# ⚛  L1 Principle — Quantum Well and Quantum Dot Electronic Structure

**ID:** `L1-496` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Semiconductor Physics — *Confined electronic states*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** wavefunctions energies of confined states
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554142

---

## 🧠 1. Introduction

**Quantum Well and Quantum Dot Electronic Structure** is a **nonlinear inverse problem** whose unknown lives in **wavefunctions energies of confined states** space, within the **Confined electronic states** sub-domain of **Semiconductor Physics**.

Measurements consist of electrons collected by an electron detector via a **optical spectroscopy PL absorption** sensing mechanism.

The forward operator applies, in order: E · strain operator; E · kp Hamiltonian operator; a fixed-point or gradient iteration on the unknown; O · states transitions operator.

Observations are corrupted by additive Gaussian noise. Well-posed; inversion for composition/strain from PL is ill-conditioned (multiple compositions produce same transition).

## ⚙ 2. Forward Model

Physical chain: **x** → E · strain → E · kp Hamiltonian → O · states transitions → **y** (detector).

```
y = `O.states_transitions` `E.kp_Hamiltonian` `E.strain` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.strain` | E · strain operator |
| `E.kp_Hamiltonian` | E · kp hamiltonian operator |
| `O.states_transitions` | O · states transitions operator |

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

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Semiconductor Physics |
| Sub domain | Confined electronic states |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | wavefunctions_energies_of_confined_states |
| Noise model | gaussian |
| Integration axis | energy |
| Difficulty delta | 5 |
| L dag | 3.6 |

## 📡 4. Measurement Model

Well-posed; inversion for composition/strain from PL is ill-conditioned (multiple compositions produce same transition).

| Metric | Value |
|---|---|
| Metric | energy_error_meV |
| Secondary | oscillator_strength_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `qdot_electronic_inversion` · **Forward operator:** `kp_schrodinger_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 4 |
| Basis n | N | 8 |
| Size nm | nm | 20 |
| Geometry | — | InGaAs_QD_on_GaAs |
| Strain pct | — | 7 |
| Composition in | — | 0.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 4 – 300 |
| Basis n | N | 4 – 20 |
| Size nm | nm | 2 – 100 |
| Strain pct | — | 0 – 10 |
| Composition in | — | 0.05 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** E_1 error <= 2 meV

| Metric | Range |
|---|---|
| Energy error mev | 0.5 – 50.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **energy_error_meV**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x8a7ba3f45d743244510b8a33be243539028ae9e9f5b48e572c16a27730f35a61`
- **Chain tx hash:** `0xbee99ef9c7180d1b64d7583953b28f2cb05f2c5a5c7d258650ded255931ae0a4`
- **Chain block:** `41554142`

---

## File Mapping

This bundle consists of: `L1-496.md`, `L1-496.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-496.md` | Source of truth — edit this | Human or LLM |
| `L1-496.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 `L1-496`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._