Non-Orientable Time and the Schrödinger Equation

Roland Riek

A physical experiment comprises along the time trajectory a start, a time evolution (duration), and an end, which is the measurement. In non relativistic quantum mechanics the start of the experiment is defined by the wave function at time 0 taking into account the starting conditions, the evolution is described by the wave function following the Schrödinger equation and the measurement by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that makes quantum mechanics statistical with all its consequences. The nature of the Born rule is thereby unknown albeit necessary since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is necessary in order to eliminate the unknown initial and end time coordinate of the experiment in reference to the beginning of the universe.

Bodo Warteenberg

The Schrödinger equation iℏ · ∂ψ/∂τ = Ĥψ contains three elements that in standard quantummechanics are not derived but postulated: the imaginary unit i, the reduced Planck actionquantum ℏ, and the Hamiltonian operator Ĥ. This supplement shows that all three follow fromthe fossil causal structure. The phase boundary surface Σ = {x ∈ M | t_m(x) · t_C(x) / t_P² = -1}delivers i through the sign change of the metric time. The action quantum ℏ follows as theminimal action of the closed oriented rotation on Σ. The Hamiltonian Ĥ is directly the fossilenergy density operator ρ_hist. The Schrödinger equation is not a postulate - it is a theorem ofthe fossil causal structure.

Rudolf Zahradník, R. Polák

Roland Riek

Abstract In terms of its time trajectory, a physical experiment is comprised of a preparation time, a start time, a time evolution (duration), and an end time (point at which the measurement actually occurs). In non-relativistic quantum mechanics the start of the experiment is described by the wave function at time t = 0 taking into account the starting conditions, the evolution is described by the wave function according to the Schrödinger equation and the measurement is described by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that renders quantum mechanics stochastic with all its consequences it entails. Still, the nature of the Born rule is unknown yet it is useful since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is therefore only necessary in order to account for the unknown preparation time and thus starting time of the experiment because only the experimental duration (time between start time and end time) can be known.

Dongmei Wei, Junxiang Wang, Hanxiu Xu +2 more

The Time-Fractional Schrödinger Equation (TFSE) is an effective tool for simulating the dynamics of non-Markovian quantum systems. The Quantum Speed Limit (QSL) time characterizes the minimum time required for the evolution of a non-Markovian quantum system. In this paper, Wei's TFSE is employed to simulate the non-Markovian quantum accelerated evolution process in the Resonant Dissipative Jaynes-Cummings (RDJC) model. By solving the QSL time of a time-fractional single-qubit open system, the enhancement mechanism of the system evolution speed induced by the non-Markovian memory effects of the environment is revealed. Further studies show that the optimized acceleration of the system evolution can be achieved by jointly regulating the fractional order, coupling strength, and photon number. Comparative analyses indicate that Wei's TFSE can accurately capture the non-Markovian accelerated dynamical features of the system over the entire fractional order range, whereas Naber's TFSE is applicable only within a limited fractional order interval. In addition, the comparisons of the average simulation time for calculating the dynamical trajectory of the excited-state probability demonstrate that Wei's TFSE has a significant simulation advantage in computational efficiency. Therefore, Wei's TFSE is more accurate and efficient for simulating the accelerated dynamics of non-Markovian quantum systems.