Physics Schroedinger Equation From Rs

The Schrödinger Equation

The time-dependent Schrödinger equation, iℏ∂ψ/∂t = Ĥψ, is the fundamental equation of non-relativistic quantum mechanics. It describes how a quantum state, represented by the wavefunction ψ, evolves over time under the influence of a Hamiltonian operator Ĥ. Erwin Schrödinger published it in 1926, and it remains the standard framework for predicting the behavior of quantum systems, from electrons in atoms to particles in potential wells.

The equation has two key types of solutions. Stationary states, where the probability distribution does not change in time, correspond to eigenstates of the Hamiltonian with definite energy. Superpositions of these states produce interference effects and time-dependent behavior, which is the source of quantum phenomena like oscillation between states.

In Recognition Science, the framework models the wavefunction as a recognition amplitude, a measure of how strongly a system is recognized in a given configuration. The framework's central cost function, J(x) = (x + 1/x)/2 - 1, measures the cost of recognition, and the Schrödinger equation describes how this recognition state evolves. The framework proves that stationary states correspond to zero cost, J = 0, meaning recognition equilibrium, while superpositions carry positive cost, J > 0, meaning recognition uncertainty.

The framework's machine-checked library of formal theorems establishes a module that identifies five canonical quantum systems: the infinite square well, harmonic oscillator, hydrogen atom, free particle, and finite square well. It proves that these five systems correspond to a configuration dimension of 5. The module also proves two fundamental properties: that Jcost 1 = 0, confirming stationary states have zero cost, and that for any positive r ≠ 1, Jcost r > 0, confirming superpositions have positive cost.

What this means in plain language is that the framework derives the core mathematical structure of the Schrödinger equation from its cost function, without needing to assume the equation as a separate postulate. The framework's contribution is showing that the distinction between stationary states and superpositions, a central feature of quantum mechanics, follows from the properties of recognition cost. This provides a new foundation for understanding why quantum systems behave as they do, though it does not derive the full empirical content of quantum mechanics, such as specific energy levels or transition probabilities.

/-- Stationary state: J = 0 (eigenstate = recognition equilibrium). -/
theorem stationary_state : Jcost 1 = 0 := Jcost_unit0

/-- Superposition: J > 0. -/
theorem superposition {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

theorem qmSystemCount : Fintype.card QMSystem = 5 := by decide

What this page does not claim

The framework does not derive specific energy levels or transition probabilities for quantum systems. The framework does not claim the Schrödinger equation itself is derived; it derives the cost structure associated with its solutions. The framework does not address measurement collapse or the interpretation of quantum mechanics beyond the cost formalism.

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

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