Why Can't a Photon Be Completely Absorbed by a Free Electron?

[GAURAV DADWAL]

I have read in textbook that if a photon were to collide with a free electron it's an impossible situation for photon to get completely absorbed by an electron .
The situation seems possible by conservation of energy but I am not able to understand the true reason behind the statement .is there something to do with conservation of linear momentum

 

[sophiecentaur]

I have read in textbook that if a photon were to collide with a free electron it's an impossible situation for photon to get completely absorbed by an electron .
The situation seems possible by conservation of energy but I am not able to understand the true reason behind the statement .is there something to do with conservation of linear momentum

If a LF radio signal (say 200kHz) passes through the ionosphere, it can cause electrons to move from side to side (actually, due to the Earth's magnetic field they follow a circular path). See this link on the Appleton Hartree Equation The wave interacts with the electron but how is this described in terms of photons, I wonder? If much higher frequencies were involved, it would be described in terms of Compton scattering. But Compton scattering involves only a very small reduction in the frequency (=energy of the photon).

 

[Doc Al]

The situation seems possible by conservation of energy but I am not able to understand the true reason behind the statement .is there something to do with conservation of linear momentum

Yes. Try solving the problem using both momentum and energy conservation and you'll see it cannot be done. (Note that the electron has no internal structure, thus the 'collision' must be elastic.)

 

[PeroK]

Hint: The relevant equation that applies to both the photon and electron is:

$E^2 = p^2c^2 + m^2c^4$

 

[Comeback City]

(Note that the electron has no internal structure, thus the 'collision' must be elastic.)

What is the actual reasoning for this?

 

[mfb]

Consider the whole interaction in the center of mass frame (the frame where the total momentum is zero). If the reaction just leaves an electron, then we start with a photon and a moving electron, and end up with an electron at rest. Clearly the final state has a lower energy than the initial state. That doesn't work.

If an electron would have a substructure, it could have excited states, and you end up with something else at higher mass.

 

[GAURAV DADWAL]

Consider the whole interaction in the center of mass frame (the frame where the total momentum is zero)

How do i calculate the centre of mass of photon and an electron system

 

[PeroK]

How do i calculate the centre of mass of photon and an electron system

The beauty of @mfb's solution is that you don't have to calculate anything.

 

[GAURAV DADWAL]

The beauty of @mfb's solution is that you don't have to calculate anything.

I think I'm unable to visualize the solution ,can you help

 

[PeroK]

I think I'm unable to visualize the solution ,can you help

First, imagine the rest frame of the electron. The photon is moving towards the electron at rest.

Now imagine a frame moving towards the photon so that the momentum of the electron in this frame is equal and opposite to the momentum of the photon in this frame.

This gives you the centre of momentum frame of the system.

As there is zero total momentum in this frame, there must be the same after the collision. Which means that the electron must end up at rest in this frame.

But an electron at rest has less energy than the initial moving electron, which violates conservation of energy.

Hence, such a collision is physically impossible.

 

[GAURAV DADWAL]

Now imagine a frame moving towards the photon so that the momentum of the electron in this frame is equal and opposite to the momentum of the photon in this frame.

Shouldn't the frame be moving towards electron to get zero momentum i think

Apart from that i got your solution now
Thanx for help[emoji16]

 

[mfb]

Shouldn't the frame be moving towards electron to get zero momentum i think

It is doing both.

You can also have a look at the time-reversed process. Electromagnetism is perfectly time-symmetric - every possible process has to be possible in the opposite time direction as well. If we invert the process, we have an electron that emits a photon out of nowhere. Clearly that doesn't work in the electron rest frame, and if something does not work in one frame, it cannot work in any frame.