Question

In a one-dimensional solid with free electrons and T-0K, assuming the free electron concentration is n, please calculate the Fermi energy EF. What is the total electron kinetic energy? ( Express your result in terms of electron mass m and other constants) (1000)

Answer 1

One - dimension solid with free electrons means free electrons gas in one dimension. So, assuming that, an electron of mass m is confined to a length L by infinite barriers. Schrodinger equation for electron in I D is F = -h^2/2 m d 62/d x^2 psi_n (x) = E_n psi_n (x) The solution for the above equation is psi (x) = A sin k x + B cos k x - (1) Applying the boundary conditions, psi (x) = 0 at x = 0 and psi (L) = 0 at x = L therefore psi (x = 0) = 0 - A sin k (0) + B cos k (0) = B = 0 therefore (1) psi (x) = A sin k x - (2) Now, psi (x = L) =) A sin K L (or) sin K L = 0 K L = n pi k = n pi/L where n is integer