<story><title>Schrödinger equation emerges mathematically from classical mechanics (2012)</title><url>https://www.researchgate.net/publication/241778960_A_Pseudo-Quantum_Triad_Schrodinger%27s_Equation_the_Uncertainty_Principle_and_the_Heisenberg_Group</url></story><parent_chain></parent_chain><comment><author>alecst</author><text>Schrodinger&amp;#x27;s reasoning was remarkable.&lt;p&gt;The high-level description of classical mechanics was formulated by Hamilton, who was starting from optics. He saw a mathematical analogy between the equations for light and the equations for mechanics. The principle of least time (Fermat&amp;#x27;s principle) for light became the principle of least action for mechanics.&lt;p&gt;But the principle of least time does not predict diffraction, just the geometric path of a light ray. It fails when the wavelength of the light is large compared to whatever it&amp;#x27;s interacting with.&lt;p&gt;At the time, the equations for mechanics were clearly failing for small systems. Here&amp;#x27;s where Schrodinger had his incredible insight: what if mechanics broke in &lt;i&gt;the same way&lt;/i&gt; as optics? Could matter itself display a kind of &amp;quot;diffraction&amp;quot; when its &amp;quot;wavelength&amp;quot; was similar in size to the objects it was interacting with? Could this explain the success of de Broglie&amp;#x27;s work, which treated small particles like waves?&lt;p&gt;Guided by that, he was able to add &amp;quot;diffraction&amp;quot; to the equations of matter and come up with the Schrodinger equation.&lt;p&gt;It&amp;#x27;s worth reading the original paper if you have a physics background -- probably grad-level (just being realistic.) I&amp;#x27;ve been wanting to write a blog post about this because the physics lore is something like &amp;quot;Schrodinger just made a really good guess&amp;quot; but that totally undersells the depth of his reasoning.</text></comment>
<story><title>Schrödinger equation emerges mathematically from classical mechanics (2012)</title><url>https://www.researchgate.net/publication/241778960_A_Pseudo-Quantum_Triad_Schrodinger%27s_Equation_the_Uncertainty_Principle_and_the_Heisenberg_Group</url></story><parent_chain></parent_chain><comment><author>bluish29</author><text>I don&amp;#x27;t know if there is a rule about science papers links, but I think using the journal paper link [1] is more suitable. The paper is open access, so no need for research gate.&lt;p&gt;[&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;iopscience.iop.org&amp;#x2F;article&amp;#x2F;10.1088&amp;#x2F;1742-6596&amp;#x2F;361&amp;#x2F;1&amp;#x2F;012015&quot; rel=&quot;nofollow noreferrer&quot;&gt;https:&amp;#x2F;&amp;#x2F;iopscience.iop.org&amp;#x2F;article&amp;#x2F;10.1088&amp;#x2F;1742-6596&amp;#x2F;361&amp;#x2F;1&amp;#x2F;0...&lt;/a&gt;]</text></comment>