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← Part III/Infinite Square Well

2. Infinite Square Well

Reading time: ~30 minutes | Pages: 8

The "particle in a box": the simplest bound state problem with discrete energy levels.

The Potential

A particle confined to $0 \leq x \leq a$:

$$V(x) = \begin{cases} 0 & 0 < x < a \\ \infty & \text{otherwise} \end{cases}$$

Boundary conditions: $\psi(0) = \psi(a) = 0$

Energy Eigenstates

Inside the well ($0 < x < a$), solve:

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$$

Applying boundary conditions gives:

$$\psi_n(x) = \sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right), \quad n = 1, 2, 3, \ldots$$

These are normalized eigenstates

Energy Levels

Quantized energies:

$$E_n = \frac{n^2\pi^2\hbar^2}{2ma^2} = \frac{n^2h^2}{8ma^2}, \quad n = 1, 2, 3, \ldots$$

Key observations:

Energy levels grow as $n^2$
Ground state energy $E_1 > 0$ (zero-point energy)
No state with $n = 0$ (would violate uncertainty principle)
Energy spacing increases with $n$

Orthonormality

$$\int_0^a \psi_n^*(x)\psi_m(x)dx = \delta_{nm}$$

The eigenstates form a complete orthonormal basis

General Solution

Any state can be expanded:

$$\Psi(x,t) = \sum_{n=1}^{\infty} c_n \psi_n(x)e^{-iE_n t/\hbar}$$

where $c_n = \langle\psi_n|\Psi(x,0)\rangle$

Properties of Eigenstates

Number of nodes: $\psi_n$ has $n-1$ nodes (zeros)

Parity:

Odd $n$: symmetric about center ($x = a/2$)
Even $n$: antisymmetric

Classical limit: For large $n$, $|\psi_n|^2 \to$ uniform (correspondence principle)

Expectation Values

Position:

$$\langle x \rangle_n = \frac{a}{2}$$

Momentum:

$$\langle p \rangle_n = 0, \quad \langle p^2 \rangle_n = \frac{n^2\pi^2\hbar^2}{a^2}$$

Interactive Simulation

Run this Python code to compute and visualize the energy eigenstates. Try changing the box width or particle mass to see how the energy levels change!

Infinite Square Well Simulation

Python

Calculate wave functions and energy levels for a particle in a 1D box

infinite_well.py87 lines

import numpy as np
import matplotlib.pyplot as plt

# Physical constants
hbar = 1.055e-34  # J·s
m = 9.109e-31     # electron mass (kg)
a = 1e-9          # box width (1 nm)

# Position array
x = np.linspace(0, a, 1000)

# Calculate energy level
def energy_level(n):
    """Energy of nth level in Joules"""
    return (n**2 * np.pi**2 * hbar**2) / (2 * m * a**2)

# Calculate wave function
def psi(n, x):
    """Normalized wave function for nth state"""
    return np.sqrt(2/a) * np.sin(n * np.pi * x / a)

# Display results
print("=" * 55)
print("Infinite Square Well: Particle in a Box")
print("=" * 55)
print(f"\nParameters:")
print(f"  Box width a = {a*1e9:.1f} nm")
print(f"  Particle mass = {m:.3e} kg (electron)")
print(f"\nEnergy Levels:")
for n in range(1, 6):
    E_n = energy_level(n)
    E_eV = E_n / 1.602e-19  # Convert to eV
    print(f"  n = {n}: E = {E_n:.3e} J = {E_eV:.3f} eV (E_{n}/E_1 = {n**2})")

# Create visualization
fig, axes = plt.subplots(1, 2, figsize=(14, 6))

# Plot wave functions with energy level offsets
colors = ['blue', 'red', 'green', 'purple', 'orange']
ax1 = axes[0]

for n in range(1, 6):
    psi_n = psi(n, x)
    E_n_eV = energy_level(n) / 1.602e-19
    # Scale and offset wave functions for visualization
    ax1.plot(x * 1e9, psi_n * 0.3e-4 + E_n_eV, label=f'n={n}', color=colors[n-1], linewidth=1.5)
    ax1.axhline(y=E_n_eV, color=colors[n-1], linestyle='--', alpha=0.3, linewidth=1)

# Draw the potential walls
ax1.axvline(x=0, color='black', linewidth=3)
ax1.axvline(x=a*1e9, color='black', linewidth=3)
ax1.fill_betweenx([0, 25], -0.1, 0, color='gray', alpha=0.3)
ax1.fill_betweenx([0, 25], a*1e9, a*1e9+0.1, color='gray', alpha=0.3)

ax1.set_xlabel('Position (nm)', fontsize=12)
ax1.set_ylabel('Energy (eV) / Wave function', fontsize=12)
ax1.set_title('Wave Functions with Energy Levels', fontsize=14)
ax1.legend(loc='upper right', fontsize=9)
ax1.grid(True, alpha=0.3)
ax1.set_xlim(-0.1, a*1e9+0.1)
ax1.set_ylim(0, 25)

# Plot probability densities
ax2 = axes[1]
for n in range(1, 6):
    psi_n = psi(n, x)
    prob = np.abs(psi_n)**2
    ax2.plot(x * 1e9, prob * 1e-9, label=f'|ψ_{n}|²', color=colors[n-1], linewidth=1.5)
    ax2.fill_between(x * 1e9, 0, prob * 1e-9, alpha=0.15, color=colors[n-1])

ax2.set_xlabel('Position (nm)', fontsize=12)
ax2.set_ylabel('Probability density |ψ(x)|² (nm⁻¹)', fontsize=12)
ax2.set_title('Probability Densities', fontsize=14)
ax2.legend(fontsize=9)
ax2.grid(True, alpha=0.3)
ax2.set_xlim(0, a*1e9)

plt.tight_layout()
plt.savefig('infinite_well.png', dpi=150, bbox_inches='tight')

print(f"\nKey Observations:")
print(f"  • Ground state (n=1): E₁ = {energy_level(1)/1.602e-19:.3f} eV")
print(f"  • Energy scales as n²: E_n = n² × E₁")
print(f"  • Wave function ψ_n has (n-1) nodes")
print(f"  • Zero-point energy prevents E = 0 (uncertainty principle)")
print("\n✓ Plot saved as 'infinite_well.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Free Particle Finite Well →

Quantum Mechanics Course