Black holes, white dwarfs and neutron stars-Shapiro, Teukolsky

[Vrbic]

Homework Statement

Hello, I try to recompute all exercises in this book and sometime I hit the snag :) One of the first is:
Exercise 2.6 (page 28)
Show that mean kinetic energy of an electron in a degenerate gas is $\frac{3}{5}E'_f$ in the nonrelativistic limit and $\frac{4}{5}E_f$ in relativistic limit. Here $E'_f=E_f-m_ec^2=p_f^2/2m_e$.

Homework Equations

The Attempt at a Solution

I know that for nonrelativistic limit $\epsilon=\frac{m_ec^2}{\lambda_e^3 \pi^2}\chi(x)$ and $\chi(x)->\frac{x^3}{3\pi^2}$. So my idea was to integrate this energy density from 0 to x-fermi, where $x=p/mc$. But result $\frac{2\pi}{3h^3}cp_f^4$ doesn't recover their.
Do have anybody some suggestions?

 

[mark726]

Hello,

Thank you for sharing your attempt at solving this exercise. It seems like you are on the right track by using the energy density equation for a degenerate gas in the nonrelativistic limit. However, I see that you are integrating from 0 to x-fermi, which may not be the correct range for the integral.

To find the mean kinetic energy of an electron in a degenerate gas, we can use the equation:

\bar{\epsilon}=\frac{\int_{0}^{p_f} \epsilon(p) p^2 dp}{\int_{0}^{p_f} p^2 dp}

Where \epsilon(p) is the energy density equation for a degenerate gas in the nonrelativistic limit.

By plugging in the appropriate values and integrating, you should be able to get the result of \frac{3}{5}E'_f.

I hope this helps and good luck with your computations!