MauroLanzini

Thank you for the tips. Browsing around I have also found http://www.scottaaronson.com/democritus/lec9.html, very interesting and mathematically very accessible, and from there http://arxiv.org/abs/quant-ph/0101012 by Lucien Hardy (more complicated, but yet accessible). Nice explanations (for what I can judge) of what (mathematically) distinguishes classic from quantum, and why complex numbers (and not reals, nor quaternions, as I learned) are so important. I'll seek out also the 'LAX hierarchy'. Thanks.

Hello again studiot. I try to explain myself with a somewhat different line of reasoning. Then I'll admit that my 'idea' is so poor that I can't even explain it and I stop, I promise. - The equations of classic physics describe a world which can be very complicated, with solutions which can be stable, unstable, static or oscillating in innumerable different ways, deterministic or not, even chaotic.. any kind of things. But they are never 'strange' in the same sense as the equations of quantum mechanics, where things can be both waves and particles, pass through two different slits at once and so on: the behaviour of the quantum equations is qualitatively different from the classical ones. I hope we can agree on this point. - The difference in behaviour must lie in the equations themselves: quantum mechanics equations should have something peculiar to them (with respect to classic ones) which allows them to behave the way they do. - Can this difference be that in classic equations there are no complex numbers in their coefficients, while Schrödinger's equation sports a nice 'i' (usually written on the left side, but it could be anywhere)? This is the key point, in effect. Did I get this right, or am I wrong? This is what I would like to know. The important things end here, all the rest (replacing the 'i' with 'eKi' and etc. etc.) is just speculation, but, well, it looked nice. And it's true that replacing the 'i' in Schrödinger's with 'eKi', with K = π/2, leaves it exactly the same, doesn't it ? Trivial for sure, but maybe can be useful in some way.

Well (blushing) it's linear because I feel a comfortable (a little) with linear equations, but not at all with non-linear ones. Sorry. A is a real number. But it's not the equation in itself which is important, that's just an example. The point is that (sorry for having to use a non-technical language) it looks to me that the 'weird' behaviour of the Schrödinger's equation stems, mathematically, from having one (or more) coefficients with an imaginary part, while in classical physics (for what I know) this never happens elsewhere (ie. in Maxwell's equations, Navier-Stokes' etc. etc.). Then replacing 'i' with 'eKi' is just a trick (which may or may not work) which allows the same equation to switch from 'quantum' to 'classical' (and in-between).

Hmm interesting question, but I need some time to prepare an answer. Well, nice discussion anyway.. all your remarks have been thought-provoking, I'm grateful. Let me try to explain what the idea is in a more informal way. Sorry if I cannot do better, always big thanks for your patience. Consider a plain 1st-grade, linear, one variable differential equation: A * d y(t)/dt + y(t) = 0 If A is real the only solutions are decaying exponentials with the general form e-t/tau (add some constants depending on initial conditions and on the right-side part being zero or something else). This for instance is the case of a charged capacitor which is discharged through a resistor: Vcapacitor(t) = Vinitial e-t/tau . After some time Vcapacitor will settle to zero (rigorously after an infinite time, but in practice after some tau's). Now what takes Schrödinger's equation apart from every other physical equation I know of is that it has an explicitly imaginary coefficient (I got my first glimpse of this browsing http://superstringtheory.com/index.html, by the way). By adding an 'i' to the same equation as above it becomes: i * A d y(t)/dt + y(t) = 0 (where A is real) But now the equation is deeply changed because it admits periodic 'wave' solutions which were previosly impossible, and this turns the equation into something weird: with the same analogy as before you cannot say any more what the final voltage on the capacitor is because it can be always oscillating, and every time you measure it (provided you can measure it more than once) you'll get a different result because you're sampling a sinewave on a random point (allow me for the sake of the explanation that there is no 'energy dissipation' and that the wave does not decay, please). So the background idea is that what turns quantum mechanics 'weird' is, mathematically, the imaginary 'i' in the Schrödinger equation. Can anyone tell me if I'm right upon this at least? Now, what if the 'i' is not actually just 'i', but rather 'eKi' ? eKi * A d y(t)/dt + y(t) = 0 (where A is real) If K = π/2 we have (obviously) the 'weird' imaginary version of the equation, with oscillations and no steady final state. But if K = 0 we have the plain vanilla equation, where things settle to a definite 'real-world' value (after some tau's). So by varying K from π/2 to 0 we go 'smoothly' from a 'quantum' to a 'classic' description of the system. As a corollary K could be (inversely) proportional to the 'strength' of the interactions with the sorrounding environment. So a system without any interactions with the outside world would have K = π/2 and it would be described in a full 'quantum' way. But when it starts to interacts with the outside world (by way of a measurement, or just because it becomes embedded say in a piece of rock) K will go towards zero and the system will 'collapse' to a well-definite state. And at least in principle, the trajectory taken by the system going from the 'quantum' to the 'classical' state could be solved (knowing the function K = K(i_don't_know_what), of course). I need not to forget the disclaimer..: I have no idea if this theory would actually work or even if it is sensible, and I'm aware it's with 99.9999% probability NOT sensible at all hehe. It just looked interesting to me, and the worse which can happen is that I'll learn something from your answers, so it's all pretty cool anyway.

