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24

The main problem is, as you say, that time is no operator in quantum mechanics. Hence there is no expectation value and no variance, which implies that you need to state what $\Delta t$ is supposed to mean, before you can write something like $\Delta E \Delta t\geq \hbar$ or similar. Once you define what you mean by $\Delta t$, relations that look similar ...


5

When you have a bunch of interrelated phenomena in physics, trying to figure out which one is the "reason" for the other ones is often just a recipe for confusion. Different people will start from different postulates, so they will disagree on which results are trivial and which aren't, but hopefully everyone agrees on what's true. In a first course on ...


4

This is the case. The uncertainty relationship with energy and time is a matter of Fourier analysis. In fact the relationship $\Delta\omega\Delta t \simeq 1$ was know in classical EM and electrical engineering before quantum physics. The use of Fourier analysis in electrical engineering had much the same uncertainty relationship as the reciprocal ...


2

The Heisenberg Uncertainty Principle has two distinct aspects: One is the identification of matter as a wave and, in particular, the relationship between a particle's momentum $p$ and its wavelength $\lambda$ through de Broglie's relationship $p=h/\lambda$. This is the crucial bit of physical input. The second one is purely mathematical, and it's the ...


1

The uncertainty principle doesn't contain the actual position and momentum measurements of quanta, it only contains the product of the standard deviations of the ensemble measurements: $\sigma_x \sigma_p>{\hbar \over 2}$. This expression is only meaningful if we perform a large number of independent experiments and then calculate the standard deviations ...


1

Yes, this is not the "true" reason. The direct reason is that they are not commuting operators. See the Robertson-Schrodinger Relation. You end up getting that the product of the uncertainties is bounded by 1/2 of the absolute value of the failure to commute. In the case of position and momentum this is $\hbar/2$.


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The basics of atomic structure are studied in the non-relativistic limit (mass not changing with velocity). Relativistic corrections are then added as perturbations to the non-relativistic states & energies (this gives you the fine structure https://en.wikipedia.org/wiki/Fine_structure). For a full quantum treatment of relativistic effects you need to go ...



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