Quantum Phase Transition in a Potential Well With Charged Walls

We report a theoretical prediction of a quantum phase transition in a one‐dimensional quantum well with charged walls. The transition is driven by the well width, which acts as a quantum control parameter. For a single electron, we obtain exact analytical solutions via Jacobi polynomials, revealing quantized critical widths corresponding to zero‐energy states. The multielectron regime exhibits a rich phase diagram where Coulomb repulsion, exchange, and correlation interactions collectively determine the transition boundary. Statistical analysis reveals unconventional thermal behavior arising when the chemical potential coincides with the band edge. The predicted critical widths for semiconductor structures suggest experimental verifiability in GaAs, InAs, and CdTe quantum wells.