Action Quantization, Energy Quantization, and Time Parametrization

Erol Karazincir

This paper argues that the quantization of energy and the relationship E = hf are not fundamental axioms of quantum mechanics but structural consequences of two postulates: that time is discrete at the Planck scale, and that each discrete temporal unit carries equal action. Planck's constant is identified as the action content of one Planck-time tick, and the Nyquist-Shannon sampling theorem is applied to derive the maximum physical frequency. The paper further argues that wave-particle duality dissolves under this framework — a photon is always a wave, and point-like detection events are what quantized energy transfer looks like at the receiving end, not evidence for particle ontology. The double-slit experiment, quantum eraser, and interaction-free measurements are re-examined and shown to require no paradoxical interpretation when particle language is abandoned. The paper is a conceptual contribution intended to motivate formal development. Prior work by Caldirola on the chronon is discussed and distinguished.

Dhruba J. Biswas

S. Nasiri, Samira Bahrami

Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.

Rubén Santamaría

Richard W. Robinett

Abstract Using both WKB approximation methods and complete eigenstate solutions, we examine four standard, textbook level, bound state problems in quantum mechanics. We show how the action integral from the Bohr–Sommerfeld quantization condition, extended to the WKB energy quantization condition, can be expressed in the language of semi-classical and exact quantum results. Relating classical time averaging and quantum expectation value methods, the action integral can be used to connect the average value of speed squared times period across such systems giving ⟨ v 2 ⟩ n τ n = ( n + C ~ ) ( h / m ) , where n is the bound state quantum number ( n = 0 , 1 , 2 , . . . ) supplemented by the Maslov correction ( C ~ ) . We then extend this analysis using exact results for an infinite hierarchy of potentials, one derived from the repeated supersymmetrization of the familiar infinite well, as a more mathematically sophisticated example. These discussions provide a novel pedagogical twist on h as the ‘quantum of action’, accessible at several levels of the undergraduate quantum mechanics curriculum.