3 ms·
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
by rcxdude 2y ago
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.
- alok-g 2y ago>> 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.
- ben_w 2y ago> 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).
- alok-g 2y ago>> 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.
- ben_w 2y agoIf 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.
- alok-g 2y agoThe 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.
- ben_w 2y agoThe net acceleration in cartesian coordinates is zero.