# Show why an isolated electron cannot emit a photon

1. Feb 17, 2013

### asciola

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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.

2. Feb 17, 2013

### Staff: Mentor

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.

3. Feb 17, 2013

### 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.

4. Feb 17, 2013

### Dick

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?

5. Feb 17, 2013

### asciola

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

6. Feb 17, 2013

### Dick

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?

7. Feb 17, 2013

### asciola

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.

8. Feb 17, 2013

### Dick

You've got it.

9. Feb 17, 2013

### asciola

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

10. Feb 17, 2013

### Dick

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