"The Field is a collection of Harmonic Oscillators." This repository implements the transition from discrete energy levels to continuous spatial fields as presented in Leonard Susskind's Advanced QM: Lecture 6.
This lab explores the mathematical "bookkeeping" required to describe systems where particle numbers are not fixed:
- The Ladder Algebra: Proving the (\sqrt{n}) scaling factors that ensure state normalization when creating/annihilating particles.
- Multiple Harmonic Oscillators: Treating a quantum field as a collection of oscillators, each representing a specific vibration mode (harmonic).
- Bosonic Occupation: Demonstrating how multiple particles can occupy the same quantum state—the defining characteristic of Bosons.
Validates the creation operator (a^\dagger) logic:
[ a^\dagger |n\rangle = \sqrt{n+1},|n+1\rangle ]
python demos/01_ladder_logic.pyTerminal Output:
--- Ladder Operator Scaling Test ---
State |0> --(a+)--> 1.000 * |1>
State |1> --(a+)--> 1.414 * |2>
State |2> --(a+)--> 1.732 * |3>
State |3> --(a+)--> 2.000 * |4>
State |4> --(a+)--> 2.236 * |5>
Visualizes the energy contribution of different modes within the system.
python demos/03_oscillator_interlude.pyObservation: Each "mode" (oscillator) contributes energy proportional to its frequency (\omega_i) and its occupation number (n_i).
The final bridge. This maps the occupation of various oscillators into a single spatial observable: the Particle Number Density (\langle \Psi^\dagger(x),\Psi(x) \rangle).
python demos/04_boson_density.pyqft-fock-lab/
├── src/fock/ # Core Logic
│ ├── ladder.py # a+ and a- scaling logic ([a, a†] = 1)
│ ├── oscillators.py # Energy management for multiple modes
│ └── field_op.py # The Bridge: Pulling system modes into space
├── demos/ # Lab Experiments
│ ├── 01_ladder_logic.py
│ ├── 03_oscillator_interlude.py
│ └── 04_boson_density.py
└── assets/ # Visual documentation (drop screenshots here)
| Demo 03 — Oscillator Interlude | Demo 04 — Boson Density |
|---|---|
 |
 |
| Multi-mode oscillator energy intuition (frequency × occupation). | Spatial particle-number density ⟨Ψ†(x)Ψ(x)⟩ from summed bosonic modes. |
In this lab, we proved that the Field Operator (\Psi(x)) is not just a mathematical convenience. By summing up the contributions of individual oscillators (Bosonic modes), we create a field that can account for:
- Vacuum Energy: The 7.5 units of energy present even when all (n_i = 0).
- Particle Locality: Where the "humps" in the magenta density plot reveal the most likely positions of the particle collection.
Inspiration: Leonard Susskind's Theoretical Minimum
Lecture Source: Advanced QM: Lecture 6 - QFT (YouTube playlist)
"The goal is to make the invisible, visible." — Leonard Susskind
Note: Outputs prioritize clarity and conceptual correctness over high-precision numerics.