Volume 11, Issue 4, pages 687-712

Tamás Kovács, Semen Ivanov, Tamás Kovács Jr. and Michail Lazos

This paper proposes a radical reformulation of quantum mechanics in which the conventional time parameter is eliminated and replaced with energy-dependent evolution. Building on the Planck-Einstein relation \( f = E/h \) and the inverse relation \( f = 1/t \), we argue that time is not a fundamental dimension but an emergent quantity derived from energy. By introducing a modified Hamiltonian \( \widehat{H} = \frac{i\hbar}{2z} \frac{\partial}{\partial E} \) and a new wavefunction \( \Psi(x, E) = \psi(x) \exp(-i E^2 z / \hbar) \), we reformulate the Schrödinger equation in the energy domain, with \( z \in \mathbb{Z}^+ \) as a discrete evolution parameter. This approach yields a framework where energy, not time, governs dynamical evolution. We demonstrate that normalization is preserved across energy evolution, and expectation values remain constant with respect to \( z \), recovering all stationary state properties without invoking temporal arguments. Canonical examples—including the infinite square well and the quantum harmonic oscillator—are solved in this modified framework, showing complete compatibility with traditional results. In the case of the harmonic oscillator, we derive ladder operators adapted to the energy-based formalism and construct the full spectrum of stationary states without time or angular frequency. Our findings offer a new lens through which the dynamics of quantum systems can be understood, suggesting a shift from time-based to energy-based evolution. This may provide deeper insights into the structure of quantum theory, the discreteness of physical processes, and the unification of dynamics across fundamental interactions.

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