Has anyone ever attempted to quantize Newtons equations, and indeed if they have, what are the equations used for? I had a stab at quantizing just for fun. I began with this classical Newtonian equation \(-kx = \frac{\partial P}{\partial t}\) To begin, I simply re-wrote this using the respective quantum operators, \(-i \hbar \frac{\partial}{\partial x} (\frac{\partial}{\partial t}) = -kx\) I hit it with a wave function given as \(\Psi = \psi(x)\phi(t)\) which produces \(-i\hbar \frac{\partial}{\partial x}(\frac{\partial \phi(t)}{\partial t} \psi(x)) = -kx \psi(x) \phi(t)\) Dividing through by \(\Psi\) gives \(-i \hbar \frac{\partial}{\partial x}(\frac{\partial \phi (t)}{\partial t}\frac{1}{\phi(t)}) = -kx\) using the method of separation of variables and dealing with the time-dependant case we end up with \(-i \hbar(\frac{\partial \phi (t)}{\partial t}) = \lambda \phi(t)\) where \(\lambda\) is just some constant. Rearranging the equation above yields \((\frac{\partial \phi (t)}{\partial t}) = -\frac{i \lambda}{\hbar} \partial t\) Integration yields \(\int \frac{\partial \phi(t)}{\phi t} = \int -\frac{i \lambda}{\hbar} \partial t \rightarrow -\frac{i \lambda}{\hbar} t + C\) The solution then is \(C e^{-\frac{i \lambda}{\hbar} t}\)
Err, no. The correct form is \(-kx=\frac{dp}{dt}\). Do you understand the difference? The above is a "nothing", definitely not an equation. Do you understand why? Do you want to continue, Reiku?
Quantum mechanics follows from classical mechanics by converting Poisson brackets into operator commutation relations. Much of the mathematical structure in one is in the other, you just have to know how to phrase it properly. As Tach has already commented, this isn't going about things in a careful, proper manner, it's just Reiku spewing out more of his nonsense. I guess he thinks if he scatter guns enough **** onto the walls something is going to stick. Aethelwulf/Reiku, given everyone knows your past posting behaviour and even in your current incarnation you've shown a complete lack of understanding for the mathematics you post and the underlying physics threads like this are not to be made in the maths/physics subforum. Until such time as you can demonstrate a working grasp of the relevant mathematical physics your 'musings' are to be put in the pseudo or alternative theories forums. Which is where this is going. You want to play physicist by spewing out stuff you think is complicated and will convince people you're not wasting your existence. We're not here to feed your delusions of mediocrity.
Because you don't equate differential operators like that. If, for example, \(\partial_{t}\phi = \partial_{x}\phi\) that doesn't mean you can write \(\partial_{t} = \partial_{x}\), such an expression is meaningless. \(\partial_{t}\phi = \partial_{x}\phi\) says that there is some field \(\phi\) where at every point in space and time it's partial derivative wrt time is equal to its partial derivative wrt space. \(\partial_{t} = \partial_{x}\) is saying "These two operators are identical, they not only have the same action on some function, they have the same action on all functions". Yes, there's the relation \(\hat{p} \leftrightarrow -i\hbar \partial_{x}\) used in quantum mechanics but this is relating two operators which are identical. You arrived at this dubious conclusion because you assumed all you had to do to go from classical to quantum is just change notation, ie assume all classical p's become quantum p's etc and then just mess with things using quantum mechanical expressions. The correct way to express the classical equation you're looking at would not to change the classical p into a quantum operator p but instead into the expectation of that operator. This is well known to anyone who has actually done quantum mechanics and not just copied it from Google hits, in that classical quantities relate to measurements of operators. Want to see how it's done, seeing as you obviously don't know how? We've got some wave function \(|\psi(t)\rangle\), which satisfies the Schrodinger equation, \(-i\partial_{t}|\psi(t) = \hat{H}|\psi(t)\rangle\). This is trivially solved to \(|\psi(t)\rangle = e^{i\hat{H}t}|\psi(0)\rangle\). We want to measure some quantity A, which is associated to a quantum mechanical operator \(\hat{A}\). This operator is time independent, it has an infinite dimensional matrix representation where all entries are constant. Therefore we get \(A(t) = \langle \psi(t)|\hat{A}|\psi(t)\rangle\). Putting in the solution for the wave function we can collect all of the time dependence into a single term, \(\langle \psi(t)|\hat{A}|\psi(t)\rangle = \langle \psi(0)| e^{-i\hat{H}t}\hat{A}e^{i\hat{H}t}|\psi(0)\rangle \equiv \langle \psi(0)| \hat{A}(t)|\psi(0)\rangle\). This is how you get different 'pictures' in quantum mechanics. The Schrodinger picture considers the operators time independent and the states time dependent, while the Heisenberg picture is the other way around. Either way we might want to ask what the time derivative of this