010 011 This is essentially the idea of limits and now I’ll apply it to differentiation and illustrate why Leibniz’ definition no longer works. As the value of “h” tends to 0 (we treat it as 0 because it will always be closer to 0 than any value we imagine and therefore even smaller than infinitesimal numbers) the derivative becomes “4x” Thus, we see that a derivative of a function is no longer a fraction but a limit. This leads us back to our initial conundrum about the chain rule as cross cancelling is no longer possible due the derivatives (dy/dx) being notations and not fractions. However, despite this, the chain rule stands as somewhere along the way the limits inherit properties of fractions which allows us to treat Leibniz notation as fractions, despite them not being fractions. Infinitesimal differentiation: As I stated before, there is a way to treat derivatives as quotients in mathematics which is via non-standard analysis. I’ll provide a brief layman summary of this topic as it is very hard to grasp. First, we need to understand the definition of an infinitesimal number which is basically a number closer to 0 than any standard real number and it has some properties of standard real numbers. Now the difference between limits and infinitesimals is that with limits we can treat these infinitely small changes as 0, and basically remove them, but with infinitesimals they are hyperreal numbers. These hyperreal numbers (the small increments dx,dy,h) must be treated separately from real numbers and can only interact with other hyperreal numbers. Through this method Leibniz’ definition will work as real numbers and hyperreal numbers form fractions of their own. Hopefully this explanation made some sort of sense. Either you can just accept that limits somehow just behave with some properties of fractions or you can research more into infinitesimal differentiation which will lead you into non-standard analysis, group theory and just a lot of pure math. Source: chapter_2b.pdf (wisc.edu), Infinitesimal - Wikipedia An introduction to quantum mechanics Shyam Yashoman The vast scope of the quantum world may be too intricate for us to ever understand completely, which is ironic given that quantum mechanics is the study of matter at the smallest scale. However, just because we are unable to make sense of the quantum world, it does not mean its value or validity should be compromised. Currently, even with our limited yet undoubtedly growing understanding of quantum mechanics, we have been able to develop extraordinary technologies from fluorescent lights to quantum computers. In this article, I will outline the core principles of quantum mechanics – beginning with the dual-slit experiment. The Dual-Slit Experiment and WaveParticle Duality Perhaps the most infamous principle relating to quantum mechanics: the dual-slit experiment. The basic dual-slit experiment consists of a light source, (typically a laser beam), being fired at a plate containing two parallel slits. The light passing through the two slits is then projected onto a screen behind the plate. This experiment was first performed by Thomas Young in 1801 and demonstrated how light exhibits both wave and particle characteristics, leading to the understanding of the wave-particle duality of light. Primarily, the dual-slit experiment illustrates how when only one of the slits is open, the electrons and photons that pass-through form a pattern on the screen in approximately the same shape as the slit (pretty obvious right). However, when both slits are open, they form an interference pattern on the screen - and not two separated rectangular strips as you might expect. Interference patterns are known to arise from the superposition of waves (where two or more waves that cross paths interact to either combine or cancel with one another). For example, if a crowd of people were to all shout simultaneously, their individual sound waves undergo superposition and result in a combined sound wave with a greater amplitude (therefore increasing the volume of the perceived sound). Coming back to the dual-slit experiment, one argument to explain this phenomenon is that before reaching the two slits, the electron splits in some way and hence passes through both slits simultaneously. The two parts of the electron then undergo superposition, before recombining and hitting the screen as a localised particle. Now, this theory could easily be evaluated by placing a detector just before the two slits. This detector would either observe which slit the electron passes through, or whether the electron passes through both slits simultaneously. However, this is where Th is e sen<a l i ea of limits and ow I’ll apply it to differen<a<o and illustrate why Leibn z’ defini<o n lo ger works. First Principle: lim "→$ ( + ℎ) − ( ) ℎ Example: