Here i would like to post the biggest mistake ever made in physics....unfortunately even greatest and most intelligent men make mistakes or simply don,t tak into account things that could have changed the world of physics or advance the science even 50 or 100 years... this was the case of Sir Willian R. Hamilton... Hamilton an Irish mathematician and Physicist famous by inventing the "Quaternions" and several works in math of physics, could have become a legend among all scientist and mankind, if only he had considered its own hypothesis and formulae beyond consideration...he and not Einstein Dirac and others..could have been the inventor of Relativity and Quantum mechanics...Still surprised?..let,s take a simple review.. RE-DISCOVERING QM AND RELATIVITY: -------------------------------------- One of the interest of Hamilton as a Mathematician..was to prove the relationship between optics (geometric) and Mechanics..(does this not sound familiar to you?..let,s go on) in fact he "proposed" invented the relationship (here,s come the hotstepper).. [tex] \Psi=e^{iS/a} [/tex] with a------->0 very small quantity (1) Maxwell and other had discovered the wave equation for electromagnetism... [tex] \frac{d^{2}\Psi}{dt^{2}}=\nabla^{2}\Psi [/tex] now if we substitute (1) into Maxwell Wave equation we get.. [tex] E^{2}=H^{2}=p^{2}c^{2} [/tex] (2) the equation of motion for a photon¡¡¡¡¡¡....for the case V=0 E=H (classical mechanics) of course for low momentum p this does not resemble the typical Hamiltonian of free particle.. [tex] E=p^{2}/2m [/tex] so you must add a term [tex] \alpha [/tex] to this so it satisfy the equation of motion... then (2) becomes.. [tex][tex] E^{2}=H^{2}=p^{2}c^{2}+\alpha [/tex] ¡¡¡the equation of the Hamiltonian for relativity¡¡¡¡....from this you can obtain the Lagrangian the momenta, and so on The same reasoning can be obtained for the Schroedinguer equation beggining from the Hamilton equation and using (1) [tex] \frac{dS}{dt}+(1/2m)(\frac{dS}{dx})^{2} [/tex] [tex] (\hbar/i)ln\Psi=S [/tex] from this two equation above and ignoring terms proportional to a------>0 you can obtain the equation of Quantum mechanics and almost the base to all quantum theory.. this is why i said the BIGGEST mistake on physics..Hamilton could have discovered the most important theories of 20-th century 100 years before..by the way i don,t think you need to be Einstein or planck to realize that a solution to Specific-Heat equation and so on is by setting [tex] E_{n}=n\hbar\omega [/tex]
What is the point of showing this? Anyway even if he had gotten to the equations he wouldn't have seen what connection they have to reality, for example, he wouldn't have known that he was describing a photon.
As an undergraduate, I used to hang around with David Finkelstein. He used to say that if Galileo had only known that the symmetry group of nature should be simple, he could have discovered the general theory of relativity. What a missed chance! :rofl: Those were good times.
-To Daniel: Of course..but he could have realized that for a particle in wich the action satisfied: [tex] |S|\sim{a} [/tex] the particle could be described by a wave and viceversa (the notation means similar or of the order of a) in fact Hamilton was a "Child-prodigy" and a gifted-man so due to his intelligence he should have realized or at least propose QM and Relativity (as i have presented to you) as something "Bizarre" limit of Classical Physics -To Physics-Monkey:Is not the same, Galileo did not know group theory but the math used in describing relativity and Quantum theory were known in the times of Hamilton is not the same critics..of course i agree with you in the term "those were good times" if math were so easy nowadays i myself (and you and your college mates and my college mates..) could discover quantum gravity -To Hurkyl: just substitute the expresion [tex] \Psi=e^{iS/a} [/tex] into the equation [tex] \frac{d^{2}\Psi}{dt^{2}}=c^{2}\nabla^{2}\Psi [/tex] then a----> so we can ignore the terms proportional to a or a*a, note that although a tends to 0 the exponential will always be finite and can be removed at the end of your calculations.. and sorry i forgot to mention that [tex] \frac{dS}{dx}=p [/tex] p=momentum
Sounds like some interesting work. Could you explain what you said in laymen terms? For example, what are the quantities in the equations?
Speaking "naively"..my "work" (well not mine...:rofl: :rofl: ) says 2 main things... a)for "small" particles with mass m<<<1 and for every particle , there is a Wave function associates to it, in a sense that classical-mechanics can be described in a "fashion" as the limit of optics...(or in modern language, every particle has a wave associate to it)-----> this is one of the most important principles of Quantum mechanics. b)For a particle with a very high "momentum" p its true Hamiltonian,Energy, and so on is: [tex] E=\sqrt{(pc)^{2}+\alpha} [/tex] for small momentum we derive the identity [tex] 2mE=p^{2} [/tex] wich is precisely classical mechanics, the identity above is precisely the relativistic Hamiltonian (wich generates all the mechanics of the particle) a) and b) for non-physicist are the main "basis2 for QM and Relativity 2 ofthe most important theories of all history together with Newtonian mechanics.
I know about that part, I meant Hamilton's equation and how you derive the relaativistic hamiltonian from it.
Einstein has called the cosmological constant his biggest mistake, and Hawking has called his statemnets that entropy must decrease in a contracting universe his biggest mistake. Both mistakes pale in comparison to that made by the creator of the universe - the universe itself, which must surely be the biggest mistake of all time. I'd like to attribute this to Douglas Adams, but I'm afraid that that may be a mistake....
haha, well, what can I say. History is so full of giants, that even some of them are bound to have a couple of missed chances. It would surely kill many of them if they were to know how close they are to some revolution. But then again, they are already dead. lol.
Not really, in the prologue to H2G2 Adams writes about the three philosopical blockbusters "Where God Went Wrong", "Some More of God's Greatest Mistakes" and "Who Is This God Person Anyway?"