A Study of Phase Transitions in a Quantized “Bead on a Rotating Hoop” Model

The "bead on a rotating hoop" system is a classical mechanical model that exhibits a bifurcation in its equilibrium point when the angular frequency of the hoop exceeds a critical threshold. This bifurcation resembles a second-order phase transition in thermodynamics, though it manifests in the system’s effective potential rather than in free energy. Since this transition occurs in the absence of thermal energy, a quantized version of the system may exhibit a quantum phase transition, alongside a possible thermal phase transition. In this study, we quantize the mechanical model using canonical quantization. To facilitate analysis, we express the nonlinear terms in the Hamiltonian as a power series, recasting the system as a perturbed harmonic oscillator. Perturbation theory is then applied to determine the corrected energy levels, which are subsequently used to evaluate the partition function and free energy. By analyzing the free energy, we investigate the presence of both quantum and thermal phase transitions in the system.