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July 12

Uncertainty principle & H atom

According to the uncertainty principle the electron has a momentum :
$\Delta p_{e}={\tfrac {h}{4\pi \Delta x_{e}}}{\text{ , }}$
$\Delta x_{e}=10^{-10}{\text{m (can be found from liquid hydrogen density)}}$ .

To conserve the atom momentum zero , the proton must have a momentum  :
$\Delta p_{p}=-\Delta p_{e}$.
So $\Delta x_{p}=\Delta x_{e}$ and the proton must also smear to atom's full size.

But Rutherford scattering experiment shows that a size of a nucleus is $10^{5}$ smaller. Why does the uncertainty principle fail? — Preceding unsigned comment added by U240700 (talk • contribs) 08:54, 12 July 2024 (UTC)

You seem to be confusing the (expectation) values of the momenta and their uncertainties. Conservation of momentum demands $\langle p_{p}\rangle =-\langle p_{e}\rangle$ and says nothing about the uncertainties. --Wrongfilter (talk) 09:24, 12 July 2024 (UTC)

Uncertainty $\Delta p_{e}$ means that $p_{e}$ can take any value within $\Delta p_{e}$. So taking the intervals like $p_{e}=0{\text{ ... }}\Delta p_{e}$ or $p_{e}=0.5\Delta p_{e}{\text{ ... }}1.5\Delta p_{e}$ is suitable for the order of magnitude. U240700 (talk) 12:19, 12 July 2024 (UTC)

The total momentum in quantum mechanics is conserved meaning that

${\hat {\mathbf {p} }}_{e}+{\hat {\mathbf {p} }}_{p}=0$,

or

$<({\hat {\mathbf {p} }}_{e}+{\hat {\mathbf {p} }}_{p})^{2}>=(\Delta \mathbf {p} _{e})^{2}+(\Delta \mathbf {p} _{p})^{2}+2<\Delta {\hat {\mathbf {p} }}_{p}\Delta {\hat {\mathbf {p} }}_{e}>=0$.

This equality holds because electron and proton momentums are anticorrelated. Ruslik_Zero 21:00, 12 July 2024 (UTC)

Last equation confirms only that $\Delta p_{p}=-\Delta p_{e}$. How does it prove or disprove $\Delta x_{p}=\Delta x_{e}$? U240700 (talk) 00:54, 13 July 2024 (UTC)

Your value $\Delta x_{e}$ comes from a different experiment, not Rutherford's, and is therefore not relevant here. It applies, for instance, to a measurement of the average charge density in an atom. This would be a low-energy experiment, taking care not to disturb the electronic structure of the atom. The uncertainty principle does not stop you from setting up an experiment to measure the instantaneous position of an electron more precisely, but the concomitant momentum uncertainty would likely ionise the atom. Rutherford's alpha particles had energies in the MeV range, if I'm not mistaken, much higher than the binding energies of electrons in atoms. They could in principle — I don't know how, and Rutherford's experiment is certainly not set up to do so — be used to locate electrons to much smaller $\Delta x_{e}$ than the value you give. --Wrongfilter (talk) 03:53, 13 July 2024 (UTC)

I have no questions for $\Delta x_{e}$, I have a question for $\Delta x_{p}$. The proton is located in center of atom in very small boundaries ($10^{-15}{\text{ m}}$) . It violates the uncertainty principle. The proton must be smeared over $10^{-10}{\text{ m}}$ and (like the electron) be detected (materialized) everywhere , not just in center. U240700 (talk) 06:57, 13 July 2024 (UTC)

Okay, like you I got too hung up on the location of protons and electrons. The Rutherford experiment does not actually measure the location of nuclei, it measures their size. The experiment does not tell you where the nuclei are (not even in relation to their electron shell). Putting them at the centre of atoms is subsequent model building. --Wrongfilter (talk) 09:14, 13 July 2024 (UTC)

The momenta of the proton and electron are only equal and opposite if the total momentum of the atom is zero. In that case, the position of the atom is completely indeterminate. In order for the atom to be localized, it must have some uncertainty in the total momentum, which means that you can't equate the momentum uncertainties of the proton and electron in this way. The proton can have a larger momentum uncertainty, and a smaller position uncertainty, than the electron. (Note also that the uncertainty principle is an inequality, and it's worth calling out that the ground state is a minimum uncertainty state.) --Amble (talk) 00:35, 16 July 2024 (UTC)

Rutherford scattering experiment did not actually measure the position of nucleus. It only measured differential cross-section, which appeared to be one of a point-like charge. Where this nucleus is located is absolutely irrelevant. Ruslik_Zero 20:20, 18 July 2024 (UTC)

If the nucleus has some uncertainty $\Delta x$, then Rutherford actually measured not size of the nucleus but size of $\Delta x$. Because the nucleus might materialize in every point within $\Delta x$. U240700 (talk) 06:11, 25 July 2024 (UTC)