some markdown updates

ModularedPresentation.ipynb

@@ -55,6 +55,242 @@
     " - Documentation + modular design\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "---\n",
+    "\n",
+    "## 0. Quantum picture: what am I actually solving?\n",
+    "\n",
+    "Before getting lost in matrices and `numpy`, I want to be clear what the **physics problem** is.\n",
+    "\n",
+    "In the postulate language of quantum mechanics:\n",
+    "\n",
+    "- A **state** of the particle is described by a wavefunction $$\\Psi(x)$$ \n",
+    "  (or more abstractly, a vector $$|\\Psi\\rangle$$ in a Hilbert space).\n",
+    "\n",
+    "- **Observables** (like energy) are represented by **Hermitian operators**.\n",
+    "  For a particle in 1D with potential $$V(x)$$, the Hamiltonian operator is\n",
+    "\n",
+    "  $$\n",
+    "  \\hat{H} = -\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2} + V(x).\n",
+    "  $$\n",
+    "\n",
+    "- The **energy eigenstates** are solutions of\n",
+    "\n",
+    "  $$\n",
+    "  \\hat{H}\\,\\psi_n(x) = E_n \\,\\psi_n(x),\n",
+    "  $$\n",
+    "\n",
+    "  with boundary conditions and normalization:\n",
+    "\n",
+    "  $$\n",
+    "  \\int |\\psi_n(x)|^2 dx = 1.\n",
+    "  $$\n",
+    "\n",
+    "Physically:\n",
+    "\n",
+    "- Each $$\\psi_n(x)$$ is a **stationary state**: if the system is prepared in $$\\psi_n$$, the probability density\n",
+    "  $$|\\psi_n(x,t)|^2$$ is independent of time (it only picks up a phase $$e^{-iE_n t/\\hbar}$$).\n",
+    "\n",
+    "- The energies $$E_n$$ are the possible outcomes of a precise **energy measurement**.\n",
+    "\n",
+    "- A general state can be expanded as\n",
+    "\n",
+    "  $$\n",
+    "  \\Psi(x,0) = \\sum_n c_n \\psi_n(x),\n",
+    "  $$\n",
+    "\n",
+    "  and then evolves as\n",
+    "\n",
+    "  $$\n",
+    "  \\Psi(x,t) = \\sum_n c_n \\psi_n(x)\\,e^{-iE_n t/\\hbar}.\n",
+    "  $$\n",
+    "\n",
+    "So my numerical solver is really a machine that takes a **chosen potential** $$V(x)$$ and returns:\n",
+    "\n",
+    "- The discrete spectrum $$\\{E_n\\}$$ (energy levels for that potential).\n",
+    "- The corresponding eigenfunctions $$\\psi_n(x)$$ (stationary states).\n",
+    "\n",
+    "Once I have those, I can in principle build **any** time-dependent solution by choosing coefficients $$c_n$$.\n",
+    "\n",
+    "\n",
+    "---\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# %% [markdown]\n",
+    "### 2.2.1 Infinite wall as a limit of a finite barrier\n",
+    "\n",
+    "Another way to see $$\\Psi(0) = \\Psi(a) = 0$$:\n",
+    "\n",
+    "- Start with a **finite** but huge barrier outside $$[0,a]$$:\n",
+    "  \n",
+    "  - $$V(x) = 0$$ for $$0 < x < a$$,\n",
+    "  - $$V(x) = V_0$$ (large but finite) for $$x \\le 0$$ and $$x \\ge a$$.\n",
+    "\n",
+    "- For a bound state with energy $$E < V_0$$, the solution in the barrier region decays exponentially\n",
+    "  (evanescent tail). As $$V_0 \\to \\infty$$, the decay length goes to zero, and the wavefunction is\n",
+    "  squeezed to exactly zero at the wall.\n",
+    "\n",
+    "So the infinite square well is an **idealized limit** where those evanescent tails are completely killed.\n",
+    "In that limit, the physically acceptable states are exactly those that vanish at the boundaries.\n",
+    "\n",
+    "### 2.2.2 Probability current and “no leakage”\n",
+    "\n",
+    "The boundary conditions are also tied to **probability conservation**.\n",
+    "\n",
+    "The probability current in 1D is\n",
+    "\n",
+    "$$\n",
+    "j(x) = \\frac{\\hbar}{m}\\,\\text{Im}\\big(\\Psi^*(x)\\,\\partial_x \\Psi(x)\\big).\n",
+    "$$\n",
+    "\n",
+    "For a particle trapped in the well, there should be **no net probability flow out** of the region.\n",
+    "Taking $$\\Psi(0) = \\Psi(a) = 0$$ ensures that $$j(0) = j(a) = 0$$, so there is no leakage of probability\n",
+    "through the walls. This is part of making $$\\hat{H}$$ self-adjoint with these boundary conditions.\n",
+    "\n",
+    "---"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# %% [markdown]\n",
+    "### 2.3.1 Physical interpretation: standing waves, nodes, and quantization\n",
+    "\n",
+    "The infinite well is a quantum version of a **string fixed at both ends**:\n",
+    "\n",
+    "- The boundary conditions $$\\Psi(0) = \\Psi(a) = 0$$ only allow certain “standing wave” shapes.\n",
+    "- Those shapes are exactly\n",
+    "\n",
+    "  $$\n",
+    "  \\Psi_n(x) = \\sqrt{\\frac{2}{a}} \\sin\\left(\\frac{n\\pi x}{a}\\right),\n",
+    "  $$\n",
+    "\n",
+    "  with integer $$n = 1,2,3,\\dots$$\n",
+    "\n",
+    "Each $$n$$ has:\n",
+    "\n",
