This Demonstration explores the solutions of the secular equations for graphene nanoribbons with zigzag and bearded edges. For graphene nanoribbons with zigzag edges, the secular equation is
, while for a ribbon with bearded edges it is
. For ribbons containing
pairs of carbon atoms in their unit cells, these equations quantize the transverse momentum of the electron,
, and couple it to the longitudinal momentum,
. The equation for a ribbon with zigzag edges is the same as equation (19) in . The equation for a ribbon with bearded edges,
, is different from the equation
given in reference . The sign in front of the second term is of no physical importance, but the minus is more convenient for mathematical treatment, as has been discussed for zigzag ribbons .