r"""
Schrodinger's wavepacket spreading
======================================

Erwin Schrodinger's 1926 wave equation,

.. math::

    i\hbar\partial_t\psi = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi,

made a radical and verifiable prediction: a localized particle's
wavefunction inevitably spreads out over time, even in free space
(:math:`V=0`). :func:`~physicskit.fields.solitons.nls_evolve` solves the
nonlinear Schrodinger equation :math:`i\partial_t\psi +
\tfrac12\partial_x^2\psi + g|\psi|^2\psi = 0` (natural units
:math:`\hbar=m=1`); switching its nonlinearity off (``g=0``) leaves
exactly Schrodinger's free-particle equation above. Starting from a
Gaussian wavepacket of initial width :math:`\sigma_0`, this reproduces
the free-particle spreading law

.. math::

    \sigma(t) = \sigma_0\sqrt{1+\left(\frac{t}{2\sigma_0^2}\right)^2}

directly from that solver.
"""

import matplotlib.pyplot as plt
import numpy as np

from physicskit.fields import nls_evolve, nls_evolve_frames, plot_field_1d

# %%
# A narrow Gaussian wavepacket, :math:`\sigma_0 = 2` (units :math:`\hbar=m=1`)
# --------------------------------------------------------------------------------

N, L, sigma0 = 2048, 200.0, 2.0
x = np.linspace(-L / 2, L / 2, N, endpoint=False)
psi0 = np.exp(-(x**2) / (4 * sigma0**2)).astype(complex)
psi0 /= np.sqrt(np.sum(np.abs(psi0) ** 2) * (x[1] - x[0]))

# %%
# Free-particle evolution: ``g=0`` keeps only Schrodinger's linear term
# --------------------------------------------------------------------------

t_final, steps = 6.0, 3000
psi = nls_evolve(psi0, x, dt=t_final / steps, steps=steps, g=0.0)

# %%
# The wavepacket spreads exactly as predicted by the free-particle
# spreading law.

dens = np.abs(psi) ** 2 / np.sum(np.abs(psi) ** 2 * (x[1] - x[0]))
sigma_num = np.sqrt(np.sum(x**2 * dens) * (x[1] - x[0]))
sigma_theory = sigma0 * np.sqrt(1 + (t_final / (2 * sigma0**2)) ** 2)

fig, ax = plot_field_1d(x, np.abs(psi0), label="t = 0")
plot_field_1d(x, np.abs(psi), ax=ax, label=f"t = {t_final}")
ax.set_title(f"sigma(t): numeric {sigma_num:.3f}, theory {sigma_theory:.3f}")
fig.tight_layout()

print(f"numeric sigma(t={t_final}) = {sigma_num:.4f}")
print(f"theory  sigma(t={t_final}) = {sigma_theory:.4f}  (sigma0 * sqrt(1+(t/2 sigma0^2)^2))")

# %%
# A space-time diagram, and sigma(t) tracked continuously against theory
# -------------------------------------------------------------------------------
# The two-instant comparison above only checks :math:`\sigma(t)` at the very
# end; recording every intermediate snapshot with
# :func:`~physicskit.fields.solitons.nls_evolve_frames` (the same free-particle
# split-step integrator, restructured only to also keep a history) shows the
# whole spreading process at once, as a widening light-cone-like wedge in
# space and time, and lets :math:`\sigma(t)` be measured from the recorded
# density at every frame, not just the last one -- tracking the free-particle
# spreading law continuously rather than checking a single endpoint.

n_frames = 60
frames, times = nls_evolve_frames(psi0, x, dt=t_final / steps, steps_per_frame=steps // n_frames, n_frames=n_frames, g=0.0)
frame_dens = np.abs(frames) ** 2
frame_dens /= np.sum(frame_dens, axis=1, keepdims=True) * (x[1] - x[0])
sigma_num_t = np.sqrt(np.sum(x[np.newaxis, :] ** 2 * frame_dens, axis=1) * (x[1] - x[0]))
sigma_theory_t = sigma0 * np.sqrt(1 + (times / (2 * sigma0**2)) ** 2)

fig2, (ax_space, ax_sigma) = plt.subplots(1, 2, figsize=(10, 4))
extent = (x.min(), x.max(), times.min(), times.max())
im = ax_space.imshow(np.abs(frames), extent=extent, origin="lower", aspect="auto", cmap="viridis")
fig2.colorbar(im, ax=ax_space, label="|psi(x, t)|")
ax_space.set_xlim(-8 * sigma_theory_t[-1], 8 * sigma_theory_t[-1])
ax_space.set_xlabel("x")
ax_space.set_ylabel("t")
ax_space.set_title("Space-time diagram: the widening wavepacket")

ax_sigma.plot(times, sigma_num_t, "o", markersize=3, label="numeric sigma(t)")
ax_sigma.plot(times, sigma_theory_t, "k--", label="theory")
ax_sigma.set_xlabel("t")
ax_sigma.set_ylabel("sigma(t)")
ax_sigma.set_title("Spreading law, tracked continuously")
ax_sigma.legend()
fig2.tight_layout()