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.
Hongpu Yang (Sun,) studied this question.