Solving the Compton Wavelength: Help from Andrew

[mess1n]

Hey, I'm stuck on part of the calculation for the compton wavelength. Basically, my notes go:

Energy of Electron

Before collision: E₀ = m_ec²

After collision: E_e = $$\sqrt{P_{e}^{2}c^{2} + m_{e}^{2}c^{4}}$$

Notes:
P_e = momentum of electron after collision
E_e = energy of electron after collision
m_e = mass of electron

I can't see why this follows as the energy after the collision. Any ideas?

Cheers,
Andrew

 

[Dale]

After collision: E_e = $$\sqrt{P_{e}^{2}c^{2} + m_{e}^{2}c^{4}}$$
...
I can't see why this follows as the energy after the collision. Any ideas?

That is the formula for the total energy of the electron. The first term is the kinetic energy which is zero before the collision, and the second term is the mass-energy or rest-energy which is unchanged, i.e. the first formula is derived from the second for the special case of the electron at rest.

 

[Doc Al]

I can't see why this follows as the energy after the collision. Any ideas?

That second equation is true for any particle and is often written like this:

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

The 'Before collision' version is just a special case where P = 0, since the electron starts out at rest.

(Edit: DaleSpam beat me to it.)

 

[mess1n]

Ah sorry maybe I should clarify my question. What I'm really wondering is where the formula for E² comes from. If it's a general equation, where can I find a derivation or explanation of it?

Cheers,
Andrew

 

[nicksauce]

Ah sorry maybe I should clarify my question. What I'm really wondering is where the formula for E² comes from. If it's a general equation, where can I find a derivation or explanation of it?

Cheers,
Andrew

Any book on special relativity.

 

[jtbell]

where the formula for E² comes from.

Are you acquainted with the usual formulas for relativistic energy and momentum?

$$E = \frac {m_0 c^2} { \sqrt {1 - v^2 / c^2}}$$

$$p = \frac {m_0 v} { \sqrt {1 - v^2 / c^2}}$$

Solve one equation for v, substitute into the other equation, and simplify.

 

[mess1n]

Apparently I'm not! Thanks for the help everyone, that's cleared it up.

 

[Vanadium 50]

Solve one equation for v, substitute into the other equation, and simplify.

You must love algebra!

It's simpler and faster to calculate E²-(cp)². Then divide out the denominator and you're done.

 

[jtbell]

Yeah, since we actually know where we want to end up, that's the easiest way to do it. I've been working out too many solutions for homework problems where you don't know the answer in advance.