observable is, \(\partial_{t}A(t)\). Well we just use the chain rule and the fact \([f(\hat{O}),\hat{O}] = 0\) always, \(\partial_{t}A(t) = \langle \psi(0) | -\hat{H}\hat{A}(t) + \hat{A}(t)\, \hat{H} |\psi(0)\rangle = \langle \psi(0) | [\hat{H},\hat{A}(t)] | \psi(0)\rangle \equiv [H,A(t)]\). In the Heisenberg picture the states are time independent so we get \(0 = \langle \psi(0)| \Big( \partial_{t}\hat{A}(t) - [\hat{A}(t),\hat{H}] \Big) | \psi(0)\rangle\). This is the Heisenberg Equation of Motion for the time dependent operators and their observables. So let's put in \(\hat{A} = \hat{p}\). Well we know that \(\hat{H} = \frac{1}{2}\hat{p}^{2} + U(\hat{x})\) and the first term goes and we're left with \(\partial_{t}\hat{p}(t) = \langle [\hat{p},U(\hat{x})] \rangle\). It's easy to show that \( [\hat{p},U(\hat{x})] |\psi(0)\rangle = -(\partial_{x}U)(\hat{x})|\psi(0)\rangle\) and thus we have not an operator equation but an equation for the observables of operators as \(\dot{p}(t) = -\nabla U(x(t))\), exactly as the Newtonian mechanics would say if you started with the classical Hamiltonian \(\frac{1}{2}m\dot{x}^{2} + U(x)\). The fact you don't know this shows how you've never done any real quantum mechanics, you would have come across this in an introductory course or textbook on quantum mechanics, as it is a nice example to illustrate quantum mathematical methods while returning a result the students should be familiar with. Yet another piece of evidence you're trying to talk about things you don't understand. It's generally the most ignorant who have the most inflated, unjustified view of their abilities. No, you read what the classical expresions where, you looked up a bunch of formulae from quantum mechanics and you indiscriminantly applied them on the assumption such usages were valid. As I just explained, they are not. No, you don't. You have an interest but not a genuine one. A genuine one wouldn't start thread after thread throwing around expressions he doesn't understand, pretending to be doing actual physics when it is neither actual physics nor demonstrating any understanding of actual physics (quite the opposite in fact). It is very very clear you do not have a working understanding of this area of physics, both from this thread and your Planck thread, never mind all your previous accounts. There's plenty of people here who don't understand quantum mechanics and most of them, the half rational ones, don't start thread after thread spewing out algebra they can't do properly, claiming its about physics they don't understand. You don't have a genuine interest, you have an interest in appearing to be genuine and informed when in fact you are neither. I get it, you want to put on the proverbial white coat, say something which sounds complicated and delude yourself you've achieved something. Unfortunately it results in precisely the opposite. You regularly try this line of flawed reasoning, ie if I don't explicitly say "This is wrong" then you assume I'm accepting it is right. The fact I might neither have the time nor the inclination to go through a post line by line doesn't mean I accept it correct. You should have learnt that by now, given all the times you've tried this, only to have me turn around and nail your post to the wall in extreme detail. Besides, even if there were no one here explaining your mistakes you'd still be practically innumerate and extremely bad at quantum mechanics and relativity. Even if this forum were filled with thousands of people posting "Wow, what a genius you are!" and "Surely you are the most insightful physicist ever!" and giving you offers of marriage or free beer and blow jobs, it wouldn't accomplish anything. No one with any mathematics or physics education would believe you, it wouldn't get you a research job or a PhD or a publication. It would accomplish sweet F.A.. Your behaviour of sticking your fingers in your ears and declaring I don't respond to certain posts or how you try to pick up on little things like "Is it an equation or an expression!" show you know you've backed yourself into a corner. If you had something good to retort with you'd do it. Instead you're stuck trying to magnify out of all reasonable proportions completely pathetic responses. Because if I reply to you then surely I cannot be replying to or reading any other threads Please Register or Log in to view the hidden image! Lousy logic. I moved this here because I do not believe you started this thread in good faith. Even if you weren't Reiku you've presented enough evidence that you don't understand the necessary physics to have a proper discussion on this stuff so starting a thread where you try to do physics, badly, is not a terribly good idea. You have shown you're happy to carry on waffling nonsense even when your mistakes are explained to you by multiple people and since you open this thread with a load of algebra it's clear you're going down the route again. Furthermore, as Tach commented and as I just explained, you've made several mistakes