Physics Wave Packet Spreading From Jcost

Wave packet spreading

In quantum mechanics, a wave packet is a localized disturbance, like a pulse of light or a cluster of particles, that spreads out as it moves. For a free particle of mass m, the width σ(t) grows with time t according to σ(t) = σ₀√(1 + (ħt/(2mσ₀²))²), where σ₀ is the initial width and ħ is the reduced Planck constant. This spreading is not a flaw in the theory; it is a direct consequence of the uncertainty principle, which says you cannot pin down both a particle's position and its momentum precisely at the same time.

The standard result, derived by Erwin Schrödinger in 1926, shows that the width grows roughly linearly after a characteristic time scale. The spread is slow for heavy particles and fast for light ones: an electron spreads noticeably, while a macroscopic object barely moves. This behavior is a central feature of quantum mechanics, explaining why quantum effects fade for large masses and why particle beams widen over distance.

In Recognition Science, a framework that derives physical structure from the cost of recognition events, the same spreading phenomenon is modeled through a cost function. The framework defines a domain cost as J(m/e), where J(x) = (x + 1/x)/2 - 1, a function that measures the mismatch between a mass m and an energy e. The module proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and a threshold constant φ - 3/2 is positive, where φ is the golden ratio.

What the module does not do is derive the wave packet spreading formula itself. The Lean file, a machine-checked collection of formal theorems, proves only the general properties of the cost function. The connection to spreading is stated in a research note, not as a theorem. To make it a theorem about wave packets, one would need to define m and e in terms of the particle's mass and energy, which the module does not do. The spreading formula remains a structural idea, not a proved result.

The practical takeaway is this: the framework's cost function behaves sensibly for positive values, vanishing when its inputs match, but the leap from that to a specific prediction about wave packet spreading is not yet formalized. The module is a template, shared across many subjects, waiting for a subject-specific definition to turn it into a real physical statement.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

This module does not prove the wave packet spreading formula itself. This module does not define m and e in physical terms. The research note about spreading is not a theorem.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/WavePacketSpreadingFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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