Schrödinger evolution in Measurement Quantization

A conditional continuum representation of discrete count redistribution. MQ posits local count redistribution on relational adjacency links with a balance law in the Internal Frame. This is not a physical spatial lattice. Given spatial realization and encoding data through the Frames mapping, a realized availability density rho subscript S is represented in the System Frame. The density is normalized over its specified domain to probability density rho subscript P, assuming a positive finite total availability.

Amplitude and phase

Psi subscript MQ is the normalized encoded complex wave function. r is realized spatial position in metres and t is physical time in seconds, not a primitive update index. Phi is a real dimensionless phase, and i is the imaginary unit. The phase is additional encoding data; density alone does not determine it. Phase is undefined at a zero amplitude, while the complex wave function remains the nonsingular variable.

ψ_{MQ} (r, t) = \sqrt{ρ_{P} (r, t)} e^{i φ (r, t)}

{| ψ_{MQ} (r, t) |}^{2} = ρ_{P} (r, t)

MQ0527, MQ0575, MQ0576 and MQ0601 supply the amplitude-phase and modulus relations. These encode a specified state rather than independently determine its phase or primitive realization.

Specified spatial operator

H subscript d is the discrete spatial Hamiltonian, Delta subscript d the dimensionless unscaled spatial second-difference sum for the selected discretization, Delta x its realized spatial grid spacing in metres, m a positive realized subsystem mass in kilograms, and V the potential energy in joules. The Hamiltonian has energy units. The diagram does not identify primitive relational adjacency automatically with Euclidean nearest-neighbor spatial adjacency.

H_{d} = - \frac{ℏ^{2}}{2 m} \frac{Δ_{d}}{{(Δ x)}^{2}} + V

MQ0569, MQ0571, MQ0579 and MQ0588 record the discrete operator and the stated continuum limit. The metric, weights, spatial encoding, kinetic form and boundary conditions require specification. A refinement limit of a realized representation is not a literal subdivision of an indivisible primitive fundamental count. Conservation by itself does not select this Laplacian or its kinetic coefficient.

Consistent upper-count action convention

Every displayed action factor in this graphic is upright hbar, the MQ upper-count action scale, in joule seconds. It is used consistently in the discrete operator and continuum evolution. l subscript f, m subscript f and t subscript f are the realized fundamental length, mass and time in metres, kilograms and seconds. n subscript M is the dimensionless realized mass count.

ℏ = \frac{{l_{f}}^{2} m_{f}}{t_{f}}

m = n_{M} m_{f}

MQ2182 and MQKB-EQ-1728 distinguish this upper-count action identity from the conventional exact SI reduced Planck constant. MQKB-EQ-1751 supplies the action/mass construction; the mass-count label is standardized from its source-local lowercase m to project notation uppercase M. The quadratic nonrelativistic kinetic dispersion is a physical premise, not a consequence of scalar count conservation. No numerical action value, independent experimental calibration or equivalence with the conventional SI action has been asserted.

Schrödinger representation

i ℏ \frac{\partial ψ_{MQ} (r, t)}{\partial t} = [- \frac{ℏ^{2}}{2 m} \nabla^{2} + V (r, t)] ψ_{MQ} (r, t)

MQ0312 records the continuum Schrödinger form. Here the same upper-count action convention is retained throughout this model illustration. The spatial Laplacian has units inverse square metres, so both sides have energy times wave-function units. The time derivative concerns realized physical time. This is a nonrelativistic scalar-state representation under its supplied potential, mass, geometry, phase and realization premises.

For a time-independent self-adjoint Hamiltonian on the stated dense domain, its strongly continuous unitary evolution preserves the norm. A time-dependent potential requires the appropriate common-domain, regularity and unitary-propagator hypotheses; formal symmetry of the differential expression alone is insufficient. CLM1085 retains representability and boundary conditions. CLM5263 explicitly distinguishes an assumed closed-System-Frame unitary group and its generator from primitive scalar adjacency balance. A Stone-type generator result does not derive the required group, geometry or specific Hamiltonian from counts alone.

The illustrations are conceptual rather than measured availability fields or a numerical solution of the displayed equation. The red/blue legend illustrates real and imaginary components only. Source authority is MQKB523.zip. The graphic summarizes a source-consistent conditional construction and does not independently validate every microscopic emergence premise.

Schrödinger Evolution

From Discrete Count Redistribution to Quantum Time Evolution

The Schrödinger equation gives the time evolution of a nonrelativistic quantum state. In the conventional formulation it is part of the theory’s dynamics. Measurement Quantization (MQ) asks how that equation can arise from discrete relational counts. See Schrödinger’s fourth communication for the original wave-mechanical treatment.