already which show you don't understand the notation and also that you haven't actually learnt any quantum mechanics. Someone not knowing quantum mechanics isn't a problem if they behave themselves, ask reasonable questions and enter into honest, informed discussion. You have shown you won't. You having a thread about quantum mechanics in the main forum will only serve to feed your delusions of competency, which I have no wish to prop up. If and when you demonstrate a sound understanding of the material you post lengthy mathematics about, then you can post such things in the maths/physics forum. You can ask qualitative questions, you can post single equations for discussions, you can reply to threads, but threads like this, where it's line after line after line of mathematics you cannot do properly, on a topic you blatantly don't understand, you cannot do. If you'd asked the question "Is there a quantum version of classical equations from Newtonian mechanics?" along with an example equation from Newtonian mechanics then the thread would have stayed. As such you attempted to develop your own answer, a task you are clearly ill equipped mathematically to do. You want to put forth your own take on things, your own derivations, your own results, do it in pseudo or alternative theories. Consider this your last 'friendly' warning in this regard.
Basic calculus says that you don't know what you are talking about. The way you divided the LHS is laughable. Like I said, give it a rest, Reiku.
I didn't say you actually said \(\partial_{t} =\partial_{x}\), I was using that as an example of what you did write, which was of the form \(\partial_{t}\partial_{x} = \ldots\). I'm wondering whether you're being obtuse deliberately or you're really so thick you didn't understand that. I did explain it in detail, about why you cannot just leave the derivatives hanging like that. Obviously my point went over your head, showing how little you understand this stuff. Well spotted, I missed that one having had my attention drawn by other mistakes he made. It's like a magic eye picture, the more you stare at his nonsense, the more things you spot. Except at the end you don't suddenly see an aeroplane or a person, you see a pile of ****.
Just as bad, reiku, just as bad. Nice try at obfuscating your errors. Thanks for the laughs, keep up the good work, it is entertaining.
Look in the mirror , Reiku. You suffer from a very rare form of graphomania : you feel compelled to write posts containing mathematical formulas that make absolutely no sense. You should consult a psychiatrist, with the appropriate drugs your affliction may be treatable. Or maybe not, your posts are immensely entertaining.
Separation of variables is a well known procedure, except you can't do it properly. This is a problem you regularly have Reiku, you think that if you attempt to use some result from the mainstream then you're not in the wrong but this assumes you can actually use the method properly. Invariably this turns out not to be the case. You're not exactly helping. The problem is you don't know how to do it properly! This is just another piece of evidence you clearly lack the necessary understanding and working knowledge of any of the necessary ground work to work with things like the Dirac equation, which you started talking about in another thread. Part of the reason we're able to call you on this is yes, it is a well known method, we know it. It's sufficiently familiar to people who went to university to do physics, rather than just lie on internet forums like you do, that mistakes in it are easily spotted. Your inability to spot such mistakes illustrates you don't understand the method, you just blindly apply formulae you find and tell yourself you couldn't possibly be wrong in doing so. Wrong. Wrong. Go on, have another go. Tach and I both commented about something you got wrong, why don't you tell us what it is.
...Aethelwulf is a sockpuppet for Reiku. Doesn't matter how many times you change your name, the BS you post gives you away. Reiku, I couldn't care less about what you think.
Better not to know what one is talking about half the time than all of the time, as seems to be the case with you. Reiku, you've been shown to have made a plethora of mistakes in just about every thread like this you've made. Just like you used to have happen with your previous accounts. On the same subjects. With the same equations. And same mistakes. And same whining. And same IP address range. When you return from your 3 day holiday (and it wasn't me who reported you this time) try to point out where your mistake is in the bit Tach quoted. You can also try to retort my lengthy explanation of why you were wrong in several ways in regards to your opening posts and how it demonstrates you haven't ever actually learnt quantum mechanics properly, you just attempt to parrot in some way which isn't obvious copy/pasting. Too bad you lack the necessary understanding to correctly alter and reshuffle equations you don't understand.
