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Effective Mass Renormalization from Energy Allocation: A Modified Schr¨odinger Equation for Non-Equilibrium Quantum Systems

Hongpu Yang

Published Apr 5, 2026• 0 citations
Publisher

Abstract

In standard quantum mechanics, the Schr¨odinger equation assumes a fixed inertial mass for the quantum particle. Here we introduce a dimensionless parameter η = E˙ resp/E˙ main that quantifies the internal energy allocation between response (exploration) and maintenance (localization) of a quantum system. We show that when η = 1, the effective inertia of the wavepacket is modified, leading to an effective mass renormalization m∗ = m/η. The modified free-particle Schr¨odinger equation becomes iℏ∂tψ = −(ℏ 2/(2m∗ ))∇2ψ, which preserves the Gaussian nature of the wavepacket while scaling the dispersion rate. For an initial Gaussian wavepacket of width σ0, the time evolution of its spatial variance is σ 2 (t) = σ0 2 + (ηℏt/(2mσ0))2 . When η = 1, standard quantum mechanics is recovered. The framework is extended to the harmonic oscillator, yielding energy levels En = (n + 1/2)ℏω0 √η (under a specific renormalization scheme). This provides a parameterized extension of quan tum mechanics that can be tested in non-equilibrium driven systems such as cold atoms in modulated optical lattices.

DOI
10.5281/zenodo.19426159

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In standard quantum mechanics, the Schr¨odinger equation assumes a fixed inertial mass for the quantum particle. Here we introduce a dimensionless parameter η = E˙ resp/E˙ main that quantifies the internal energy allocation between response (exploration) and maintenance (localization) of a quantum system. We show that when η = 1, the effective inertia of the wavepacket is modified, leading to an effective mass renormalization m∗ = m/η. The modified free-particle Schr¨odinger equation becomes iℏ∂tψ = −(ℏ 2/(2m∗ ))∇2ψ, which preserves the Gaussian nature of the wavepacket while scaling the dispersion rate. For an initial Gaussian wavepacket of width σ0, the time evolution of its spatial variance is σ 2 (t) = σ0 2 + (ηℏt/(2mσ0))2 . When η = 1, standard quantum mechanics is recovered. The framework is extended to the harmonic oscillator, yielding energy levels En = (n + 1/2)ℏω0 √η (under a specific renormalization scheme). This provides a parameterized extension of quan tum mechanics that can be tested in non-equilibrium driven systems such as cold atoms in modulated optical lattices.
AbstractWe demonstrate a formal equivalence between Gaussian Resonance R(x) and the squared quantum wave function |ψ(x)|², and derive a connection to the Schrödinger equation. The key step: replacing the real Gaussian parameter ε² with the imaginary quantity i transforms the Gaussian Resonance metric into the quantum probability ℏ amplitude. The time evolution of this amplitude under an Hamiltonian operator yields exactly the Schrödinger equation. Gaussian Resonance is therefore the real (dissipative) version of quantum mechanics; quantum mechanics is the complex (conservative) extension of Gaussian Resonance. Natural systems oscillate within a resonance neighborhood — not at a fixed point — and this oscillation is governed by the same principle at all scales, from fluid dynamics to quantum mechanics, with ε as the interpolating parameter between classical and quantum regimes.Central result: the substitution ε² i in the Gaussian Resonance → ℏ metric R(x) = exp(-f(x)²/2ε²) yields the quantum amplitude ψ(x) = exp(-f(x)²/2i ). The time evolution of ψ under a Hamiltonian H gives ℏ ℏ i ∂ψ/∂t = Hψ — the Schrödinger equation
It is proposed that the Schrodinger equation for a free point particle has non-linear corrections which depend on the mass of the particle. It is assumed that the corrections become extremely small when the mass is much smaller or much larger than a critical value (the critical value being related to but smaller than Planck mass). The corrections become significant when the mass is close to this critical value and could play a role in explaining wave-function collapse. It appears that such corrections are not ruled out by present day experimental tests of the Schrodinger equation. Corrections to the energy levels of a harmonic oscillator are calculated.
Modern quantum mechanics traditionally treats the internal frequencies of quantum systems as static parameters or time-independent eigenvalues, assuming they can be altered instantaneously without resistance. In this paper, we propose the principle of Quantum Frequency Inertia (QFI), establishing that internal transition and resonance frequencies possess intrinsic inertia and relaxation dynamics when subjected to rapid modulation. By formulating an extended Lagrangian and Hamiltonian framework where frequency acts as an independent dynamical coordinate, we derive the governing equation of motion incorporating an inertial coefficient and a characteristic relaxation time scale. Microscopic analysis confirms that this frequency retardation stems from non-Markovian environmental back-action and finite propagation speeds of mediating interactions. We analyze manifestations of QFI across diverse platforms—including atomic clocks, optical cavities, and superconducting quantum circuits—and outline testable predictions such as frequency response resonance peaks and underdamped transient ringing. This framework redefines frequency as an active dynamical degree of freedom, offering new insights into ultra-fast quantum control and metrological limits.
Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free particle (with or without spin-orbit interaction). For the simple harmonic oscillator, however, the fact that the wavepacket remains a Gaussian as time progresses has to be inferred on the basis of the skewness and kurtosis as opposed to a study of all the relevant moments.
In standard quantum mechanics, the visibility of double-slit interference is determined solely by coherence and decoherence. Here we introduce a dimensionless parameter η = Ė_resp / Ė_main that quantifies the internal energy allocation between response (exploration) and maintenance (localization) of a quantum system. We show that even in the absence of decoherence, the visibility becomes v(η) = 2√η/(1+η) when the system is prepared in a non-equilibrium steady state with η ≠ 1. The derivation is based on an effective mass renormalization m* = m/η that preserves the standard Schrödinger form while scaling the dispersion rate. A concrete experimental realization using ultracold atoms in an optical lattice with amplitude modulation is proposed, together with a full statistical protocol to falsify the prediction. This provides a clean, experimentally accessible test of energy-efficiency theory (EET) against standard quantum mechanics.