MQ starts from one axiom, “measure is discrete.” From that starting point it develops logical count relations and uses a right-triangle construction to geometrically anchor an expanding frame of information in its cosmological description. The Internal Frame is the discrete domain of relational count configurations. The Frames mapping relates those configurations to the System Frame, the realized description in which continuous relational separations and observable quantities are represented. Physical reference measures enter through realization. The cosmological construction and the one-body continuum limit discussed here are distinct applications of that architecture.

The four inputs to MQ’s fundamental-measure and constant reconstruction are the fine-structure constant, the speed of light, the electron mass, and the Bohr radius. The fine-structure constant selects the branch for our universe by calibrating the dimensionless radial coefficient $θ_{s i}$ . The speed of light anchors the realized length–time relation, with its exact SI numerical value set by definition. The electron mass and Bohr radius participate in the cross-dimensional normalization. The four roles concern the selected universe and its physical normalization, and the inputs can be related. In the cosmological construction the cosmic microwave background temperature separately selects an epoch through the adopted age prescription. The law-form claim concerns the logical construction, while particular derivations retain their mapping, normalization, action, and domain conditions.

Interpretive boundary

The Schrödinger reconstruction uses closed-domain conservation, admissible adjacency-local redistribution, a continuous phase encoding, and a controlled nonrelativistic limit. Discreteness is the starting axiom. The reconstruction also depends on the conditions that specify this physical realization.

For the background, see Establishing the Discreteness of Measure, Fundamental Measures, and Determinism and the Foundations of Quantum Behavior.

The State Before the Wave Function

In the MQ model, a relational element $ξ$ in an admissible context $A$ carries a nonnegative integer count $n_{I} (ξ, A)$ . Counts add across disjoint subregions. Local redistribution preserves the total in a closed domain with zero net boundary flux. This gives a count description before any probability amplitude is introduced.

In the one-body realization, let $r$ denote realized position and $t$ denote System Frame time. The Frames mapping admits a continuous availability density $ρ_{S} (r, t)$ in this realization. Dividing by its finite, positive integral $N_{1} (t)$ gives the normalized density $ρ_{P} (r, t) = \frac{ρ_{S} (r, t)}{N_{1} (t)}$ .

Let $φ (r, t)$ be the dimensionless realized phase and $i$ the imaginary unit. The MQ wave function $ψ_{MQ} (r, t)$ encodes the normalized density and the phase together.

ψ_{MQ} = \sqrt{ρ_{P} (r, t)} e^{i φ (r, t)}

The squared modulus is ${| ψ_{MQ} (r, t) |}^{2} = ρ_{P} (r, t)$ . MQ obtains this density by normalizing realized availability before encoding it as a wave function. The source’s minimal continuous phase encoding uses reversible, linear, norm-preserving rotations. Under those assumptions it gives the complex unit phase, with the choices of global phase and phase orientation left open.

From Adjacency to the Laplacian

On a regular realized lattice, composing the forward and backward first differences along each direction gives a centered second difference. Summing these differences over the three realized spatial directions with equal weights gives the unscaled discrete Laplacian $Δ_{d}$ . For smooth realized fields, dividing by the square of the realized lattice spacing $Δ x$ gives the discretization that converges to the continuum Laplacian $\nabla^{2}$ . The lattice coordinates, derivatives, and continuous time belong to the System Frame description.

Let $m$ be the positive realized subsystem mass, $ħ$ the reduced Planck action scale used in the upper-count calibration, and $V$ the real multiplicative potential in the admissible conservative realization. The mapped lattice Hamiltonian $H_{d}$ has the form $H_{d} = - \frac{ħ^{2}}{2 m} \frac{Δ_{d}}{{(Δ x)}^{2}} + V$ . A Hermitian lattice Hamiltonian gives a norm-preserving Cayley time step. The simple forward step generally fails to preserve the norm exactly.

Why the Kinetic Coefficient Is Not Inserted by Hand

Let $l_{f}$ , $t_{f}$ , and $m_{f}$ denote the fundamental measures of length, time, and mass. At the upper count limit, the action scale is $ħ = \frac{{l_{f}}^{2} m_{f}}{t_{f}}$ . The realized mass count $n_{m}$ sets the subsystem mass through $m = n_{m} m_{f}$ . The source uses this mass-count notation for the quantum subsystem.

In the source construction, the mapped local metric and causal invariants select a Lorentz/Poincaré kinematics under the stated realization assumptions. Its nonrelativistic contraction supplies the Galilean boost law. With the phase-action and Hamilton–Jacobi characteristic relations, the boost law fixes the free inertial Hessian. In the rest frame with the free energy zero chosen at rest, the kinetic energy $T$ is $T = \frac{p^{2}}{2 m}$ , where $p$ is the momentum magnitude.

Writing $\nabla$ for the spatial gradient, the corresponding momentum operator is $\hat{p} = - i ħ \nabla$ . Applying it in the quadratic kinetic expression gives $- \frac{ħ^{2}}{2 m} \nabla^{2}$ .

