Mastering Compton Scattering Squaring for Scientists

[xinlan]

Homework Statement

How to square this : (Pi+m.c-Pf)^2 ?

Homework Equations

The Attempt at a Solution

I did : Pi^2 + m^2*c^2 + Pf^2...
but then I don't know the rest of the completion..

please help me..

Thank you in advance..

 

[Doc Al]

Please state the full problem that you're trying to solve. Are you trying to derive the formula for Compton scattering? If so, you'll need both conservation of momentum and energy. (What do Pi and Pf represent? What's "mc" stand for?)

 

[xinlan]

Yes, I'm trying to derive the formula for Compton Scattering.
There are 7 parts for deriving the formula.

here is part A
Consider a photon with initial momentum pi on a collision course with a stationary electron of mass m. What is the total energy Ei of these two particles before the collision? Don't forget to include the rest mass energy of the electron.
Express your answer in terms of the pi, the speed of light c , and m .

Here is part B

After the collision, the photon exits with a momentum pf at an angle phi from its initial momentum vector. The electron scatters off with a momentum pe. What is the total energy after the collision? In this case, do not forget to include the relativistic energy of a particle.
Express your answer in terms of pf, pe , m , and c.


Here is part C
The conservation of energy requires that the energy of the system before the collision Ei and the energy after the collision Ef be equal to one another. We want to manipulate this conservation equation to arrive at the result for Compton scattering. The first step is to take Ei as one side of the equation and Ef as the other side of the equation. Divide both sides by c and isolate the square root on one side by subtracting the term pf from both sides of the equation. You should now be left with a trinomial on one side. Complete the right-hand side of the equation with these three terms.
Express your answer in terms of pi, pf, m, and c.

Here is part D
Square both sides of the equation obtained in Part C and choose the result of squaring the right-hand side (the one you entered in the last part) as your answer. Your unreduced answer should have nine terms in it. If you further reduce your answer you should have at least six terms.

Here is part E
You can use conservation of momentum to eliminate the term pe from the equation. If you recall that vector of pe, pi, and pf are all vectors you can use vector addition to state one of the vectors in terms of the other two. Give the value for vector of pe in terms of vector of pi and pf .

Here is part F
You now wish to get a value for pe^2 in terms of the scalar quantities pi, pf, and the angle phi between the two vectors. Recall that when squaring a vector it is necessary to use the dot product: pe^2=(pe)^2 = vector of pe . vector of pe.
Express your answer in terms of pi, pf, and phi.

Here is part G
Substitute the value you obtained for pe^2 in Part F into the equation obtained in Part D. Eliminate terms, and divide by 2 so that one side of the equation reads pi*pf-pi*pf*cos phi. Supply the other side of the equation in the answer box.
Express your answer in terms of pi, pf, m, and phi .

I got part A till C and part H as well.
I don't understand part D and E
I figure it out if I can't do part D, then I can't do the rest of them..

so please help me..
thanks so much..

 

[Doc Al]

Show what you did for the first three parts.

 

[Doc Al]

for part A = Ei = pi.c+mc^2
part B = Ef = pf.c + square root of pe^2*c^2 + m^2*c^4
part C = square root of pe^2 + m^2*c^2 = pi+m*c-pf

Good. So just do the squaring as instructed:
$$p_e^2 + m^2c^2 = (p_i + mc - p_f)^2 = (p_i + mc - p_f)(p_i + mc - p_f)$$

Just multiply that right hand side out, term by term. You'll get $p_i^2 + p_i m c - p_i p_f + ...$ (9 terms, initially.)

 

[xinlan]

oh.. let me try it first.. thanks so much..

 

[xinlan]

I got it.. thanks..
could you explain what does the question mean on part E ?
because i don't understand the question is..

thanks so much..

 

[Doc Al]

Question E is asking you to start by writing conservation of momentum as a vector equation, using
$$\vec{p}_i$$, $$\vec{p}_f$$, $$\vec{p}_e$$.

 

[xinlan]

does it mean that the answer that I got from part D needs to be rewrited as a vector equation ?

 

[Doc Al]

does it mean that the answer that I got from part D needs to be rewrited as a vector equation ?

No. In part E, you write the conservation of momentum vector equation. In part F, you'll square that equation. In part G, you'll use the results to eliminate p_e in the equation from part D. (Just follow the instructions, step by step.)

 

[xinlan]

I got it.. :)

thank you for Doc Al so much..
I really appreciate it..

 

[Doc Al]

My pleasure. Keep at it!