Editing Condensed Matter Physics (section)

== <span style="color: #FFFFFF;">Applying</span> ==
'''Simulating Electron Density in a 1D Periodic Potential (Kronig-Penney Model):'''
<syntaxhighlight lang="python">
import numpy as np
import matplotlib.pyplot as plt

def solve_1d_periodic(n_cells, points_per_cell, potential_strength):
    """
    Simplified simulation of electron energy bands in a crystal.
    """
    n_total = n_cells * points_per_cell
    x = np.linspace(0, n_cells, n_total)
    dx = x[1] - x[0]
    
    # Potential: series of delta-like peaks at cell boundaries
    V = np.zeros(n_total)
    V[::points_per_cell] = potential_strength
    
    # Hamiltonian: -1/2 * d^2/dx^2 + V
    main_diag = np.ones(n_total) / (dx**2) + V
    off_diag = -0.5 * np.ones(n_total - 1) / (dx**2)
    H = np.diag(main_diag) + np.diag(off_diag, k=1) + np.diag(off_diag, k=-1)
    
    # Periodic boundary conditions
    H[0, -1] = H[-1, 0] = -0.5 / (dx**2)
    
    energies = np.linalg.eigvalsh(H)
    return energies

# Compare free electron vs. periodic potential
free_energies = solve_1d_periodic(10, 50, 0)
crystal_energies = solve_1d_periodic(10, 50, 100)

plt.figure(figsize=(10, 4))
plt.scatter(range(len(free_energies)), sorted(free_energies), s=5, label='Free Space')
plt.scatter(range(len(crystal_energies)), sorted(crystal_energies), s=5, label='In Crystal')
plt.ylabel("Energy Levels")
plt.xlabel("State Number")
plt.title("Emergence of Energy Bands and Gaps")
plt.legend()
plt.show()
</syntaxhighlight>

; Technological Impact
: '''Transistors & LEDs''' → The entire digital revolution (Silicon, Gallium Nitride).
: '''High-Strength Materials''' → Carbon nanotubes, Graphene, advanced alloys.
: '''MRI Magnets''' → Superconducting coils.
: '''Data Storage''' → Giant Magnetoresistance (GMR) in hard drives.
: '''Solar Cells''' → Photovoltaics.
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