Tag: science

Rolling the Dice: A Derivation of Schrodinger’s Equation from the Wave Equation

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The quantum world reveals an astonishing truth: light, electrons, and indeed all matter in the universe — including ourselves — are described by mathematical wavefunctions that represent the probabilities of finding each particle in a particular location. These wavefunctions remain spread out until a measurement or interaction with the environment, which causes them to entangle with the environment, destroying observable interference and yielding a discrete location for the particle.

How the Curl of Curl Gives Light on Electromagnetic Waves and other Phenomena

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It is amazing that light is actually an electromagnetic wave, but how can we actually "illuminate" this through mathematics? It was James Clerk Maxwell back in the 19th century who built upon his predecessors' scientific observations of electric and magnetic fields and then applied mathematical analysis to these observations to enlighten us all about the … Continue reading How the Curl of Curl Gives Light on Electromagnetic Waves and other Phenomena

Divergence, Curl and the Taylor Series Approximation

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I wanted to examine two crucial principles in understanding the physics behind how systems flow- divergence and curl- and to fully explain both the mathematical assumptions and intuitions that underlie these bedrock foundations of field motion. I think what is most often overlooked in the explanations of divergence and curl is the fundamental importance in … Continue reading Divergence, Curl and the Taylor Series Approximation

Of Unwavering Importance: The Wave Equation Derivation

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(Also, note that to achieve these boundary conditions, the sine waves across L must all be increments of half wavelengths- this is covered more in the Appendix section.) (Note: in the above the imaginary number i was dropped from multiplying the Dn coefficient. This can be done by constructing the original Cn and Dn coefficients … Continue reading Of Unwavering Importance: The Wave Equation Derivation

The Tiniest Changes Add Up: How Infinitesimals Preserve Constant Area in Geometry and Unit Elasticity in Economics

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In our previous post, we found that area cannot be maintained through equal but opposite percent changes in length and width using algebraic techniques. We determined that when we increased length by a certain percentage and decreased width by the same percentage, our new area calculation was derived through a downward sloping parabola of the … Continue reading The Tiniest Changes Add Up: How Infinitesimals Preserve Constant Area in Geometry and Unit Elasticity in Economics