Do electrons accelerate when transitioning from one energy state to another?

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vanhees71
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The question doesn't make sense. What do you mean by "most probable". You can characterize the probability distribution for the relative position observable in many ways. One way is to look at moments, among which the average (expectation value) is the most simple. All you know about observables in quantum theory given the state the system is prepared in are probability distributions for the outcome of measurements of these observables.

Generally the expectation values for the distance between proton and electron in a hydrogen atom gets larger when exciting the atom from its ground state.

PeterDonis
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I'm struggling to express the question in terms of mathematics
Yes, and that should be a red flag to you that the question that seems fine to you when expressed in ordinary language, isn't actually well-defined. The only way to fix that is to learn the math and figure out what question expressed in math you want to ask. @vanhees71 gave one example of such a question, and gave the answer to it.

DrDu
For bound electrons (the ones that move from one energy state to another in an atom) it makes no sense to talk about their position, speed, or acceleration.
Why not? Position, speed and acceleration are observables in QM, too, and the squared wavefunctions in the respective representations yield the probabilities to observe a given position, speed or acceleration.

PeterDonis
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Position, speed and acceleration are observables in QM
Yes, but they don't all commute, so no quantum state has exact values for all of them.

Also, unless you are actually observing them, you can't say any of them have definite values, just as for any QM observable. And we don't actually observe the positions or speeds or accelerations of bound electrons; we only measure the changes in energy as they go from one energy level to another. (We don't even measure the energy levels themselves directly, only the differences between them, as shown in the frequencies of the photons emitted or absorbed in the transitions.)

weirdoguy
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I don't understand. In quantum physics the electron is a point particle (true or false?). So it arises the question how does it move? Does it move like a point particle along a curved path (mith multiple zig zags e.t.c) which we just cant determine (because we don't have yet the appropriate theory) and so we can speak only with probabilities about its location and movement? Or what does it hold about the movement of electron in the regime of quantum physics?
If you take an electron in the ground state of the Hydrogen atom and measure its total angular momentum you get 0 with 100% probability.

The expected value of its kinetic energy is, however, non zero.

What sort of orbit is that, you might ask? Well, it's a quantum mechanical "orbit", which cannot be reasonably explained in classical terms. In particular, in this system it makes little sense to think of the electron "moving" at all.

weirdoguy
Delta2
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If you take an electron in the ground state of the Hydrogen atom and measure its total angular momentum you get 0 with 100% probability.

The expected value of its kinetic energy is, however, non zero.

What sort of orbit is that, you might ask? Well, it's a quantum mechanical "orbit", which cannot be reasonably explained in classical terms. In particular, in this system it makes little sense to think of the electron "moving" at all.
Some things don't seem to make sense in quantum mechanics. I am sure you ll tell me that they don't make "classical" sense but they make "quantum mechanical" sense. Seems to me one has to redefine fundamental concepts such as the concept of movement in order for QM to make sense.

PeterDonis
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Seems to me one has to redefine fundamental concepts such as the concept of movement in order for QM to make sense.
And if you can point to something in the math that you think deserves to be called "movement" and give a good argument, you might get such a redefinition accepted. But you're not going to do it by just saying "seems to me".

Delta2
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And if you can point to something in the math that you think deserves to be called "movement" and give a good argument, you might get such a redefinition accepted. But you're not going to do it by just saying "seems to me".
No there isn't anything in the math about movement, but somethings just don't make sense. Like we talk about position and momentum in HUP, but a particle doesn't have definite position and velocity and it is like we are forbidden to talk about its "movement". How does this makes sense to you I don't know but it doesn't seem to make sense to me. Maybe you understand it as the particle being simultaneously in many places with a different probability in each place. But this understanding certainly doesn't make classical sense, might make quantum mechanical sense though.

PeterDonis
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we talk about position and momentum in HUP
That's one pair of non-commuting observables to which the HUP applies, but it's by no means the only such pair.

this understanding certainly doesn't make classical sense
You're right, it doesn't. Welcome to quantum mechanics, where the first lesson is: the world is not classical. Classical physics is an approximation that works well in some domains, but that's all it is. You should not expect everything to make sense in classical terms.

Delta2
PeterDonis
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The OP appears to be gone and the thread topic has been thoroughly covered. Thread closed.