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Comment by rcxdude | Hacker News Reader

The opposing effect is the kinetic energy of the electron: as it get squeezed into a smaller area its kinetic energy must go up, assuming energy is constant (a classical system would more or less be the same, think how an elliptical orbit speeds up as it gets closer to the barycenter, but a quantum mechanical system doesn't necessarily have a defined trajectory).

The real sleight of hand in this explanation is it doesn't really explain why the quantum-mechanical electron doesn't radiate energy like the classical electron. That's a bit more subtle, but it's basically due to the fact that the electron is wave-like and confined within the atom (i.e. it does not have enough energy to escape the nucleus). This means that the wave-function has discrete solutions as a series of 'standing waves', which is what causes the quantization in quantum mechanics.

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>> The real sleight of hand in this explanation is it doesn't really explain why the quantum-mechanical electron doesn't radiate energy like the classical electron. That's a big more subtle, but it's basically due ...

This subtle part is exactly the piece that I am missing.

So far, we have electron in an orbital. The orbital in itself seems stationary (standing waves), so it may seem nothing is accelerating. I presume that is not it for the answer.

If it is, that still feels like, Oh, I solved Schroginger's equation, and it yields these standing waves for the solution, electron is not really having a certain position and momentum as the wave function hasn't collapsed, and thereby there is no acceleration for the electron to be radiating energy.

Asking a total newbie question. I heard that QM has operators, so momentum also has an operator. Is there one for acceleration? Would it yield a zero?

If so: I see Coulomb's force. Is there another force balancing it? Or is Newton's third law not valid anymore.

> Would it yield a zero?

The electron's position is smeared out over the entire orbital, so in this sense it's being accelerated in all cartesian directions equally — sure, in spherical coordinates, it's always in the radially inwards direction, but it can't go "in" any further because its own wave function is already there, blocking it.

(Operator specifically: IDK if there is one for acceleration, but this is why electrons can't be pushed in once they're at the final energy level).

>> but it can't go "in" any further because its own wave function is already there, blocking it.

This is good food for thought for me. Though I am still not fully clear. If the above were the setup with classical mechanics and electrostatics, with nuclear charge at the center and electronic charge smeared out, I would use Gauss theorem to find electric field and still find that the smeared charge would feel a net force towards the center and want to move that way.

If you squeeze the wavefunction in space, you spread it out in velocity; the classical analogy would be that beyond a certain point, the more it actually moves inwards, the faster it has to be moving outwards.

The relevant piece here is acceleration. I understand the rest. The problem in classical explanation is acceleration resulting in energy loss to radiations. With QM, we have an explanation without that radiation, but I am not clear how and why that is, as posed in my questions above.

The net acceleration in cartesian coordinates is zero.

Am genuinely trying to understand.

It's not zero in other coordinate systems? Or you are saying it does come out zero in all the systems. If the latter, I would love to read more including the maths. :-)

Thanks.

Coordinate grids are arbitrary, for any system you can always put a different grid on the same physical reality.

It's just intuitively clear with a cartesian system that there's no net acceleration in a way it might not be for spherical coordinates. If you imagine a sphere, attach to all points of that sphere a normal unit vector pointing inwards, then all points on the sphere are "pointing inwards"; but as each point on the sphere has a corresponding point opposite it, where "inwards" means the exact opposite in cartesian coordinates, the sum of all those inward vectors is exactly zero. With spherical coordinates, you have to convince yourself that the vectors really do cancel. But they're the same vectors described differently.

Just to be clear though: these are still all euclidian geometry, just with different gridlines drawn on them. As yet, nobody has fully solved QM for curved spaces.

I understand. Thanks!

Will try to find the math that shows acceleration in the cartesian coordinates (and with Euclidian geometry) comes out to be zero. That then hopefully will help me understand how is QM solving the original problem.

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