# Examples — Quantum Mechanics  
### TriadicFrameworks /docs/theories/quantum_mechanics/examples.md

These examples illustrate Quantum Mechanics (QM) as an  
**amplitude‑first operator grammar**, not a particle model and not a wave  
model. All examples avoid classical drift and remain strictly within the  
R1 substrate regime.

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# 1. Basis Decomposition Example  
### Decomposing a State in the Energy Basis

Given:

|ψ⟩ = (1/√3)|0⟩ + (√2/√3)|1⟩

Interpretation:

- |0⟩ and |1⟩ are **basis states**, not physical states of matter  
- coefficients encode **amplitude + phase**  
- probabilities are |cᵢ|²  

Probabilities:

- P(0) = 1/3  
- P(1) = 2/3  

No particles.  
No waves.  
Pure amplitude geometry.

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# 2. Measurement Example  
### Measuring an Observable with Eigenbasis {|i⟩}

Observable Ô has eigenstates |i⟩ with eigenvalues λᵢ.

Measurement rule:

Pᵢ |ψ⟩ = cᵢ |i⟩  
Probability = |cᵢ|²

Interpretation:

- measurement is **projection**, not revelation  
- outcome depends on the chosen observable  
- basis‑relative, not absolute  

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# 3. Time Evolution Example  
### Evolving a State Under a Hamiltonian

Given Hamiltonian H:

U(t) = e^{-iHt}

State evolution:

|ψ(t)⟩ = U(t)|ψ(0)⟩

Interpretation:

- evolution is **unitary**  
- preserves norm  
- rotates amplitudes in Hilbert space  
- not motion through space  

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# 4. Position ↔ Momentum Basis Example  
### Fourier Transform as Basis Change

ψ(x) ↔ φ(p)

Relation:

φ(p) = (1/√2π) ∫ ψ(x) e^{-ipx} dx

Interpretation:

- this is a **unitary basis transformation**  
- not a wave turning into a particle  
- not a physical process  
- the state does not change — only its coordinates do  

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# 5. Ladder Operator Example  
### Harmonic Oscillator Transitions

a |n⟩ = √n |n−1⟩  
a† |n⟩ = √(n+1) |n+1⟩

Interpretation:

- not creation/destruction of particles  
- purely algebraic transitions in amplitude structure  
- defines energy‑level geometry  

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# 6. Expectation Value Example  
### Computing ⟨Ô⟩

Given:

⟨Ô⟩ = ⟨ψ|Ô|ψ⟩

Interpretation:

- expectation value is **amplitude‑weighted**, not deterministic  
- not a classical average  
- depends on basis representation  

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# 7. Entanglement Example  
### Two‑Qubit Bell State

|Φ⁺⟩ = (1/√2)(|00⟩ + |11⟩)

Interpretation:

- entanglement is **correlation in amplitude space**  
- not communication  
- not influence  
- not a physical connection  

Reduced density matrix of either subsystem:

ρ = (1/2)I

Shows maximal mixing due to entanglement.

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# 8. Mixed State Example  
### Decoherence Producing a Mixed State

ρ = p |0⟩⟨0| + (1−p)|1⟩⟨1|

Interpretation:

- not ignorance about hidden variables  
- represents **loss of phase coherence**  
- describes open‑system behavior  

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# 9. Uncertainty Example  
### Position–Momentum Incompatibility

[x, p] = i

Interpretation:

- uncertainty arises from **operator algebra**, not disturbance  
- no classical analogue  
- reflects incompatibility of observables  

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# 10. Tensor Product Example  
### Building a Composite System

|ψ⟩ ⊗ |φ⟩

Interpretation:

- defines multi‑system amplitude structure  
- enables entanglement  
- basis‑dependent  

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# Summary

These examples show QM as:

- an **amplitude‑first operator grammar**  
- structured by **basis geometry**  
- governed by **unitary evolution**  
- interpreted through **measurement projection**  
- enriched by **entanglement and mixed states**  
- coherent only in **R1**  

QM is the substrate from which QFT emerges and to which QFT collapses  
when excitations lose stability.