+    "- $$n-1$$ internal **nodes** (points where $$\\Psi_n(x) = 0$$ inside the well).\n",
+    "- Increasing **curvature** of the wavefunction, which corresponds to higher kinetic energy.\n",
+    "\n",
+    "This is why the energies scale like\n",
+    "\n",
+    "$$\n",
+    "E_n \\propto n^2.\n",
+    "$$\n",
+    "\n",
+    "The key physics point:\n",
+    "\n",
+    "- The **discreteness** of the spectrum (energy quantization) is not put in by hand.\n",
+    "- It **emerges** from:\n",
+    "  - the wave nature of the particle, and\n",
+    "  - the boundary conditions (particle confined in a finite region).\n",
+    "\n",
+    "My numerical solver is reproducing that spectrum purely from the operator\n",
+    "\n",
+    "$$\n",
+    "\\hat{H} = -\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2} + V(x)\n",
+    "$$\n",
+    "\n",
+    "and the boundary conditions encoded in the matrix.\n",
+    "---\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## 6. Using the numerical eigenstates as actual quantum states\n",
+    "\n",
+    "Once I have the eigenvalues $$E_n$$ and eigenvectors $$\\psi_n(x_i)$$ on the grid, I can treat them as\n",
+    "**full quantum states** and compute observables.\n",
+    "\n",
+    "On the grid, an integral becomes a sum:\n",
+    "\n",
+    "$$\n",
+    "\\int f(x)\\,dx \\approx \\sum_i f(x_i)\\,\\Delta x.\n",
+    "$$\n",
+    "\n",
+    "So for a normalized eigenstate $$\\psi_n(x)$$:\n",
+    "\n",
+    "- Position expectation value:\n",
+    "\n",
+    "  $$\n",
+    "  \\langle x \\rangle_n \\approx \\sum_i x_i\\,|\\psi_n(x_i)|^2\\,\\Delta x.\n",
+    "  $$\n",
+    "\n",
+    "- Position variance:\n",
+    "\n",
+    "  $$\n",
+    "  \\Delta x_n^2 = \\langle x^2 \\rangle_n - \\langle x \\rangle_n^2.\n",
+    "  $$\n",
+    "\n",
+    "For momentum, in the position representation\n",
+    "\n",
+    "$$\n",
+    "\\hat{p} = -i\\hbar \\frac{d}{dx}.\n",
+    "$$\n",
+    "\n",
+    "On the grid, I can approximate the derivative with finite differences, e.g.\n",
+    "\n",
+    "$$\n",
+    "\\left.\\frac{d\\psi}{dx}\\right|_{x_i} \\approx \\frac{\\psi(x_{i+1}) - \\psi(x_{i-1})}{2\\Delta x},\n",
+    "$$\n",
+    "\n",
+    "and then build\n",
+    "\n",
+    "$$\n",
+    "\\langle p \\rangle_n \\approx \\sum_i \\psi_n^*(x_i)\\,\\left(-i\\hbar \\frac{d\\psi_n}{dx}\\bigg|_{x_i}\\right)\\Delta x.\n",
+    "$$\n",
+    "\n",
+    "Similarly for $$\\langle p^2 \\rangle_n$$, which lets me compute $$\\Delta p_n$$ and check the\n",
+    "uncertainty relation numerically:\n",
+    "\n",
+    "$$\n",
+    "\\Delta x_n\\,\\Delta p_n \\ge \\frac{\\hbar}{2}.\n",
+    "$$\n",
+    "\n",
+    "So the solver is not just a “pretty plotter” of eigenfunctions; it is a tool to compute\n",
+    "**quantum expectation values** for arbitrary observables once they are discretized on the grid.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "### 3.5 Operator viewpoint: matrix = Hamiltonian in a finite basis\n",
+    "\n",
+    "Conceptually, what I’ve done is:\n",
+    "\n",
+    "- Start from the abstract operator equation\n",
+    "\n",
+    "  $$\n",
+    "  \\hat{H}|\\psi_n\\rangle = E_n |\\psi_n\\rangle,\n",
+    "  $$\n",
+    "\n",
+    "- Choose the **position basis** sampled on a grid\n",
+    "  $$\\{|x_1\\rangle, |x_2\\rangle, \\dots, |x_{N-1}\\rangle\\},$$\n",
+    "- Represent $$\\hat{H}$$ as an $$N\\times N$$ **matrix** in that basis using finite differences for the kinetic term\n",
+    "  and a diagonal matrix for $$V(x)$$.\n",
+    "\n",
+    "Because $$\\hat{H}$$ is Hermitian (and my discrete approximation respects that symmetry), the matrix is:\n",
+    "\n",
+    "- Real symmetric,\n",
+    "- Has **real** eigenvalues,\n",
+    "- And its eigenvectors form an **orthonormal basis** of the grid space.\n",
+    "\n",
+    "This is exactly what I see in the overlap matrix: numerically, it is very close to the identity.\n"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -1381,6 +1617,17 @@
    "source": [
     "plot_dead(E=E_vals, psi=psi_vecs, V=V_full, x=x, nos=5);\n"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Deploy:\n",
+    "\n",
+    "\n",
+    "https://huggingface.co/spaces/AhiBucket/Hand-wave\n",
+    "\n"
+   ]
   }
  ],
  "metadata": {

app.py

@@ -419,7 +419,7 @@ elif page == "Benchmarks & Verification":
     This solver has been rigorously tested against known analytical solutions and external libraries to ensure physical accuracy.
     """)
     
-    tab1, tab2, tab3 = st.tabs(["Analytical Benchmarks", "QMSolve Comparison", "Hamiltonian Check"])
+    tab1, tab2, tab3 = st.tabs(["Analytical Benchmarks", "QMSolve Comparison", "Code"])
     
     with tab1:
         st.subheader("1. Infinite Square Well")