Timeline for Why do sums become integrals in quantum mechanics of infinite basis states?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 20, 2023 at 17:28 | vote | accept | TunaSandwich | ||
| Apr 19, 2023 at 19:41 | answer | added | E. Anikin | timeline score: 3 | |
| Apr 19, 2023 at 19:40 | comment | added | ACuriousMind♦ | see physics.stackexchange.com/a/359982/50583 | |
| Apr 19, 2023 at 19:37 | history | edited | Qmechanic♦ |
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| Apr 19, 2023 at 19:15 | comment | added | LPZ | Intuitively, the small $dx$ comes from the fact that the kets $|x\rangle$ have infinite norm. Rather than being orthonormal, they satisfy:$$\langle y|x\rangle=\delta(x-y)$$ By dimensional analysis you can see that the discrete sum needs to be corrected by the small $dx$ to let the norms diverge. Naturally, this is a heuristic, but can be made rigorous by considering say a periodic box and letting its volume go to infinity. | |
| Apr 19, 2023 at 18:34 | comment | added | FlatterMann | Technically it doesn't follow all that trivially. Physicists like to handwave these things into existence, but in mathematical reality you have to study the solution theory of quantum mechanical equations very carefully. To give you an idea of what is really involved in this, you can take a look at en.wikipedia.org/wiki/Spectrum_(functional_analysis). The complexity of the phenomenon shouldn't discourage you to learn more about the handwaving methods of physics. It's just to show that "proving" that they make sense is not easy. | |
| Apr 19, 2023 at 18:14 | comment | added | naturallyInconsistent | You would have to construct a limiting process for each case. e.g. the case for momentum would be a different limiting process than for position. And yes, the factor of length will change the wavefunction's units and magnitude to compensate. | |
| S Apr 19, 2023 at 18:10 | review | First questions | |||
| Apr 19, 2023 at 18:13 | |||||
| S Apr 19, 2023 at 18:10 | history | asked | TunaSandwich | CC BY-SA 4.0 |