The $k$ -space region of wave vectors up to a given magnitude is a disk of area $\pi k^2$ . The $k$ -space area occupied by a single particle state is $\frac{1}{2} \frac{2 \pi}{L_x} \frac{2 \pi}{L_y} = \frac{2 \pi^2}{A},$ where the factor of $\frac{1}{2}$ is due to the spin degeneracy $g_s = 2 s + 1 = 2$ . The number of states with wave vector magnitude smaller than $k$ is $N(k) = \frac{\pi k^2}{2 \pi^2/A} = \frac{A k^2}{2 \pi}.$ $k$ -space is used extensively in solid state physics e.g. to visualize the energy bands of a material as a function of electron momentum. In position space, the position-energy dispersion would just be a probability blur.

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