This conversion from a sum over a discrete grid of $k$-space states to a volume integral is one of the extremely commonly used ideas in solid state physics: it provides us a way to count all the possible waves.
This conversion from a sum over a discrete grid of $k$-space states to a volume integral is one of the extremely commonly used ideas in solid state physics: it provides us a way to count all the possible waves.
We can use this approximation and rewrite the total energy as an integral: