Timeline for Is it possible to formulate the Schrödinger equation in a manner that excludes imaginary numbers?
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23 events
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| Jul 26 at 2:16 | comment | added | tparker | @James Electrons DO "travel very strangely in our real world", and it HAS "shown up in experiments that electrons don't obey Newton's laws" for well over a century now. Respectfully, this is covered in detail in literally every QM textbook. I have no idea why you think that electrons follow Newton's laws. They don't. | |
| Jul 25 at 8:13 | comment | added | James | Conversely, perhaps you could show me definitively how it is not possible for us to detect the electron wavefunction behavior's deviation from Newton 1st or 2nd law? | |
| Jul 25 at 8:00 | comment | added | James | @tparker Thanks for indulging. It really bothers me for some reason that the wavefunction time evolution does not obey inertia and $F=ma$... It's because the electron cloud behavior (in position basis) can be measured repeated in experiments to get an overall picture of the cloud distribution, right? If this cloud is not following inertia or $F=ma$, then the electron would travel very strangely in our real world, and it would surely have shown up in experiments that electrons don't obey Newton's laws, would you think? | |
| Jul 25 at 4:38 | comment | added | tparker | So I would say that macroscopic solid objects, EM fields, and (idealized) single isolated quantum particles all live in the same real space, even though they have very different mathematical descriptions and evolve according to very different dynamical laws. | |
| Jul 25 at 4:37 | comment | added | tparker | @James I guess to some degree it depends on exactly how you define "space", but your last sentence makes no sense to me. Why must an object "in real space" obey Newton's laws? Electromagnetic fields don't obey Newton's laws either, but I would certainly say that they exist "in real space". To me, it makes more sense to logically separate out the nature of the space that an object lives in from the dynamical laws of motion that govern its time evolution. | |
| Jul 24 at 17:09 | comment | added | James | I suppose in 1 sentence, my confusion is essentially this: if the electron cloud $\Psi$ does not obey Newton's laws under time evolution (which can be experimentally measured statistically, so can be proven in real space either true or false), then it cannot be time evolving in real physical space, so it must be operating in some theoretical configuration space. | |
| Jul 24 at 16:58 | comment | added | James | contradicts Newton's law that a particle with momentum $p$ must move due to inertia. Second case: suppose we apply a linear potential gradient $V$ to our electron. then $\frac{\partial \Psi}{\partial t}=\frac{i}{2m}\frac{\partial \Psi}{\partial x}+V\Psi$ which also does not accelerate correctly according to Newton's $F=ma$ upon analysis. So, this leads me to believe the 1-electron wavefunction in the Schrodinger equation is not operating in real physical space, but it is in its configuration space where Newton's & other laws may not apply. Please do help to sort out any misunderstanding. | |
| Jul 24 at 16:52 | comment | added | James | thank you. i thought too that "configuration space for a single-electron wavefunction is just the real space", but (as Sabine would say) this changed my mind: let's have a free electron with momentum $p$ moving in the x direction. This has an instantaneous wavefunction with some spatial distribution, correct? Then we ask $\frac{\partial \Psi}{\partial t}=\frac{i}{2m}\frac{\partial \Psi}{\partial x}$ to time evolve this electron. But this electron is not moving with momentum $p$ according to $\frac{\partial \Psi}{\partial t}$, the wave simply disperses in place instead of traveling, which... | |
| Jul 24 at 14:18 | comment | added | tparker | Or, put another way: the one-electron wave function is a complex-valued field defined on a real manifold (which is just real space in the case of a single particle). | |
| Jul 24 at 14:15 | comment | added | tparker | @James Respectfully, I think you're a bit confused. The configuration space for a single-electron wavefunction is just real space. Newton's laws don't apply to the Schrodinger equation, but that has nothing at all to do with whether or not the configuration space is the same as real space. The configuration space is not "complex valued" - that concept doesn't even make sense. The configuration space is a manifold, not a function. The one-electron wave function is a map from configuration space (which is a real manifold) to the complex numbers. | |
| Jul 24 at 5:15 | comment | added | James | on configuration space ("theoretical diagram") of the process, and the precise mapping between configuration space to real physical space is not entirely clear to begin with? | |
| Jul 24 at 5:14 | comment | added | James | thank you. I am particularly interested in the 1-electron wavefunction (which precludes multi-particle Hilbert entangled states), as even this 1-particle wavefunction exists in its own configuration space, since it can be shown that Newton's laws don't apply to evolutions of the Schrodinger equation, correct? Given that this configuration space is complex valued, and its mapping to our real physical space is unclear, how valid it is do you think to say that "oh, such and such process happens faster than the speed of light" in our real space, whereas the calculations for it is done [contd..] | |
| Jul 24 at 4:57 | comment | added | tparker | But some philosophers think that the fact that the wavefunction is defined on configuration space rather than on real space makes it less "real" than, say, an electromagnetic field, and more likely that it just represents a state of knowledge rather than an actual "physical" object. | |
| Jul 24 at 4:56 | comment | added | tparker | it's a fundamental postulate of QM which is the "origin" of why QM is so weird, rather than a consequence of anything else. As to your second question - does it have anything to do with the fact that the wavefunction is complex-valued - no, I think that's a totally separate fact about QM. In fact, the same paper that I linked to in my answer points out that you could imagine a "real-valued" version of QM, in which the wavefunction is real-valued rather than complex valued. Such a theory would be almost the same as standard QM - certainly much closer to standard QM than to classical physics. | |
| Jul 24 at 4:53 | comment | added | tparker | @James The question of why a quantum wavefunction is defined on configuration space rather than real space is a very deep one, which gets at the heart of the difference between quantum and classical physics. The implication is that the size of the Hilbert space scales exponentially with the number of particles, which certainly isn't true of a classical electromagnetic or gravitational field. That fact is what underlies quantum entanglement and all the weird stuff that can result from that (like the enormous computational power of quantum computing). IMO, there's no way to derive this fact - | |
| Jul 23 at 17:22 | comment | added | James | Nice answer! (+1) Could you elaborate on why Schrodinger equation needs to operate in its own configuration space, while Maxwell equations can operate in our real physical space, please? Is this is something to do with the Maxwell equations being all real, so is applicable to our real physical space, while the Schrodinger equation by default cannot apply to real physical space because physical space is real (i.e. there is no imaginary component to our space)? Is there a real-valued construction of the Schrodinger equation that can operate in real physical space like the Maxwell equations? | |
| Nov 16, 2016 at 15:06 | comment | added | user64742 | to be fair though, if the computer can use complex numbers better than someone ought to make a complex number data type. | |
| Nov 16, 2016 at 6:59 | comment | added | tparker | Some hard-core digital physics proponents might argue that "deep down," the laws of quantum mechanics do only use real numbers, and physicists' use of complex numbers are just a convenient mathematical shortcut. | |
| Nov 16, 2016 at 6:50 | comment | added | user64742 | Someone needs to create a binary data type for complex number... | |
| Nov 16, 2016 at 2:33 | history | edited | tparker | CC BY-SA 3.0 |
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| Nov 16, 2016 at 2:24 | history | edited | tparker | CC BY-SA 3.0 |
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| Nov 16, 2016 at 2:23 | vote | accept | Haru Fujimura | ||
| Nov 16, 2016 at 2:18 | history | answered | tparker | CC BY-SA 3.0 |