Quiz 2#

Physics 141/241: Computational Physics I

Instructor: Javier Duarte

Spring 2023

Due date: Monday, April 24, 2023 8pm

Total points: 10

Submission Instructions#

Problem 1 (141/241) [10 points]#

In lecture 8, we showed that starting from the distribution function

\[\begin{split} f = \begin{cases} F(-E)^{7/2} & \mathrm{if}~E<0\\ 0 & \mathrm{if}~E\geq0 \end{cases}~, \end{split}\]

we could integrate over the velocity components \(\rho = \int d^3v f\) to derive the relation \(\rho = c_p(-\Phi)^5\).

From the last line of slide 8, we saw that the constant \(c_p\) is given by the integral

\[ c_p = 2^{7/2}\pi F \int_{0}^{\pi/2}d\theta \sin\theta \cos^2\theta~\left(1-\cos^2\theta\right)^{7/2}~. \]

Evaluate this integral to derive \(c_p = 2^{5/2}\pi^2 F\frac{7!!}{10!!}\), where \(!!\) denotes the double factorial (not to be confused with the factorial function iterated twice).

Hint: Use the trigonometric identity \(\sin^2\theta + \cos^2\theta = 1\) and \(\int_0^{\pi/2}\sin^{2m}\theta d\theta = \frac{\pi}{2}\frac{(2m-1)!!}{(2m)!!}\).