How to Verify the Energy of a Scattered Photon Formula?

[Reshma]

An electron of mass 'm' and speed 'v' collides with a gamma ray photon of initial energy hf₀, as measured from the laboratory frame. The photon is scattered in the electron's direction of travel. Verify that the energy of the scattered photon, as measured in the laboratory frame, is:
$$E = hf_0\left(1 + \frac{2hf_0}{mc^2}\sqrt{\frac{1 + v/c}{1 - v/c}}\right)^{-1}$$

Well this also seems to be a Compton effect problem. The relativistic Doppler equation for frequency is given by:
$$f = f_0\sqrt{\frac{1 - \beta}{1 + \beta}}$$
where $\beta = v/c$

Need guidance to apply this formula to obtain above result...

 

[Andrew Mason]

An electron of mass 'm' and speed 'v' collides with a gamma ray photon of initial energy hf₀, as measured from the laboratory frame. The photon is scattered in the electron's direction of travel. Verify that the energy of the scattered photon, as measured in the laboratory frame, is:
$$E = hf_0\left(1 + \frac{2hf_0}{mc^2}\sqrt{\frac{1 + v/c}{1 - v/c}}\right)^{-1}$$

Well this also seems to be a Compton effect problem. The relativistic Doppler equation for frequency is given by:
$$f = f_0\sqrt{\frac{1 - \beta}{1 + \beta}}$$
where $\beta = v/c$

Need guidance to apply this formula to obtain above result...

Use the Compton formuala to determine the energy of the photon after the collision in the frame of the electron before the collision. Then apply the Doppler equation to get the energy in the lab frame.

AM

 

[BlazzedTroll]

Thank you for the hints here on how to solve it but I have been working on this problem for several days now and I keep getting to similar places after using the compton scattering and doppler effect but I can't see how sqrt((1-v/c)/1+v/c)) is becoming sqrt((1+v/c)/(1-v/c))
can someone please explain this?

 

[Andrew Mason]

Thank you for the hints here on how to solve it but I have been working on this problem for several days now and I keep getting to similar places after using the compton scattering and doppler effect but I can't see how sqrt((1-v/c)/1+v/c)) is becoming sqrt((1+v/c)/(1-v/c))
can someone please explain this?

v is negative if the observer is moving away from the source and positive if the observer is moving toward the source.

AM