Timeline for Where does $\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$ come from?
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| May 24, 2016 at 16:13 | comment | added | ACuriousMind♦ | This answer is correct, but I would like it even more if it emphasized the point more that the very definition (already classically) of "momentum" is that it is the conserved quantity associated to translations, and that the conserved quantities generate their symmetries in the Hamiltonian formulation. | |
| May 24, 2016 at 15:56 | vote | accept | nougako | ||
| May 24, 2016 at 11:19 | comment | added | nougako | @knzhou, thanks for the reference. But could you please mention which part in that manuscript precisely talks about fitting atomic spectra? | |
| May 24, 2016 at 8:59 | comment | added | CuriousOne | Looks like you are getting some eyes rolling when you say that it was wild guesses back then. :-) Maybe one should step back a little further and look at the timescale... it took about 240 years between Newton explaining how matter moves to the founders of quantum mechanics telling us with some hand waving why stable matter even exists. We still don't know exactly what matter really is... even the period of guessing that you are covering here took, in reality, about 20 years. | |
| May 24, 2016 at 8:36 | comment | added | knzhou | @nougako You got me to dig around; have a look at this! It was to fit atomic spectra. Some motivation came from the 'old quantum theory'. Some parts are surprisingly prescient. But lots of it is redundant, or just logically in the wrong place. | |
| May 24, 2016 at 8:25 | comment | added | nougako | So, there was actually an aspect of gambling behind some of the development of physics and technologies in this time? I don't believe there is zero reason in physicists proposing a certain postulate. There must be some motivation inspiring them to postulate $[x,p] \propto \hbar$. | |
| May 24, 2016 at 8:21 | comment | added | knzhou | @nougako Nope, that was discovered by Dirac a few years afterward (see here). The further back in history you go, the less sense it makes! | |
| May 24, 2016 at 8:19 | comment | added | nougako | Alright, at least your bringing up that the commutation relation being hypothesized gives me some light about the history. But I am sure it was not really "out of nowhere". To me there must have been some educated guess behind the proposal of that commutation. Could it be that they were actually trying to imitate the result from classical mechanics about the Possion bracket between $q$ and $p$ into QM to see if it worked? | |
| May 24, 2016 at 8:17 | comment | added | knzhou | @nougako But the point of my answer is that there really isn't logic going on here: the content is just postulated. It's inspired guesses. | |
| May 24, 2016 at 8:15 | comment | added | knzhou | @nougako Another concurrent historical development was Heisenberg's matrix mechanics, where they postulated $[x, p] = i \hbar$ out of nowhere. (This is 100% equivalent to $p = -i\hbar \partial_x$ by the Stone-von Neumann theorem.) | |
| May 24, 2016 at 8:09 | comment | added | nougako | I am still not getting how I should connect your first explanation about the de Broglie hypothesis and the rest of your explanation. As to my own view of your answer, it seems like the first paragraph suffices to explain the history. I imagine de Broglie hypothesis was first tested, and turns out to be a success. This implies that momentum operator acting on a plane wave is given by the derivative of that plane wave. Then people made an induction, what if we generalize this relation to any wavefunction and test it. It turns out to be successful as well so people use this relation up till now. | |
| May 24, 2016 at 8:05 | comment | added | knzhou | (I think a similar misconception is behind your asking for a "historical development". You might think it's like math, where people accumulate lemmas and eventually prove a big theorem. That's backwards: QM wasn't historically developed. Schrodinger postulated his equation out of nowhere. The development was filled in afterward.) | |
| May 24, 2016 at 8:00 | history | answered | knzhou | CC BY-SA 3.0 |