Reinterpretation of the Hydrogen Ground-State Radius from Phase Criticality

In quantum mechanics, quantization conditions have traditionally been introduced as postulates, such as the Bohr condition, without a clear physical necessity. In this study, we reinterpret the quantization condition using a dimensionless phase criticality index defined as α = rω/c. Within this framework, the stability of the electron in the hydrogen atom is determined by the simultaneous satisfaction of phase self-consistency and the balance of Coulomb interaction. Without invoking the de Broglie standing-wave assumption, we show that the ground-state radius naturally emerges as r = ℏ/(mcα), corresponding to the Bohr radius. This result suggests that the quantized structure of the hydrogen atom can be understood as a numerical consequence of electromagnetic interaction and phase criticality, rather than an independent quantization postulate.