For isotropic one-dimensional systems with parabolic energy dispersion, the density of states is D 1 D ( E ) = 1 2 π π ℏ ℏ ( 2 m E ) 1 / 2 . In two dimensions the density of states is a constant D 2 D = m 2 π π ℏ ℏ 2 , while in three dimensions it becomes D 3 D ( E ) = m 2 π π 2 ℏ ℏ 3 ( 2 m E ) 1 / 2 .