Girvin, Yang Modern Condensed Matter Physics 2019 SSP · Problem 12.11

In a 2D crystal with periodic boundary conditions, Bloch wave vectors are quantized in both directions—just like standing waves on a string. Each component of $\vec{K}$ can take only discrete values spaced by $2\pi$ over the system size, so the number of allowed states in $k$-space is simply the product of the number of allowed values in each direction. Counting these carefully and comparing to the real-space lattice sites shows that momentum space and real space each contain exactly the same number of independent quantum states—a fundamental consistency check in solid-state physics.

In a 2D crystal with periodic boundary conditions, Bloch wave vectors are quantized in both directions—just like standing waves on a string. Each component of $\vec{K}$ can take only discrete values spaced by $2\pi$ over the system size, so the number of allowed states in $k$-space is simply the product of the number of allowed values in each direction. Counting these carefully and comparing to the real-space lattice sites shows that momentum space and real space each contain exactly the same number of independent quantum states—a fundamental consistency check in solid-state physics.