The core idea is to present the project not just as a piece of code, but as a solution to a well-defined problem. Categorising the requirements into three main areas: Functional, Non-Functional, and System/Architectural.

1. Introduction: Problem Statement & Scope

Goal: To develop a medium-high-performance, user-friendly Python library, Dirac-Solver , for simulating the time evolution of a single-particle spinor field $\psi(t, \vec{r})$ governed by the (3+1)D Dirac equation.
Motivation: Address the need for an accessible tool for studying relativistic quantum phenomena in contexts like graphene, topological insulators, or for purely pedagogical purposes. The combination of a high-level Python API with a low-level C++ core aims to provide both ease of use and computational efficiency.
Core Equation: The solver will find numerical solutions to the time-dependent Dirac equation: $$i\hbar \frac{\partial \psi(\vec{r}, t)}{\partial t} = \hat{H} \psi(\vec{r}, t)$$ where the Dirac Hamiltonian $\hat{H}$ is: $$\hat{H} = c\vec{\alpha} \cdot \hat{\vec{p}} + \beta m_0 c^2 + V(\vec{r})I_4$$ Here, $\vec{\alpha}$ and $\beta$ are the standard $4 \times 4$ Dirac matrices, $\hat{\vec{p}} = -i\hbar\nabla$ is the momentum operator, and $V(\vec{r})$ is a scalar potential.
Scope: The initial release will focus on 1D and 2D systems for a single, non-interacting particle. The scope explicitly excludes many-body effects and field quantization (QED effects).

2. Functional Requirements (What the System Must Do)

A. Physics Engine Core (The C++ Backend)

FR-1: Discretization Scheme: Implement a Finite-Difference Time-Domain (FDTD) scheme to solve the Dirac equation. A staggered-leapfrog algorithm for the spatial and temporal derivatives is required to ensure numerical stability and second-order accuracy.
FR-2: Boundary Conditions: The user must be able to specify boundary conditions. At a minimum, Perfectly Matched Layers (PML) or Absorbing Boundary Conditions (ABC) must be implemented to prevent spurious reflections from the grid edges. Simple periodic boundary conditions should also be an option.
FR-3: Supported Potentials: The solver must correctly model the following potential types:
- Free Particle: $V(\vec{r}) = 0$.
- Scalar Potentials: User-definable functions $V(\vec{r})$ representing barriers, steps, and quantum wells.
- Coulomb Potential: $V(r) = -\frac{Z e^2}{4\pi\epsilon_0 r}$, with appropriate handling of the singularity at $r=0$ through regularization or a specialized grid.
FR-4: Initial State: The library must provide helper functions to initialize the spinor field $\psi(\vec{r}, t=0)$ as a Gaussian wavepacket with user-specified momentum, position, and spin polarization.

B. Python API (The User Interface)

FR-5: Simulation Setup: Users must be able to define, configure, and run a simulation through a "Pythonic" API. This includes setting grid parameters (dimensions, spacing $dx, dy, dz$), time step $dt$, total simulation time, choice of potential, and initial conditions.
FR-6: Solver Execution: A clear and simple method, e.g., solver.run(), to execute the time-evolution loop. The method should provide progress feedback for long simulations.
FR-7: Data Retrieval: Post-simulation, the user must be able to access the full 4-component complex spinor field $\psi(\vec{r}, t)$ at specified time intervals.

C. Outputs and Analysis (What the System Must Produce)

FR-8: Observable Calculations: The library must compute and provide access to key physical observables derived from $\psi$:
- Probability Density: $\rho(\vec{r}, t) = \psi^\dagger \psi$
- Probability Current Density: $\vec{j}(\vec{r}, t) = c\psi^\dagger \vec{\alpha} \psi$
- Expectation Value of Energy: $\langle E \rangle(t) = \int \psi^\dagger \hat{H} \psi \, d^V$
FR-9: Data Persistence: Simulation results (the final spinor field and time-series of observables) must be exportable to a standard, high-performance file format like HDF5 or a simple set of NumPy binary files (.npy).
FR-10: Visualization: The library must include a visualization module that uses matplotlib to generate:
- 2D plots (e.g., probability density $\rho(x, t)$ for a 1D simulation).
- Animations of the time evolution of $\rho(x,y,t)$.
- Vector field plots for the probability current density $\vec{j}$.