f(x) = 2 ! + 1 lim "→$[2( + ℎ)! + 1] − [2 ! + 1] ℎ lim "→$2 ! + 4 ℎ + 2ℎ! + 1 − 2 ! − 1 ℎ lim "→$ℎ(4 ℎ + 2ℎ) As the value of “h” tends t ( e treat it as 0 because it will always be clos r to 0 than any v lue we imagine and therefore even smaller than infinitesimal umb rs) the derivaFve becomes “4x” Thus, we see that a deriva<ve of a func<on is no longer a frac<on but a limit. This leads us back to our ini<al conundrum about the chain rule as cross cancelling is no longer possible due the deriva<ves (dy/dx) being nota<ons and not frac<ons. However, despite this the chain rule stands as somewhere along the way the limits inherit proper<es of frac<ons which allows us to treat Leibniz nota<on as frac<ons despite them not being frac<ons. Infinitesimal differen<a<on: As I stated before, there is a way to treat deriva<ves as quo<ents in mathema<cs which is via non-standard analysis. I’ll provide a brief layman summary of this topic as it is very hard to grasp. First, we need to understand the defini<on of an infinitesimal number which is basically a number closer to 0 than any standard real number and it has some proper<es of standard real numbers. Now the difference between limits and infinitesimals is that with limits we can treat these infinitely small changes as 0 and basically remove them but with infinitesimals they are hyperreal numbers. These hyperreal numbers (the small increments dx,dy,h) must be treated separately from real numbers and can only interact with other hyperreal numbers. Through this method Leibniz’ defini<on will work as real numbers and hyperreal numbers form frac<ons of their own. Hopefully this expla a<on m de some sort of sense. Either you can just accep that limits somehow just behave with some proper<es of frac<ons or you can research more into infinitesimal differen<a<on which will lead you into non-standard analysis, group theory and just a lot of pure math. Reference: chapter_2b.pdf (wisc.edu), Infinitesimal - Wikipedia An Introduc+on To Quantum Mechanics The vast scope of the quantum world may be too intricate for us to ever understand completely, which is ironic given that quantum mechanics is the study of ma:er at the smallest scale. However, just because we are unable to make sense of the quantum world does not mean its value or validity should be compromised. Currently, even with our limited yet undoubtedly growing understanding of quantum mechanics, we have been able to develop extraordinary technologies from fluorescent lights to quantum computers. In this arCcle, I will outline the core principles of quantum mechanics – beginning with the dual-slit experiment. The Dual-Slit Experiment and Wave-Par7cle Duality Perhaps the most infamous principle relaCng to quantum mechanics: the dual-slit experiment. The basic dual-slit experiment consists of a light source (typically a laser beam) being fired at a plate containing two parallel slits. The light passing through the two slits is then projected onto a screen behind the plate. This experiment was first performed by Thomas Young in 1801 and demonstrated how light exhibits both wave and parCcle characterisCcs, leading to the understanding of the wave-parCcle duality of light. Primarily, the dual-slit experiment illustrates how when only one of the slits is open, the electrons and photons that pass-through form a pa:ern on the screen in approximately the same shape as the slit (pre:y obvious right?). However, when both slits are open, they form an interference pa:ern on the screen - and not two separated rectangular strips as you might expect. Interference pa:erns are known to arise from the superposiCon of waves (where two or more waves that cross paths interact to either combine or cancel wi h on another). For example, if a crowd of p ople were to all shout simul an ously, their individual s und w ves undergo s perp siCon and result in a combin d sou d wave with a grea er amplitude (therefore increasing the volume f the perceived sound). Coming back to the dual-slit experiment, one argument to explain this phenomenon is that before reaching the two slits, the electron splits in some way and hence passes through both slits simultaneously. The two parts of the electron then undergo superposiCon, before recombining and hiQng the screen as a localised parCcle. Now, this theory could easily be evaluated by placing a detector just before the two slits. This detector would either observe which slit the electron passes through, or whether the electron passes through both slits simultaneously. However, this is where the profound peculiarity of quantum mechanics begins. By acCvely observing the path of each electron, you will find that they will no longer form the interference pa:ern shown above but will instead form two individual strips.
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