The extended Schr(o)dinger equation for the Kirchhoff elastic rod with noncircular cross section is derived using the concept of complex rigidity. As a mathematical model of supercoiled DNA, the Schr(o)dinger equation for the rod with circular cross section is a special case of the equation derived in this paper. In the twistless case of the rod when the principal axes of the cross section are coincident with the Frenet coordinates of the centreline, the Schr(o)dinger equation is transformed to the Duffing equation. The equilibrium and stability of the twistless rod are discussed, and a bifurcation phenomenon is presented.