D Advanced Performance Requirements (Optional)

FR-11: Parallelization with OpenMP in C++ to accelerate simulations.
FR-12: Future support for GPU (CUDA/OpenCL) for large-scale problems.
FR-13: Logging of memory usage and execution time for each simulation.

3. Non-Functional Requirements (How Well the System Performs)

NFR-1: Performance: For a benchmark 2D problem (e.g., a $512 \times 512$ grid), the C++ core must achieve a target number of timesteps per second on standard consumer hardware. This defines the "medium-high-performance" goal.
NFR-2: Accuracy & Validation: The solver's accuracy must be verifiable.
- It must demonstrate conservation of total probability ($\int \psi^\dagger \psi \, dV = 1$) to within a specified tolerance for bound states.
- Results for canonical problems (e.g., Zitterbewegung for a free particle, Klein tunneling for a potential step) must agree with analytical solutions to within a predictable error margin related to the grid spacing ($O(\Delta x^2, \Delta t^2)$).
- Robust error handling (e.g., invalid user inputs, numerical divergences).
NFR-3: Usability:
- The API must be well-documented with docstrings and a tutorial in the form of a Jupyter Notebook.
- Installation must be straightforward via pip.
NFR-4: Portability: The software must compile and run on major operating systems (Linux, macOS, and Windows) with common C++ compilers (GCC, Clang, MSVC) and supported Python versions (e.g., 3.8+).
NFR-5:Documentation: User guides (Getting Started, Advanced Tutorial), Technical manual of the C++ backend (including class diagram and FDTD algorithm flow) and use Doxygen for documentation.
NFR-6: Community: Code published on GitHub with a clear license (e.g., MIT or GPL). Issues and pull requests managed with priority labels.

4. System & Architectural Requirements

SR-1: Hybrid Architecture: A two-layer architecture is required:
- Backend: A C++ library containing the FDTD algorithm, data structures for the fields, and physics computations. This ensures maximum performance.
- Frontend: A Python wrapper that provides the high-level API for users with examples.
SR-2: Interoperability Layer: pybind11 (or a similar tool like SWIG/CFFI) will be used to create the bindings between C++ and Python. This layer must efficiently handle the transfer of NumPy arrays to and from the C++ backend without significant memory copying overhead.
SR-3: Testing Framework: A comprehensive testing suite is required, using Google Test for the C++ backend and pytest for the Python API and integration tests. Continuous Integration (CI) via GitHub Actions should be set up to run these tests automatically.
SR-4: Modularity: Each module (potentials, boundary conditions, visualization) must be independent and loosely coupled.
SR-5: Versioning: Define a semantic versioning scheme (SemVer) to plan future releases.

5. Project Management Requirements

PM-1: Initial timeline:
- Phase 1: 1D solver + minimal API.
- Phase 2: Extension to 2D + visualization , magnetic vector potentials implementation.
- Phase 3: Documentation and public release.
PM-2: Quality metrics (test coverage percentage, reproducible benchmarks).
PM-3: Risk plan (e.g., numerical instability, memory bottlenecks).

<strong>1. Introduction: Problem Statement & Scope</strong>

<strong>2. Functional Requirements (What the System <em>Must Do</em>)</strong>

<strong>A. Physics Engine Core (The C++ Backend)</strong>

<strong>B. Python API (The User Interface)</strong>

<strong>C. Outputs and Analysis (What the System <em>Must Produce</em>)</strong>

<strong>D Advanced Performance Requirements (Optional)</strong>

<strong>3. Non-Functional Requirements (How <em>Well</em> the System Performs)</strong>

<strong>4. System & Architectural Requirements</strong>

<strong>5. Project Management Requirements</strong>