I'm reading a solid state physics book and there's something which is confusing me, related to the free electron gas.
After solving Schrodinger's equation with $V = 0$ and with periodic boundary conditions, one finds out that the allowed values of the components of $\mathbf{k}$ are:
$$k_x = \dfrac{2n_x\pi}{L}, \quad k_y=\dfrac{2n_y \pi}{L}, \quad k_z = \dfrac{2n_z\pi}{L}.$$
In the book I'm reading the author says that it follows from this that: there is one allowed wavevector – that is, one distinct triplet of quantum numbers $k_x,k_y,k_z$ – for the volume element $(2\pi/L)^3$ of $\mathbf{k}$ space.
After that he says that this implies that in the sphere of radius $k_F$ the total number of states is
$$2 \dfrac{4\pi k_F^3/3}{(2\pi/L)^3}=\dfrac{V}{3\pi^2}k_F^3 = N,$$
where the factor $2$ comes from spin.
Now, why is that the case? Why it follows from the possible values of $k_x,k_y,k_z$ that density of points in $k$-space? I really can't understand this properly.
Best Answer
Consider $k_x$, $k_y$, and $k_z$ defining the three orthogonal axes of a three dimensional space. This is what he calls $\boldsymbol{k}$ space.
The allowed values of $\boldsymbol{k}$, that is $k_i = 2 n_i \pi / L$, are represented by evenly spaced points in this $\boldsymbol{k}$ space. Each allowed point is separated from its closest neighbours by a distance $2\pi/L$ along each axis. To see this, just compute the separation between points with consecutive integers $n_i$ along each axis:
$$\frac{2(n+1)\pi}{L} - \frac{2n\pi}{L} = \frac{2\pi}{L}$$
Therefore there is a $\boldsymbol{k}$ space volume of $(2\pi/L)^3$ for each allowed point.
You could imagine each allowed point being the centre of a cube with side length $2\pi/L$. All these cubes would exactly fill the space.
Then he postulates that allowed states are occupied for $|\boldsymbol{k}| \leq k_F$. The $\boldsymbol{k}$ space volume of all such states is just the volume of a sphere with radius $k_F$ (assuming that $k_F \gg 2\pi/L$), that is, $4\pi k_F^3/3$. The total number of states is the total volume divided by the volume of each state, multiplied by $2$ for spin, which is your final formula.