Lesson 6 — From Generator to Hamiltonian

What Does the Generator Become?

We discovered that smooth transport has a clear initial direction called the generator. QM2 now shows that, because the transport preserves the state geometry, this generator has a special form. That form gives us the Hamiltonian.

Generator A

A describes how the transport begins.

Why Is the Hamiltonian Special?

The transport preserves the state geometry.

Because of that preservation, the generator has the correct structure for H to be Hermitian.

Hermitian

A Hermitian Hamiltonian is the form required for geometry-preserving quantum evolution.

From Generator to Hamiltonian

The generator of geometry-preserving transport can be rewritten in the Hermitian form used throughout quantum mechanics.

Why the Generator Is Hermitian

DERIVED

Because transport preserves the Hermitian geometry, its infinitesimal generator has the Hermitian property required of the quantum Hamiltonian.

  1. U(t)†U(t) = I

    Unitary transport (from D2).

  2. d/dt [ U(t)†U(t) ] = 0

    Differentiate, using D4.

  3. U(dt) ≈ I − iH dt

    The local form near the identity.

  4. H† = H

    The generator is forced to be Hermitian.

The QM2 Chain

  1. Continuous Transport
  2. Geometry Preserved
  3. Generator A
  4. Hamiltonian H
  5. Continuous Quantum Evolution

What QM2 Has Shown

Under D1–D4, continuous structure-preserving transport is generated by a Hermitian operator H. This is the mathematical role played by the Hamiltonian in quantum mechanics; the specific physical form of H requires additional physical structure.

What QM2 has not yet shown

  • • where a specific physical Hamiltonian comes from
  • • measurement or probability
  • • collapse or decoherence
  • • non-trivial multi-state dynamics from the current one-dimensional QM1 sector

Scope of This Result

SCOPE CONDITION

The derivation shown here establishes the result within the mathematical setting specified by QM2. It must not automatically be extrapolated to arbitrary higher-dimensional or interacting quantum systems without the additional reconstruction required by later work.

QM2 reconstructs the form of continuous quantum transport under its assumptions. The deeper question of uniquely selecting underlying relational dynamics belongs to the broader reconstruction programme.

Build the Correct QM2 Sequence

Tap the cards in the correct conceptual order. Use the arrows to rearrange them.

Available cards

Your sequence

    Nothing placed yet.

← Back to Lesson 5