Unformatted text preview: θ
θ L the momentum transfer is now only 2p cos , and also it takes longer between
t = 2cos : Therefore P = cos2 u :
Adding up all the photons is the same as averaging an isotropic distribution
over cos2 : hcos2 i cos2 d sin d'
d sin d'
cos2 sin d
= 2 20 sin d =
R R R 1
3 (where we set = cos ). Another way to get this result is by symmetry:
Consider a randomly oriented unit vector, (x; y; z), with x2 + y2 + z2 = 1.
Since there is nothing special about the x; y, or z directions with respect to
the unit sphere, we must have hx2i = hy2i = hz2i :
But hx2i + hy2i + hz2i = hx2 + y2 + z2i = h1i = 1 ;
so each one must be 1/3. Note that in spherical coordinates z = cos , and
we are done.
View Full Document
- Fall '09
- unit sphere, randomly oriented unit, sin 12 d 12, d 12 sin