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.

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.

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

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

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

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

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

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.

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

9. Uncertainty Example#

Position–Momentum Incompatibility#

[x, p] = i

Interpretation:

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

10. Tensor Product Example#

Building a Composite System#

|ψ⟩ ⊗ |φ⟩

Interpretation:

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

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.