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Chapter 56
Schrödinger Flow Equation

Quantum Coherence and Boundary Systems
Author: Mark Anthony Higgins
© Mark Anthony Higgins 11 October 2025
Revised Date: 23 February 2026
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Flow Algebra 
flow "Schrödinger Flow Equation" v1.0

# ---------- PURPOSE ----------
# Models the Schrödinger equation as a coherence-preserving flow in complex phase space.
# The equation governs time evolution of a wavefunction ψ under a potential V(x,t):
# iħ ∂ψ/∂t = [ - (ħ² / 2m) ∇² + V(x,t) ] ψ
# In the coherence framework, this expresses local conservation of phase curvature — the continuous evolution of boundary memory.
# IB tracks coherence integrity of the wavefunction (1 = perfectly coherent, <1 = partial decoherence or environmental coupling).

# ---------- PARAMETERS ----------
params:
  units: "SI"
  ħ: real = 1.054_571_817e-34       # reduced Planck constant [J·s]
  m: positive                        # particle mass [kg]
  V: func?(x:real,t:real)->real      # potential energy function [J]
  IB: 0.0..1.0 = 1.0                 # wavefunction coherence index
  λ_decoh: nonnegative = 0.0         # decoherence rate [s^-1]
  dt: positive?                      # timestep [s]
  include_probability_flow: bool = true

# ---------- STATE ----------
state Quantum:
  ψ: complex                         # wavefunction ψ(x,t)
  ρ: real                            # probability density |ψ|²
  J: real                            # probability current density [A/m²]
  Hψ: complex                        # Hamiltonian operator applied to ψ
  phase: real                        # instantaneous phase arg(ψ)
  IB_rate: real                      # rate of coherence change
  norm: real                         # normalisation ∫|ψ|² dx
  energy_expectation: real           # ⟨E⟩ = ⟨ψ|H|ψ⟩
  regime: str                        # "stationary", "dynamic", or "decohering"

# ---------- INVARIANTS ----------
invariants:

  # Schrödinger time evolution (differential form)
  Quantum.Hψ = -(params.ħ**2 / (2 * params.m)) * ∇²(Quantum.ψ) + params.V(x,t) * Quantum.ψ

  iħ ∂ψ/∂t = Quantum.Hψ
  Quantum.ψ(t+dt) = Quantum.ψ(t) + (-i/params.ħ) * Quantum.Hψ * (params.dt or 1)

  # Probability density and current (continuity equation)
  Quantum.ρ = |Quantum.ψ|^2
  if params.include_probability_flow:
    Quantum.J = (params.ħ / params.m) * Im(conjugate(Quantum.ψ) * ∇Quantum.ψ)

  # Continuity relation (local conservation)
  ∂Quantum.ρ/∂t + ∇⋅Quantum.J = 0

  # Normalisation check
  Quantum.norm = ∫Quantum.ρ dx

  # Expected energy
  Quantum.energy_expectation = Re(∫conjugate(Quantum.ψ) * Quantum.Hψ dx)

  # Coherence decay (optional)
  if params.λ_decoh > 0 and params.dt is defined:
    params.IB = clamp(params.IB - params.λ_decoh * (1 - params.IB) * params.dt, 0, 1)
    Quantum.IB_rate = -params.λ_decoh * (1 - params.IB)

  # Regime classification
  if params.IB == 1:
    Quantum.regime = "stationary"
  elif params.IB > 0.5:
    Quantum.regime = "dynamic"
  else:
    Quantum.regime = "decohering"

# ---------- TRANSFORMS ----------
transforms:

  evolve_time_step:
    comment: "Advance ψ(x,t) one timestep via unitary evolution operator."
    Quantum.ψ = exp(-i * H * (params.dt or 1) / params.ħ) * Quantum.ψ

  apply_potential:
    comment: "Update ψ(x,t) under a new potential landscape."
    Quantum.Hψ = -(params.ħ**2 / (2 * params.m)) * ∇²(Quantum.ψ) + params.V(x,t) * Quantum.ψ

  normalise_wavefunction:
    comment: "Renormalise ψ to ensure total probability = 1."
    Quantum.ψ = Quantum.ψ / sqrt(∫|Quantum.ψ|² dx)
    Quantum.norm = 1

  decay_coherence:
    comment: "Apply environmental decoherence to wavefunction coherence index."
    params.IB = clamp(params.IB - 0.01 * (1 - params.IB), 0, 1)

