Editing Quantum Mechanics (section)

== <span style="color: #FFFFFF;">Applying</span> ==
'''Simulating a simple 1D Quantum Well (Particle in a Box):'''
<syntaxhighlight lang="python">
import numpy as np
import matplotlib.pyplot as plt

def solve_infinite_well(L, n_points, n_states):
    """
    Solve for the energy levels and wavefunctions of an infinite potential well.
    L: Width of the well
    n_points: Spatial resolution
    n_states: Number of eigenstates to return
    """
    x = np.linspace(0, L, n_points)
    dx = x[1] - x[0]
    
    # Construct the Hamiltonian matrix using finite difference
    # H = -hbar^2 / (2m) * d^2/dx^2
    # Using units where hbar = 1 and m = 1
    main_diag = np.ones(n_points) / (dx**2)
    off_diag = -0.5 * np.ones(n_points - 1) / (dx**2)
    H = np.diag(main_diag) + np.diag(off_diag, k=1) + np.diag(off_diag, k=-1)
    
    # Solve eigenvalue problem
    energies, wavefunctions = np.linalg.eigh(H)
    
    return x, energies[:n_states], wavefunctions[:, :n_states]

L = 1.0
x, E, psi = solve_infinite_well(L, 500, 3)

# Plotting the probability densities |psi|^2
plt.figure(figsize=(10, 6))
for i in range(3):
    plt.plot(x, psi[:, i]**2 + E[i], label=f'State n={i+1}')
plt.title("Probability Densities in an Infinite Potential Well")
plt.xlabel("Position (x)")
plt.ylabel("Energy / Probability Density")
plt.legend()
plt.show()
</syntaxhighlight>

; Quantum mechanics in technology
: '''Semiconductors''' → Transistors, microchips (built on band theory).
: '''Lasers''' → Stimulated emission of radiation.
: '''MRI''' → Nuclear magnetic resonance (spin manipulation).
: '''Atomic Clocks''' → Transition frequencies of atoms (GPS timing).
: '''Quantum Computing''' → Qubits, Grover's algorithm, Shor's algorithm.
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