Show why an isolated electron cannot emit a photon

[asciola]

Homework Statement

"Use the laws of conservation of relativistic momentum and energy to show that an isolated electron cannot emit a photon."

Homework Equations

The Attempt at a Solution

So far I have tried to answer using the p = p' + q where q is the momentum of the photon, along with E^2 = p^2*c^2 + m^2*c^4, but am lost on how to prove that it is not possible to emit a photon.

 

[mfb]

You can assume that it emits a photon and get a contradiction based on that.
As the choice of a reference frame is arbitrary, you can choose to have the electron at rest, initially, for example.

 

[asciola]

so if the electron is initially at rest that gives

p = 0 so 0 = p' + q

therefore

-p' = q and p'² = q²

rewriting as

E²/c² - m²*c² = (h/λ)²

I feel like this isn't going in the right direction.

 

[Dick]

so if the electron is initially at rest that gives

p = 0 so 0 = p' + q

therefore

-p' = q and p'² = q²

rewriting as

E²/c² - m²*c² = (h/λ)²

I feel like this isn't going in the right direction.

Ok, so the initial electron at rest has total energy mc^2. The final electron has some momentum p. What's the total energy of the final electron in terms of p?

 

[asciola]

Ok, so the initial electron at rest has total energy mc^2. The final electron has some momentum p. What's the total energy of the final electron in terms of p?

the only equation i can think of is E² = p²c² + m²c⁴

 

[Dick]

the only equation i can think of is E² = p²c² + m²c⁴

That's fine. So if you assume p is nonzero, which is larger, the initial energy of the electron or the final energy of the electron?

 

[asciola]

That's fine. So if you assume p is nonzero, which is larger, the initial energy of the electron or the final energy of the electron?

I think i know where this is going. So if the final energy is larger, then in order for energy to be conserved the energy of the photon would have to be negative, which is not possible.

 

[Dick]

I think i know where this is going. So if the final energy is larger, then in order for energy to be conserved the energy of the photon would have to be negative, which is not possible.

You've got it.

 

[asciola]

You've got it.

Thanks! I was thinking it would involve more equations to prove.

 

[Dick]

Thanks! I was thinking it would involve more equations to prove.

Not this one. This one is easy. Taking the frame to be the rest frame of electron actually makes it pretty obvious.