Compton Scattering- finding photon wavelength

[PhysicsInNJ]

Homework Statement

An electron moving to the left at 0.8c collides with an incoming photon moving to the right. After the collision, the electron is moving to the right at 0.6c and an outgoing photon moves to the left. What was the wavelength of the incoming photon?

Homework Equations

p=γmv
E_γ=pc=h/λ

λ'-λ = (h/m_ec)(1-cosθ)

The Attempt at a Solution

*Defining left as positive*
I have the following expressions for the initial and final momentum of the system, and I set them equal to each other.

P_initial= p_e + p_γ
Pi = -γ_im_ev_ei + E_γi/c
P_f = γ_fm_ev_ef - E_f/c

given Ei/c = h/λ

h/λ - γ_im_ev_ei = γ_fm_ev_ef - h/λ'

h(1/λ + 1/λ') = γ_fm_ev_ef + γ_im_ev_ei

I can plug in all the values for the electron and get some number, but I am still left with λ and λ'. I'm not sure if my approach is correct, but if it is, I don't see how I can find just λ (the initial wavelength).

 

[Chandra Prayaga]

Is the total momentum the only quantity that is conserved?

 

[PhysicsInNJ]

Is the total momentum the only quantity that is conserved?

I presume energy would also be conserved, but writing out that expression also leaves me with λ and λ'

(γ_im_ev_i²)/2 +m_ec² - hc/λ = hc/λ' - (γ_fm_ev_f²)/2 - m_ec²

 

[Chandra Prayaga]

Your expression for the energy of the electron is wrong

 

[PhysicsInNJ]

Would E= √(p²c² + m₀²c⁴) be more accurate?

 

[Chandra Prayaga]

That is correct