4. Consider a 1-dimensional electron gas confined in a region of length L and containing N electrons. [36 points] (a) Fi

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4 Consider A 1 Dimensional Electron Gas Confined In A Region Of Length L And Containing N Electrons 36 Points A Fi 1 (42.17 KiB) Viewed 37 times

4. Consider a 1-dimensional electron gas confined in a region of length L and containing N electrons. [36 points] (a) Find the Fermi energy &F of this 1-D electron gas. (12 points) (b) Calculate the mean energy per electron at temperature T = 0 K. Express your result in terms of EF. (6 points) (c) Calculate the mean energy per electron at temperatures 0 < T << TF, where TF = &F /k. Express your result in terms of & and KT. (You may assume that for T <<< TF, the chemical potential μ = EF.) (10 points) (d) Calculate the heat capacity Cy of the entire N-electron gas at low temperatures (0 < T << Tp). Express your result in terms of TF and N. (3 points) (e) Using the thermodynamic relation between Cy and S, find the entropy of the system at low temperatures. (5 points)