Spin–orbit coupled particles in an infinite helical box

Calculating the electronic states of a quantum mechanical particle in a potential well with infinitely high walls is probably the most readily accessible, still analytically solvable textbook example of quantisation via spatial confinement. As such, it is an ideal starting point for including additional terms to the Hamiltonian in order to adapt the model to a specific experimental setting, e.g. by introducing multiple particles or external fields. Here we present an extension that assumes an externally given spatial modulation of the potential well with helical characteristics as boundary condition. We derive analytical solutions for quantum-mechanical spin $1/2$ -particles in a helical spin–orbit-split environment, which is the most simple approach the inital situation encountered in investigations of the chirality-induced spin selectivity effect. We show that the model extension steps lead to a Rashba-type splitting of the spin eigenstates with perfect time-reversal symmetry between right- and left-handed helices, which indicates that additional external perturbations of the system are required to model the CISS effect.