Energy Quantization

Jorge Henrique de Oliveira Sales, Suzuki, A. T., D. S. Bonafe

We present in this work a pedagogical way of quantizing the atomic orbit for the hydrogen's atom model proposed by Bohr without using his hypothesis of angular momentum quantization. In contrast to the usual treatment for the orbital quantization, we show that using energy conservation, correspondence principle and Plank's energy quantization Bohr's hypothesis can be deduced from and is a consequence of the Planck's energy quantization.

J. Michael Hollas

S. Venkatesan

This is the first publication/preprint of discovery of "Energy Quantization" effects in classical subatomic particle simulations. This study discovers the Quantum effects were already enclosed in classical mechanics. This discovery was made in the year 2026.

Dhruba J. Biswas

Ciann‐Dong Yang, Chung-Hsuan Kuo

Probability interpretation is the cornerstone of standard quantum mechanics. To ensure the validity of the probability interpretation, wavefunctions have to satisfy the square-integrable (SI) condition, which gives rise to the well-known phenomenon of energy quantization in confined quantum systems. On the other hand, nonsquare-integrable (NSI) solutions to the Schrödinger equation are usually ruled out and have long been believed to be irrelevant to energy quantization. This paper proposes a quantum-trajectory approach to energy quantization by releasing the SI condition and considering both SI and NSI solutions to the Schrödinger equation. Contrary to our common belief, we find that both SI and NSI wavefunctions contribute to energy quantization. SI wavefunctions help to locate the bifurcation points at which energy has a step jump, while NSI wavefunctions form the flat parts of the stair-like distribution of the quantized energies. The consideration of NSI wavefunctions furthermore reveals a new quantum phenomenon regarding the synchronicity between the energy quantization process and the center-saddle bifurcation process.