# ⚛  L1 Principle — Time-Independent Schrödinger Equation

**ID:** `L1-312` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *Stationary quantum eigenproblem*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** wavefunctions and eigenvalues
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554079

---

## 🧠 1. Introduction

**Time-Independent Schrödinger Equation** is a **linear inverse problem** whose unknown lives in **wavefunctions and eigenvalues** space, within the **Stationary quantum eigenproblem** sub-domain of **Quantum Mechanics**.

Measurements consist of electrons collected by an electron detector via a **spectroscopic transition** sensing mechanism.

The forward operator applies, in order: E · hamiltonian operator; computes eigen-pairs of a linear operator; O · spectra wavefunctions operator.

Observations are corrupted by additive Gaussian noise. Eigenvalues real under self-adjointness; inversion of V(r) from spectrum is the inverse Sturm-Liouville problem (ill-posed for incomplete data).

## ⚙ 2. Forward Model

Physical chain: **x** → E · hamiltonian → O · spectra wavefunctions → **y** (detector).

```
y = `O.spectra_wavefunctions` `E.hamiltonian` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.hamiltonian` | E · hamiltonian operator |
| `O.spectra_wavefunctions` | O · spectra wavefunctions operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `E.eigensolve` | Computes eigen-pairs of a linear operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Quantum Mechanics |
| Sub domain | Stationary quantum eigenproblem |
| Carrier | electron |
| Problem class | linear_inverse |
| Solution space | wavefunctions_and_eigenvalues |
| Noise model | gaussian |
| Integration axis | energy |
| Difficulty delta | 5 |
| L dag | 3 |

## 📡 4. Measurement Model

Eigenvalues real under self-adjointness; inversion of V(r) from spectrum is the inverse Sturm-Liouville problem (ill-posed for incomplete data).

| Metric | Value |
|---|---|
| Metric | energy_error_eV |
| Secondary | wavefunction_L2_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `time_independent_schrodinger` · **Forward operator:** `schrodinger_stationary_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N basis | — | 1000 |
| N eigen | — | 20 |
| Box size a | — | 10 |
| Potential type | — | harmonic |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N basis | — | 100 – 100000 |
| N eigen | — | 1 – 1000 |
| Box size a | — | 1 – 1000 |
| Potential type | — | harmonic, coulomb, square_well, double_well |

## 🎯 6. Tolerance (ε)

**Center tolerance:** E_n error <= 0.01 eV

| Metric | Range |
|---|---|
| Energy error ev | 0.001 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **energy_error_eV**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xab0f3547902121b91cd7d7ed4ca2213f07a0e1a6b61e44b79c9739ff9fe2b819`
- **Chain tx hash:** `0x573f7d6dbd01c915c0c21a66d22fb71fc0c3ff3d1a0a01ac884d144356ec230c`
- **Chain block:** `41554079`

---

## File Mapping

This bundle consists of: `L1-312.md`, `L1-312.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-312.md` | Source of truth — edit this | Human or LLM |
| `L1-312.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-312`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._