Schrödinger Potential Fields with no Energy Quantisation?

[stevendaryl]

you already mentioned that its due to conservation laws

That was the answer to the question: Why can't a free electron emit a photon? You can observe free electrons in other ways besides looking for photons that they emit.

 

[greswd]

That was the answer to the question: Why can't a free electron emit a photon? You can observe free electrons in other ways besides looking for photons that they emit.

Yeah, I was thinking from a photonic, spectroscopy POV

 

[stevendaryl]

Well, here's something to consider when thinking about spontaneous emission. If you have an electron in a high-energy state, then intuitively you understand why it would emit a photon and fall into a lower-energy state: It's natural for systems to want to lower their energy, in the same way that it's natural for water to want to flow downhill. But this intuitive answer doesn't actually make any sense, by itself. When an electron emits a photon, its energy goes down, but the energy in the electromagnetic field goes UP. The total energy is unchanged. So the real question is not: why does the electron's energy go down, but why does nature prefer to give its energy to photons, as opposed to electrons?

Well, we can understand that through entropy, which amounts to counting states. There is only one (or a small number) of ways that a bound electron can absorb a quantity of energy, because there are only a few states associated with a given energy. In contrast, there are infinitely many ways that photons can absorb a quantity of energy, because there are continuum-many photon states. So if you pick a way to split energy up between an electron and the electromagnetic field, it's overwhelmingly more likely that most of the energy will go to the electromagnetic field. So what we observe is that electrons tend to radiate their energy away.

Now, if the electron itself has continuum-many states, then this counting argument doesn't apply. Now, there is no good reason, as far as entropy, for the electron to give up its energy to the electromagnetic field.

 

[greswd]

Oh I see. A probabilistic way of looking at things.

 

[stevendaryl]

For a free particle, there is just no way to choose $\vec{K}$ and $\vec{p'}$ so that both energy and momentum are conserved.

This is easiest to see in the rest frame of the particle. Then initially, the momentum is zero. Initially, the energy is $mc^2$. After it emits a photon, the energy of the particle must still be at least $mc^2$ (because that's the lowest possible energy of the particle), which means that the energy of the photon has to zero (or negative!) to get energy to balance.

 

[greswd]

This is easiest to see in the rest frame of the particle. Then initially, the momentum is zero. Initially, the energy is $mc^2$. After it emits a photon, the energy of the particle must still be at least $mc^2$ (because that's the lowest possible energy of the particle), which means that the energy of the photon has to zero (or negative!) to get energy to balance.

oh yeah. there is no incoming photon that exists which can collide with the electron or something.