# ---------- MONITORS ----------
monitors:
  probability_conservation:
    assert abs(d/dt(Quantum.norm)) <= 1e-9
    comment: "Total probability conserved under unitary evolution."
  energy_consistency:
    assert Quantum.energy_expectation >= 0
  coherence_bounds:
    assert 0 <= params.IB <= 1
  phase_tracking:
    comment: "Track continuous phase evolution of ψ across time."

# ---------- SCENES ----------
scenes:

  free_particle:
    given params:
      m: 9.11e-31
      V: (x,t)->0
      IB: 1.0
    expect:
      Quantum.regime = "stationary"
      comment: "Free evolution of plane wave — uniform phase velocity."

  harmonic_oscillator_ground_state:
    given params:
      m: 9.11e-31
      V: (x,t)->0.5 * m * (ω₀^2) * x^2
      IB: 1.0
    expect:
      Quantum.energy_expectation = 0.5 * ħ * ω₀
      comment: "Stationary Gaussian wavefunction — minimal phase curvature."

  wavepacket_dispersion:
    given params:
      m: 9.11e-31
      V: (x,t)->0
      IB: 1.0
    expect:
      Quantum.regime = "dynamic"
      comment: "Wavepacket spreads over time — curvature unfolds coherently."

  decohering_wavefunction:
    given params:
      m: 9.11e-31
      V: (x,t)->0
      IB: 0.7
      λ_decoh: 0.05
      dt: 1.0
    enable:
      - decay_coherence
    expect:
      Quantum.regime = "decohering"
      comment: "Environmental coupling diffuses phase — probability conserved but coherence fades."

# ---------- NOTES ----------
notes:
  - **Schrödinger Equation:**  
    \[
    i\hbar\frac{\partial\psi}{\partial t}
    = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(x,t)\right]\psi
    \]
    Describes the continuous evolution of the wavefunction ψ in time.

  - **Probability Continuity:**  
    \[
    \frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{J} = 0, \quad
    \rho = |\psi|^2, \quad
    \mathbf{J} = \frac{\hbar}{m}\Im(\psi^*\nabla\psi)
    \]
    Ensures conservation of total probability across all space.

  - **Energy Expectation:**  
    \[
    \langle E \rangle = \int \psi^*(x) H \psi(x)\,dx
    \]

  - **Coherence Law:**  
    \[
    \dot{I_B} = -\lambda (1 - I_B)
    \]
    Models gradual decoherence from environmental interaction.

  - **Interpretation (HST–Flow):**  
    The Schrödinger equation is not just a temporal rule — it is **phase continuity in curvature form**.  
    The wavefunction ψ describes a field of recursive curvature coherence, with time evolution preserving the total boundary memory (probability).  
    When \(I_B = 1\), the flow is unitary — a perfect coherence loop.  
    When \(I_B < 1\), the loop opens — phase information leaks outward, and interference visibility decays.

  - **Visualization:**  
    - Left: ψ(x,t) as a complex plane wavefront (pure state).  
    - Middle: ψ evolving under potential (curvature modulation).  
    - Right: ψ amplitude fading with phase blur as IB decreases (decoherence).
Overview

The Schrödinger equation is not just the engine of quantum mechanics — it is the grammar of coherence through time.
It describes how phase information flows continuously across a boundary without loss, defining the dynamic equilibrium between curvature, potential, and probability.
In the HST view, it expresses recursive phase continuity — the way reality remembers its own curvature through the wavefunction.

🔹 1. Classical Form — Time Evolution of the Wavefunction

The time-dependent Schrödinger equation governs how a system’s state evolves:

\[ i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(x,t) \right] \psi \]

It ensures that the probability density ρ=∣ψ∣2\rho = |\psi|^2 remains constant across time, preserving total normalisation.
This is captured by the continuity equation:

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0, \qquad \mathbf{J} = \frac{\hbar}{m} \Im(\psi^* \nabla \psi) \]

where J\mathbf{J} is the probability current — the flux of coherence through configuration space.