I think I've understood what you mean, studiot: the entries in the H matrix can be real numbers, but they can also be differential operators. Thus what I said is wrong: it's not true that: " the imaginary 'i' is what allows a first-order differential equation to have periodic, 'wave' solutions, while the same equation with strictly real coefficients only admits decaying exponentials as solutions (as it happens in the 'real' world, for instance with the charging at constant voltage of an electric capacitor). " I should have written: " the imaginary 'i' is what allows the equation to exhibit the 'weird' quantum behaviour: for instance if the coefficients of H are all real numbers the Schrödinger's equations admits periodic, 'wave' solutions, while the same equation with a strictly real coefficient on the left side only admits decaying exponentials as solutions (as it happens in the 'real' world, for instance with the charging at constant voltage of an electric capacitor). " So I'm not dead yet I think.... I try to resume: replacing 'i' with 'eKi' does not change the Schrödinger's equation if we take K = pi/2, but as K approaches zero the quantum 'behaviour' of the equation becomes smaller and smaller and totally vanishes when K = 0. So this is a mathematical trick which could allow to smoothly go from a fully-quantum to a fully-classical description of the system (I'm sure about the fully-quantum side, it's pretty obvious, I can only guess it may work also for the fully-classical side).

Hehe imatfaal, thanks a lot. Yes I knew I was missing an h-bar (a character hard to find on the keyboard) I lumped it together with the H. The whole basic idea of changing 'i' to 'eKi' comes of course from Euler's equation (that one at least is inside my math skills), as you guessed. Schrödinger's equation stays the same but as a special case of a generalized equation where the 'quantumness' (that is to say the oscillations, that is to say the imaginary coefficient), can be tuned at will. This if I'm not badly wrong of course, but both 'i' and 'eKi' are unit vectors, so I think it stands up. And I wholly agree with you.. no hope to understand without much more math skills than I have. That's why I posted in the first place.. to ask someone with more knowledge than me if this is just a crazy idea or if it could possible make sense (or maybe if it's simply useless/not original). Eagerly waiting for an answer . Thanks for the links too. I knew the wikipedia entry on operators.. eh.. not easy at all to understand without help. I'll check out Hyperphysics. Not that I ever hope to understand enough math for quantum mechanics, but I'm sure it'll make an interesting reading.

Hello studiot, well as I said my maths doesn't go too far, so I have an idea of what an Hamiltonian (or a Lagrangian) are, I know they represent the total energy of the system and in the case of the Lagrangian I understand how it can be used to derive the laws of motion, but I don't go much further (I have a degree in electronics egineering, that's where my math comes from). But of course multiplying by eKi on the left side is the same as multiplying the H by e-Ki ... is this what you wanted to say?

Yes imatfaal you are reason of course, thank you (I amended my post). The basic idea stays the same though (just a matter of constants), and actually many functions which give as a result i at one extreme and 1 at the other one could be used instead of eKi

Hello. I'm not sure I'm posting in the right forums, please excuse me in case. I may also be saying something very silly, my math is not that bad but it's quite below what is needed to grab quantum theory, so please excuse me again in case. This is a form of Schrödinger's equation: i * dΨ/dt = H * Ψ In my understanding the imaginary 'i' is what allows a first-order differential equation to have periodic, 'wave' solutions, while the same equation with strictly real coefficients only admits decaying exponentials as solutions (as it happens in the 'real' world, for instance with the charging at constant voltage of an electric capacitor). What about writing Schrödinger's equation this way: eKi * dΨ/dt = H * Ψ Now if K = π/2 then eKi = i and we have the normal Schrödinger's. But if K = 0 then eKi = 1, there are no more imaginary coefficients and we have an equation where the solutions (after some time constants) will set to a well-definite value (of course many different functions could be used in place of eKi, just it looked the simplest one). K could vary with time, and could be made (inversely) proportional to some measure of the interactions with the sorrounding environment, and so it would (possibly ) bridge neatly the gap from a fully-quantum to a fully-classical description of a system. My math skills stop about here, so no idea on how to elaborate on that. But I thank whoever will let me know if this looks interesting, or why it's utterly silly (or, more probably yet, who already thought the same and why it did not work). Thanks for reading.