Squaring the energy

mess1n

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

Energy of Electron

Before collision: E0 = mec2

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

Notes:
Pe = momentum of electron after collision
Ee = energy of electron after collision
me = mass of electron

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

Cheers,
Andrew

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Dale

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

Mentor
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

Homework Helper
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

Mentor
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.

Staff Emeritus
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

Mentor
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. :yuck:

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