Every solution to the Schrödinger equation represents a stable coherence flow — a pattern of phase curvature sustained by balance between potential energy and wave dispersion.

🔸 2. The Coherence Interpretation — Curvature as Memory

In the HST–Flow framework, the wavefunction ψ\psi is not a mathematical abstraction but a field of curvature memory.
Its amplitude represents spatial distribution of coherence, and its phase represents the local curvature orientation of that coherence in time.

The term −ℏ22m∇2ψ-\frac{\hbar^2}{2m}\nabla^2 \psi describes how curvature diffuses spatially, while V(x,t)ψV(x,t)\psi defines how the environment folds that curvature locally.
The imaginary unit ii introduces rotation in the phase plane — a continual reorientation of curvature memory through time.

Thus, Schrödinger evolution is the rotation of coherence in complex space, where amplitude and phase exchange dynamically without loss.

🔹 3. Probability Flow — The Conservation of Curvature

The continuity equation is not merely statistical; it expresses the conservation of total curvature memory.
The “probability current” J\mathbf{J} is a flux of coherence, transporting information across the field while maintaining global balance.

When the boundary is closed (no external interaction), this flow is perfectly unitary — coherence recirculates within the system, and IB=1I_B = 1.
When the system couples to an environment, boundary information leaks outward, lowering IBI_B.
This is decoherence — open-boundary curvature diffusion rather than collapse.

🔸 4. Stationary and Dynamic States — Internal vs. External Balance

A stationary state satisfies:

\[ H\psi = E\psi \]

and evolves only through global phase rotation:

\[ \psi(x,t) = \psi(x,0)e^{-iEt/\hbar} \]

Its curvature field is static — the internal boundary is in perfect self-equilibrium.

By contrast, a non-stationary wavepacket evolves dynamically.
Its curvature unfolds in time, redistributing coherence while still obeying the conservation of total probability.
This dynamic interplay between dispersion and potential defines temporal curvature modulation — the way reality maintains stability while constantly transforming itself.

🔹 5. Decoherence — Opening the Loop

No real system is perfectly isolated.
When environmental interaction introduces phase noise or entanglement with unobserved degrees of freedom, coherence gradually decays

\[ \dot{I_B} = -\lambda (1 - I_B) \]

The wavefunction no longer evolves purely unitarily — its internal curvature begins to blur.
Amplitude remains conserved (probability continuity still holds), but the sharpness of phase relation between points decreases.
The once-precise interference pattern becomes diffused, marking a transition from deterministic evolution to mixed behaviour.

In HST terms, this is the boundary opening its loop — curvature spreading into adjacent domains.
Decoherence is not destruction, only redistribution.

🔸 6. Unified Coherence Framework — The Boundary as Phase Engine

At its deepest level, the Schrödinger equation defines the flow of coherence between boundaries of curvature:

  • The kinetic term diffuses curvature spatially (∇2\nabla^2).
  • The potential term focuses curvature locally (VV).
  • The phase term (iℏ∂ti\hbar \partial_t) rotates coherence in time.

Together they form a balanced recursive loop: energy, curvature, and phase continually transform while preserving global integrity.
In HST language, this loop is the phase-space heartbeat of reality — the smallest dynamical process that can remember itself.

🧠 Summary

The Schrödinger equation is a law of phase continuity, not probability mysticism.
It ensures that curvature memory (ψ) evolves coherently through time, preserving the total boundary condition of existence.

In the HST–Flow interpretation:

  • ψ is a coherence field — a record of curvature memory.
  • iħ∂ψ/∂t = Hψ ensures recursive, phase-aligned evolution.
  • IB tracks coherence integrity (1 = unitary, <1 = open).
  • Decoherence expands the boundary — phase diffuses, but total energy remains conserved.

Thus, the Schrödinger equation describes the time flow of coherence —
a self-consistent recursion where the universe continually rewrites itself without ever breaking its own symmetry.

The wavefunction does not predict the future — it preserves the memory of curvature as time unfolds.

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