Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

โ† 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:

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

Computational Example

Let's compute and visualize the first few energy eigenstates numerically:

Infinite Square Well: Wave Functions and Energies

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

python
import numpy as np
import matplotlib.pyplot as plt

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

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

# Calculate first 4 energy levels
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)

# Compute and display
print("Infinite Square Well (a = 1 nm)")
print("=" * 50)
for n in range(1, 5):
    E_n = energy_level(n)
    E_eV = E_n / 1.602e-19  # Convert to eV
    print(f"n = {n}:")
    print(f"  Energy: {E_n:.3e} J = {E_eV:.3f} eV")
    print(f"  E_{n}/E_1 = {n**2}")
    print()

# Create visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# Plot wave functions
for n in range(1, 4):
    psi_n = psi(n, x)
    ax1.plot(x * 1e9, psi_n * 1e-4.5, label=f'n={n}')
ax1.set_xlabel('Position (nm)')
ax1.set_ylabel('ฯˆ(x) (nm^-1/2)')
ax1.set_title('Wave Functions')
ax1.legend()
ax1.grid(True, alpha=0.3)

# Plot probability densities
for n in range(1, 4):
    psi_n = psi(n, x)
    prob = psi_n**2
    ax2.plot(x * 1e9, prob * 1e-9, label=f'n={n}')
ax2.set_xlabel('Position (nm)')
ax2.set_ylabel('|ฯˆ(x)|ยฒ (nm^-1)')
ax2.set_title('Probability Densities')
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('infinite_well.png', dpi=150, bbox_inches='tight')
print("Plot saved as 'infinite_well.png'")

๐Ÿ’ก This example demonstrates the computational approach to solving physics problems

โ† Free ParticleFinite Well โ†’