# Solution to Exercise 2.1-1 Free Electron Gas with Constant Boundary Conditions

| Consider the free electron gas model but let the boundary conditions be: y(0) = y(L) = 0, i.e. we have fixed boundary conditions.
|  | Derive the solution to the Schrödinger equation and the density of states for this case.
|  | Show that the number of states is the same as for the periodic boundary conditions as given in the backbone.

| For the basic solution and the dispersion relation (energy vs. momentum / wave vector), due to the boundary conditions we now obtain
y (x, y, z) | = | Ak · | sin(kx · x) · sin(ky · y) · sin(kz · z)
--- | --- | --- | ---
E = | 2 · k2 2 m
--- | ---
|  | Obviously, the boundary conditions y (x = 0) = y (x = L) = 0 (and analogously so for y and z) are satisfied by
kx | = | nx · p L
--- | --- | ---
nx | = | 1, 2, 3, ...
|  | and analogously so for the y and z direction. The amplitude factor Ak follows from the condition that the probability to have the electron in the "box" equals 1:
1 = |A k|2 | L ∫ 0 | sin2 (kx · x) d x | L ∫ 0 | sin2 (ky · y) dy | L ∫ 0 | sin2 (kz · z) dz
--- | --- | --- | --- | --- | --- | ---
|  | Hence, here we have |Ak |2 = (2/L) 3 , independent of k.
| The number of states Z(k) up to a wave vector k is generally given by
Z (k) | = | Volume of sphere with radius E(k) Volume of state
--- | --- | ---
| For fixed boundary conditions we have
Z(k) | = | 1 8 | · | 4/3 (p · k)3 (p/ L)3 | = | (k · L)3 6p2
--- | --- | --- | --- | --- | --- | ---
| This is exactly what we would get for the periodic boundary conditions – thanks to the factor 1/8.
|  | Where does this factor come from? Easy – since the quantum numbers n are restricted to positive integers in this case, we can not count states in 7 of the 8 octants of the complete sphere and must divide the volume of the complete sphere by 8.
|  | This becomes clear if we look at a drawing of the possible states in phase space:

With frame

Exercise 2.1-1 Free Electron Gas with Constant Boundary Conditions

Exercise 2.3.5-1

© H. Föll (Semiconductors - Script)