From f32cdca7fa84653544599c0c82a7971f861eb08b Mon Sep 17 00:00:00 2001 From: Anay Date: Sat, 27 Dec 2025 20:11:25 +0000 Subject: [PATCH] =?UTF-8?q?Add=201D=20Schr=C3=B6dinger=20equation=20exampl?= =?UTF-8?q?e?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- schrodinger1D.py | 159 ++++++++++++++++++++++++++++++++++++++++++ simulation_result.png | Bin 0 -> 171180 bytes 2 files changed, 159 insertions(+) create mode 100644 schrodinger1D.py create mode 100644 simulation_result.png diff --git a/schrodinger1D.py b/schrodinger1D.py new file mode 100644 index 00000000..0eb09d81 --- /dev/null +++ b/schrodinger1D.py @@ -0,0 +1,159 @@ +""" +Quantum Tunneling Simulation +============================ + +This example demonstrates how to solve the Time-Dependent Schrödinger Equation +using Diffrax. It simulates a Gaussian wave packet tunneling through a +potential barrier. + +Key JAX features used: +- `jax.jit` for compilation +- `diffrax.Tsit5` solver +- Complex number arithmetic in ODEs + +![Simulation Result](simulation_result.png) +""" + +import dataclasses +import jax +import jax.numpy as jnp +import diffrax +import matplotlib.pyplot as plt + + +# configure sim we use @dataclass to not write the __init__ method +@dataclasses.dataclass(frozen=True) +class SimConfig: + N: int = 500 # Grid points + L: float = 40.0 # Box size + t0: float = 0.0 # Start time + t1: float = 4.0 # End time + num_frames: int = 200 # Number of frames to save (Fixed integer is safer than dt) + + # dx function can be called without () because of @property + @property + def dx(self): + return self.L / self.N + + @property + def x_grid(self): + return jnp.linspace(-self.L / 2, self.L / 2, self.N) + + +# Physics + + +def get_potential(x_grid): + return jnp.where((x_grid > -0.5) & (x_grid < 0.5), 20.0, 0.0) + + +def build_hamiltonian(config: SimConfig): + x = config.x_grid + V_arr = get_potential(x) + V_matrix = jnp.diag(V_arr) + + main_diag = jnp.full(config.N, -2.0) + off_diag = jnp.ones(config.N - 1) + + M = jnp.diag(main_diag, k=0) + jnp.diag(off_diag, k=1) + jnp.diag(off_diag, k=-1) + KE = (-1.0 / (2.0 * config.dx**2)) * M + + H = KE + V_matrix + return H, V_arr + + +def get_initial_psi(config: SimConfig, x0=-10.0, k0=5.0, sigma=1.0): + x = config.x_grid + norm = 1.0 / (jnp.pi * sigma**2) ** 0.25 + psi = norm * jnp.exp(-((x - x0) ** 2) / (2 * sigma**2)) * jnp.exp(1j * k0 * x) + return psi + + +def vector_field(t, psi, args): + H = args + return -1j * (H @ psi) + + +# compile jit +@jax.jit +def run_simulation(hamiltonian, psi0, t0, t1, save_times): + + term = diffrax.ODETerm(vector_field) + solver = diffrax.Tsit5() + + # Use the passed-in array instead of creating one inside + saveat = diffrax.SaveAt(ts=save_times) + + step_controller = diffrax.PIDController(rtol=1e-8, atol=1e-8) + + sol = diffrax.diffeqsolve( + term, + solver, + t0, + t1, + dt0=0.01, + y0=psi0, + args=hamiltonian, + saveat=saveat, + stepsize_controller=step_controller, + ) + return sol + + +# Visualization + + +def create_plot(sol, config, V_arr, psi0): + x = config.x_grid + psi_final = sol.ys[-1] + + fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 8), sharex=True) + + # Plot 1 + ax1.set_title(f"Quantum Tunneling (t={config.t1})", fontsize=14) + ax1.fill_between( + x, 0, V_arr * 0.01, color="gray", alpha=0.3, label="Potential Barrier (Scaled)" + ) + ax1.plot(x, jnp.abs(psi0) ** 2, "b--", alpha=0.6, label="Initial |$\psi$|$^2$") + ax1.plot(x, jnp.abs(psi_final) ** 2, "r-", linewidth=2, label=f"Final |$\psi$|$^2$") + ax1.set_ylabel("Probability Density") + ax1.legend(loc="upper right") + ax1.grid(True, alpha=0.3) + + # Plot 2 + ax2.set_title("Wave Function Components", fontsize=12) + ax2.plot(x, jnp.real(psi_final), "g-", alpha=0.7, label="Real Part") + ax2.plot(x, jnp.imag(psi_final), "orange", alpha=0.7, label="Imaginary Part") + ax2.set_xlabel("Position ($x$)") + ax2.set_ylabel("Amplitude") + ax2.legend(loc="upper right") + ax2.grid(True, alpha=0.3) + + plt.tight_layout() + plt.savefig("simulation_result.png", dpi=150) + print("Plot saved as simulation_result.png") + plt.show() + + +# main +def main(): + print("Initializing Quantum Simulation...") + config = SimConfig(N=500, L=40.0, t0=0.0, t1=4.0, num_frames=200) + + print("Building Hamiltonian...") + H_matrix, V_array = build_hamiltonian(config) + psi0 = get_initial_psi(config) + + # always initialize outside the helper functions because helper functions are for abstractions + save_times = jnp.linspace(config.t0, config.t1, config.num_frames) + + print("Solving Schrödinger Equation (JIT Compiled)...") + # Pass 'save_times' into the function + sol = run_simulation(H_matrix, psi0, config.t0, config.t1, save_times) + + print("Generating Plots...") + create_plot(sol, config, V_array, psi0) + + +if __name__ == "__main__": + 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