bluepilotg-2_07
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- Homework Statement
- For the 1D electron gas at zero temperature derive the average energy and show that it can be written as E_avg = d/d+2 where d is the dimension of the system.
- Relevant Equations
- E_avg = E/N
E_F = ((h_bar)^2 * (k_F)^2)/2m
Fermi energy is given by $$\epsilon_F = \frac{\hbar ^2 k_F ^2}{2m}$$
##N = \frac{2k_F L}{2\pi} \Rightarrow \frac{k_F L}{\pi}## the factor of two in the numerator comes from the electrons having two spins.
$$E=\frac{2L}{2\pi} \int_{0}^{k_F} \frac{\hbar^2 k^2}{2m}\, 2\,dk$$ The two in front of the integral comes from the spin and the ##2\, dk## comes from the dimension of the system.
$$E = \frac{L\hbar^2 k_F^3}{3\pi m}$$
$$E/N = \frac{\hbar^2 k_F^2}{3m} \Rightarrow \frac{2\epsilon_F}{3}$$
The answer should be ##\epsilon_F/3##
##N = \frac{2k_F L}{2\pi} \Rightarrow \frac{k_F L}{\pi}## the factor of two in the numerator comes from the electrons having two spins.
$$E=\frac{2L}{2\pi} \int_{0}^{k_F} \frac{\hbar^2 k^2}{2m}\, 2\,dk$$ The two in front of the integral comes from the spin and the ##2\, dk## comes from the dimension of the system.
$$E = \frac{L\hbar^2 k_F^3}{3\pi m}$$
$$E/N = \frac{\hbar^2 k_F^2}{3m} \Rightarrow \frac{2\epsilon_F}{3}$$
The answer should be ##\epsilon_F/3##