TMFKAN64
Mar18-10, 07:17 PM
Hello, I'm getting a bit bogged down in constant factors while reading Griffiths' "Introduction to Elementary Particles", 1st edition. In particular, I'm having problems with equation 6.15:
d\Gamma = |M|^{2}\frac{S}{2\hbar m_{1}}\left[
\left(\frac{c\, d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\right)
\left(\frac{c\, d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\right)\cdots
\left(\frac{c\, d^{3}p_{n}}{(2\pi)^{3}2E_{n}}\right)\right]\times
(2\pi)^{4}\delta^{4}(p_{1}-p_{2}-p_{3}\cdots-p_{n})
I've found some slides online at http://www-pnp.physics.ox.ac.uk/~libby/Teaching/Lecture4.pdf that discuss normalizing the wave function and phase space so it is Lorentz invariant, but this only explains part of my problem. What I still don't understand is:
Where do the c in the numerators of the momentum terms come from?
Where does the (2\pi)^{4} term before the delta function come from?
Where did the \hbar^{3n} in the density of states go? (The \frac{1}{(2\pi\hbar)^{3n}} is responsible for the 2\pi factors in the momentum terms, right?)
Why doesn't the first fraction have m_{1}c^{2} in the denominator?
Any assistance would be greatly appreciated.
d\Gamma = |M|^{2}\frac{S}{2\hbar m_{1}}\left[
\left(\frac{c\, d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\right)
\left(\frac{c\, d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\right)\cdots
\left(\frac{c\, d^{3}p_{n}}{(2\pi)^{3}2E_{n}}\right)\right]\times
(2\pi)^{4}\delta^{4}(p_{1}-p_{2}-p_{3}\cdots-p_{n})
I've found some slides online at http://www-pnp.physics.ox.ac.uk/~libby/Teaching/Lecture4.pdf that discuss normalizing the wave function and phase space so it is Lorentz invariant, but this only explains part of my problem. What I still don't understand is:
Where do the c in the numerators of the momentum terms come from?
Where does the (2\pi)^{4} term before the delta function come from?
Where did the \hbar^{3n} in the density of states go? (The \frac{1}{(2\pi\hbar)^{3n}} is responsible for the 2\pi factors in the momentum terms, right?)
Why doesn't the first fraction have m_{1}c^{2} in the denominator?
Any assistance would be greatly appreciated.