Why the coefficient is constrained

The action calibration, realized subsystem mass, and Galilean inertial relation determine the coefficient $\frac{ħ^{2}}{2 m}$ . The controlled limit of the centered differences gives the spatial operator that it multiplies.

See Fundamental Measures and Newton & Planck Constants for the measure and constant relations.

Schrödinger Evolution as a System Frame Realization

In the admissible nonrelativistic continuum description, the realized Hamiltonian ${\hat{H}}_{MQ}$ consists of the kinetic operator and the potential energy $V (r, t)$ . With the upper-count action notation used above, the explicit evolution law is

i ħ \frac{\partial ψ_{MQ}}{\partial t} = [- \frac{ħ^{2}}{2 m} \nabla^{2} + V] ψ_{MQ}

Equivalently, $i ħ \frac{\partial ψ_{MQ}}{\partial t} = {\hat{H}}_{MQ} ψ_{MQ}$ with ${\hat{H}}_{MQ} = - \frac{ħ^{2}}{2 m} \nabla^{2} + V (r, t)$ . The equation describes the realized evolution of the encoded state.

For the whole-space one-body realization, the volume element is $d^{3} r$ . With the appropriate boundary behavior and self-adjoint generator, the encoded state preserves its norm.

\frac{d}{d t} \int_{ℝ^{3}} {| ψ_{MQ} (r, t) |}^{2} d^{3} r = 0

For a time-independent self-adjoint Hamiltonian, the strongly continuous unitary group has time-evolution operator $U (t) = \exp (- \frac{i {\hat{H}}_{MQ} t}{ħ})$ . Stone’s theorem applies once the self-adjoint generator is specified.

The Realization-Appropriate Reduced Planck Scale

MQ distinguishes the upper-count action calibration from the finite-count calibration at the Fine Structure demarcation. Let $n_{L}$ denote the realized length count and $Q_{L} (n_{L})$ the corresponding length realization term. In the dimensional action relation the calibrated momentum realization of the radial coefficient $θ_{s i}$ is written $p_{s i}$ . Its units are momentum. Using that dimensional realization explicitly, the count-resolved action scale is

ℏ (n_{L}) = \frac{p_{s i} l_{f}}{Q_{L} (n_{L}) n_{L}}

Here $Q_{L} (n_{L}) = \sqrt{1 + {n_{L}}^{2}} - n_{L}$ . At the Fine Structure demarcation, the finite-count expression gives MQ’s electromagnetic action calibration. As the length count tends to infinity, the product $Q_{L} (n_{L}) n_{L}$ tends to one half, giving the upper-count action scale $ħ$ . The italic action symbol in the finite-count expression distinguishes it from the upright upper-count value.

The quantum source also writes the same evolution law in the electromagnetic realization, using the finite-count action value consistently in both the time derivative coefficient and the kinetic term. MQ associates that calibration with ordinary electromagnetic quantum measurements. The source proposes gravitationally bound quantum states as a test of whether the action scale differs across realizations. Its proposed transition-frequency displacement is conditional on the action scale applicable to the gravitational preparation. Observation of gravitational quantum states establishes the test setting. See Nesvizhevsky and colleagues’ neutron experiment.

What MQ Derives and What It Conditions

The reconstruction specifies how the normalized density, phase encoding, discrete spatial differences, action calibration, realized mass, and inertial kinematics enter the continuum generator. It requires regular local lattice representability, sufficient smoothness for the difference limit, isotropic summation, suitable boundary behavior, an admissible self-adjoint domain, and a real conservative multiplicative potential. Conservation fixes the closed-domain total. The admissible update and its continuum realization supply the additional structure needed to obtain the evolution law.

Evolution and Definite Measurement Records

In MQ, Schrödinger evolution describes the encoded state during closed evolution between completed records. For the ideal restriction model, a completed record identifies an admissible outcome domain of positive weight, and the realized state is restricted and renormalized there. The source treats this as a change of the realized description while keeping the underlying count law. This conditioning rule specifies the update for the stated context. A physical detector implementation requires its own coupling and record specification.

The wave function carries the phase and amplitudes used to describe interference, momentum and expectation values in the realized frame. Deterministic redistribution is part of the proposed count model, and probabilities describe normalized realized availability and records. See Determinism and the Foundations of Quantum Behavior and the MQ preprint Measurement Quantization: A Count-Based Resolution of the Quantum Measurement Problem.

From Count Evolution to Quantum Evolution

Under the stated conditions, local redistribution on the discrete count domain gives a realized density, a phase encoding, and a spatial generator. The realized mass, action scale and free inertial relation fix the kinetic coefficient. Schrödinger evolution is the resulting nonrelativistic System Frame description.

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