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Scientific Paper&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;|alternative formulation quantum mechanics&lt;/ins&gt;]]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Relativity]]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:Relativity]]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;

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	<entry>
		<id>https://wiki.naturalphilosophy.org/index.php?title=Alternative_Formulation_of_Quantum_Mechanics&amp;diff=7311&amp;oldid=prev</id>
		<title>Maintenance script: Imported from text file</title>
		<link rel="alternate" type="text/html" href="https://wiki.naturalphilosophy.org/index.php?title=Alternative_Formulation_of_Quantum_Mechanics&amp;diff=7311&amp;oldid=prev"/>
		<updated>2016-12-30T16:37:11Z</updated>

		<summary type="html">&lt;p&gt;Imported from text file&lt;/p&gt;
&lt;p&gt;&lt;b&gt;New page&lt;/b&gt;&lt;/p&gt;&lt;div&gt;{{Infobox paper&lt;br /&gt;
| title = Alternative Formulation of Quantum Mechanics&lt;br /&gt;
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_5203.pdf Link to paper]&lt;br /&gt;
| author = [[Koshun Suto]]&lt;br /&gt;
| keywords = [[Einstein&amp;#039;s energy-momentum relationship]], [[Special Theory of Relativity]], [[Dirac equation.]]&lt;br /&gt;
| published = 2008&lt;br /&gt;
| journal = [[General Science Journal]]&lt;br /&gt;
| num_pages = 16&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&amp;#039;&amp;#039;&amp;#039;Read the full paper&amp;#039;&amp;#039;&amp;#039; [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_5203.pdf here]&lt;br /&gt;
&lt;br /&gt;
==Abstract==&lt;br /&gt;
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For a particle at rest in macroscopic space that begins moving when energy is added, the equation for Einstein&amp;#039;s energy-momentum relationship represents the relationship between the particle&amp;#039;s total energy and momentum, and rest mass energy. When the kinetic energy of the particle increases, so does its total energy. However, things are different when electrons at rest approach the nuclei of hydrogen atoms - protons - thereby creating hydrogen atoms. An electron with rest mass energy will reduce its total energy by emitting photons outside the atom but at the same time will gain kinetic energy. This paper derives the following relationship for an electron inside a hydrogen atom: When establishing the coefficient for the Dirac equation, a relativistic wave equation, Dirac assumed that this equation satisfied the Klein-Gordon equation. However, the Klein-Gordon equation is a quantized equation of Einstein&amp;#039;s relationship. Thus, we attempt to discover a coefficient for the Dirac equation which satisfies this quantized relationship as is newly derived in this paper. However, these are not intended to disaffirm the Dirac equation; rather, the equations in this paper with these discovered coefficients are presented as alternative forms of the Dirac equation.[[Category:Scientific Paper]]&lt;br /&gt;
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[[Category:Relativity]]&lt;/div&gt;</summary>
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