New insights from an analogy between a 1d quantum particle and a 3d dynamically and helically buckled rod

In this talk I will show how a minimally coupled nonrelativistic quantum particle in 1d is isomorphic to a much heavier, vibrating, very thin Euler-Bernoulli rod in 3d, whose ratio of bending modulus to linear density is $(hbar/2m)^2$. Axial body forces and terminal twisting couples acting on the rod play the role of scalar and vector potentials, respectively, and within the semiclassical approximation, rod inextensibility plays the role of normalization. This isomorphism leads to some new insights, particularly following the direction from quantum to classical. Among these, orbital angular momentum quantized in units of $hbar/2$ emerges when the force and couple-free inextensible rod is formed into a ring, and the ring vibrates in a toroidal helix mode. Such a ring also extends the Byers-Yang theorem to a classical object. Additionally, when the rod's axial body force is periodic, due, for example, to an alternating pattern of charged and neutral monomer blocks in a stiff polyelectrolyte, the isomorphism yields a new classical analog of a 1d Bloch electron in a magnetic field. A Zak phase can potentially be realized in this system by adiabatically varying the rod's twist. I will end with a discussion of what the isomorphism has to say about mode selection and mode coarsening in dynamical buckling.