Special Relativity/Energy/Photon Question

[orangepeelsnice]

The Process of a single, isolated electron initially at rest emitting a photon and recoiling in the opposite direction clearly violates the conservation of energy. On the other hand, a single, isolated, moving electron has kinetic energy, and can easily emit a photon. Or can it? Explain.

 

[Orodruin]

The Process of a single, isolated electron initially at rest emitting a photon and recoiling in the opposite direction clearly violates the conservation of energy. On the other hand, a single, isolated, moving electron has kinetic energy, and can easily emit a photon. Or can it? Explain.

No, it cannot. Either energy or momentum conservation (or both) would be violated.

 

[PeroK]

The Process of a single, isolated electron initially at rest emitting a photon and recoiling in the opposite direction clearly violates the conservation of energy. On the other hand, a single, isolated, moving electron has kinetic energy, and can easily emit a photon. Or can it? Explain.

What is the fundamental difference, if any, between a "moving" electron and an electron "at rest"?

 

[orangepeelsnice]

One has kinetic energy, one does not?

 

[PeroK]

One has kinetic energy, one does not?

Would all observers agree on that? Could you have a frame of reference, perhaps, where both electrons were moving?

 

[Orodruin]

Would all observers agree on that? Could you have a frame of reference, perhaps, where both electrons were moving?

Of course you can make this argument and you and me might agree that it is more elegant. However, from the point of view of a layman, I think it might seem a bit like magic and cheating somehow.

My point is that you can make the argument regardless of the frame you are considering by trying to satisfy both energy and momentum conservation at the same time. If you do this (regardless of frame) you will find that it leads to an inconsistency.

 

[PeroK]

Of course you can make this argument and you and me might agree that it is more elegant. However, from the point of view of a layman, I think it might seem a bit like magic and cheating somehow.

@orangepeelsnice could you confirm your level of knowledge. Is this a problem from a physics course?

 

[orangepeelsnice]

@PeroK I see that an electron at rest has zero momentum and clearly would violate conservation of momentum if it started to recoil and emit a photon. in fact, it would doubly violate the the conservation of momentum, because now the electron and the photon have momentum when initially there was none.

However the case where the the electron is moving has both momentum and kinetic energy, why then, can't it emit a photon? ( emits a photon equal to the KE energy lost by the electron )

Finally, I like the question you formed about a third observer who is moving and possibly becomes the observer. If this third observer became the reference frame, effectively causing both electrons to have KE and P (i'm going to stop typing out the word momentum at his point ) why can't the electron that was initially at rest now emit a photon? it seems to me, now its possible to have the at rest electron conserve momentum by emitting a high energy photon in one direction while recoiling in the opposite direction and a lower KE.

I'm a 4th semester undergrad student in the last class of a calculus based physics series.

 

[Nugatory]

However the case where the the electron is moving has both momentum and kinetic energy, why then, can't it emit a photon? ( emits a photon equal to the KE energy lost by the electron )

Try it... The electron has initial momentum $p_e$ and energy $E_e$. Suppose it were to emit a photon with energy $\Delta{E}$. What is the momentum of that photon, and and what is the momentum of an electron with energy $E_e-\Delta{E}$?
(You can find the relationship between the energy and momentum of a photon online if you don't already know it).

However, there's an easier way of solving this problem: The motion or even the presence of the observer is irrelevant to the behavior of the electron. (If this is not clear, you should google for "Galilean relativity, and consider that the lab in which the electron is apparently at rest is attached to the surface of the Earth which is moving through space at many kilometers per second relative to the sun, and many more kilometers per second relative to the Andromeda galaxy, and ...). Thus, once you've worked out the behavior of the electron using coordinates in which the electron is at rest, you have worked out its behavior using coordinates in which it is moving. It doesn't emit a photon when the observer isn't moving relative to it, so it won't emit a photon when the observer is moving relative to it.

 

[jbriggs444]

it would doubly violate the the conservation of momentum, because now the electron and the photon have momentum when initially there was none

Momentum is a vector quantity. Two oppositely directed vectors can add to zero.

 

[Dale]

in fact, it would doubly violate the the conservation of momentum, because now the electron and the photon have momentum when initially there was none.

As @jbriggs444 mentioned, this is actually not the problem. If the electron starts out with 0 initial momentum then as long as the final momentum of the electron were equal and opposite the photons final momentum then the total system momentum would still be 0.

The real problem is that the electron's invariant mass is fixed. An atom's mass is not fixed, so an atom can emit a single photon whereas an electron cannot. Being at rest or moving doesn't matter.

 

[PeroK]

@PeroK I see that an electron at rest has zero momentum and clearly would violate conservation of momentum if it started to recoil and emit a photon. in fact, it would doubly violate the the conservation of momentum, because now the electron and the photon have momentum when initially there was none.

However the case where the the electron is moving has both momentum and kinetic energy, why then, can't it emit a photon? ( emits a photon equal to the KE energy lost by the electron )

Finally, I like the question you formed about a third observer who is moving and possibly becomes the observer. If this third observer became the reference frame, effectively causing both electrons to have KE and P (i'm going to stop typing out the word momentum at his point ) why can't the electron that was initially at rest now emit a photon? it seems to me, now its possible to have the at rest electron conserve momentum by emitting a high energy photon in one direction while recoiling in the opposite direction and a lower KE.

I'm a 4th semester undergrad student in the last class of a calculus based physics series.

The simplest answer is that if an electron emits a photon, then it emits a photon in any reference frame (the photon emission is a physical fact). Therefore, it must emit one in its own rest frame (or the reference frame of an observer moving along with the electron, if you prefer).

In general, for a large range of problems it is simplest to analyse things in the rest frame of one of the particles involved. And this is one of those problems.