How Do You Derive the Compton Equation?

[latitude]

Homework Statement

Derive the compton equation.

Homework Equations

$$\lambda$$` - $$\lambda$$ = h/ mc (1 - cos$$\theta$$)
E = hf = hc/$$\lambda$$

The Attempt at a Solution

Okay, I'm sorry this is so long, I'll try and make it as concise as it is possible for a whole blather of random crap to be :]

Conservation of momentum components:
h/$$\lambda$$ = h/$$\lambda$$`(cos$$\theta$$) + Pe(cos$$\psi$$)
0 = h/$$\lambda$$`(sin$$\theta$$) - Pe(sin$$\psi$$)

After some combining, squaring, and the like (getting rid of $$\psi$$):
Pe² = (h/$$\lambda$$)² - (h/$$\lambda$$`)<sup>2</sup>cos<sup>2</sup>$$\theta$$ + (h/$$\lambda$$`)²sin²$$\theta$$ - (h/$$\lambda$$)(h/$$\lambda$$`)cos$$\theta$$


E² = p²c² + E_R²
So
P² = (E² - E_R²)/c²

So I plug that into my momentum (I'm not going to write the righthand side of the equation while i show what I did w/ that)

(E² - E_R²)/c² = ...
((hc/$$\lambda$$)² - (mc²)²)/c² = .
I tried to get rid of the denominator 'c'...
(h/$$\lambda$$)² - m²c² = ...

(m²c²$$\lambda$$)/h = $$\lambda$$/$$\lambda$$` - h/$$\lambda$$`cos$$\theta$$

After some more fiddling I get to this:

$$\lambda$$` = h/m²c² - (h/$$\lambda$$)(h/m²c²)cos$$\theta$$

It's kind of close but not really... I can write out all the steps I made if that is necessary, but I'm kind of hoping I made one nice, simple-to-fix error that is glaringly obvious to the more experienced :)

Thank you :)

 

[tiny-tim]

Hi latitude!

(have a lambda: λ and a theta: θ and a psi: ψ )

Pe² = (h/$$\lambda$$)² - (h/$$\lambda$$`)<sup>2</sup>cos<sup>2</sup>$$\theta$$ + (h/$$\lambda$$`)²sin²$$\theta$$ - (h/$$\lambda$$)(h/$$\lambda$$`)cos$$\theta$$

eugh …

the first - should be a + ​

 

[thomate1]

Hi there,

Don't worry, deriving equations can be tricky and it's common to get stuck. Let's take a look at the steps you've already done:

1. You started with conservation of momentum components, which is a good starting point.
2. You squared the equations and added them together, which is also a valid step.
3. You used the equation E = pc to substitute for p.
4. You tried to simplify the equation by getting rid of the denominator c, which is a good idea.

However, from there, I can see where you went off track. Let's take a closer look at your final equation:

\lambda` = h/m2c2 - (h/\lambda)(h/m2c2)cos\theta

The first term on the right-hand side is correct, but the second term is not. Let's break it down:

(h/\lambda)(h/m2c2)cos\theta = (hc/\lambda)(h/m2c2)cos\theta

Notice that you have c and \lambda in the numerator, but in your previous steps, you were trying to get rid of them. Also, the second term should have a minus sign, not a plus sign.

Here's a hint to help you move forward: try substituting E = hc/\lambda into your equation and see what you get. Remember, you want to end up with an expression for \lambda` in terms of \lambda and \theta.

I hope this helps and good luck with your derivation!