>[!warning] >This content has not been peer reviewed. # Quantization — Code documentation ## Purpose `rst_quantization.py` derives energy quantization from the $L^n$ norm geometry. The $L^n$ kinetic energy $E = \|p\|^n / (nm^{n-1})$ produces energy levels $E_k \propto k^n$ in a box. At $n = 2$, this recovers the particle-in-a-box spectrum exactly ($R^2 = 1$). ## Key functions | Function | Description | |:---|:---| | `ln_ball_boundary(n)` | 2D unit ball $\|x\|^n + \|y\|^n = 1$ boundary points | | `ln_ball_area_2d(n)` | Area via $4\Gamma(1+1/n)^2 / \Gamma(1+2/n)$ | | `lacunarity_2d(n)` | $\mathcal{L} = 4 / A(n)$ | | `min_uncertainty_product(n)` | $\hbar^2 / 2^{2/n}$ | | `energy_ratios(n, k_max)` | $E_k/E_1 = k^n$ | | `fit_power_law(k, ratios)` | Fit $k^n$ to arbitrary energy ratios | ## Modes | Flag | Description | Output | |:---|:---|:---| | `--geometry` | L^n ball shapes + lacunarity spectrum | 2 figures | | `--uncertainty` | min(dx*dp) vs n, Heisenberg at n=2 | 1 figure | | `--spectrum` | Energy level bars + log-log power law | 2 figures | | `--discovery` | Fit n for particle-in-box, harmonic osc., hydrogen | 1 figure | Running without flags executes all modes. ## Dependencies - Python 3.8+ - NumPy, SciPy, Matplotlib ## Execution ```bash cd "about systems" python "expanded theory applied/further applications/Quantization/rst_quantization.py" python "expanded theory applied/further applications/Quantization/rst_quantization.py" --discovery ``` ## Output | File | Description | |:---|:---| | `quantization_geometry_balls.png` | L^n ball shapes (3x3 grid) | | `quantization_geometry_lacunarity.png` | Lacunarity spectrum vs n | | `quantization_uncertainty.png` | Minimum uncertainty product vs n | | `quantization_spectrum_levels.png` | Energy level bars for six n values | | `quantization_spectrum_loglog.png` | Log-log energy ratios | | `quantization_discovery.png` | Discovery: fitted n for QM systems | ## Links - **Theory:** [[Quantization (RST)]] - **Results:** [[Quantization Results]]