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← Part III/Quantum Harmonic Oscillator

4. Quantum Harmonic Oscillator

Reading time: ~45 minutes | Pages: 12

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Course: 0% complete

The quantum harmonic oscillator is the most important exactly solvable problem in quantum mechanics - it appears everywhere from molecular vibrations to quantum field theory.

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Video Lecture

Quantum Harmonic Oscillator - MIT OCW

Prof. Allan Adams presents the complete solution of the quantum harmonic oscillator using ladder operators, demonstrating why this system is so fundamental to modern physics.

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

The Hamiltonian

Potential energy (parabolic well):

$$V(x) = \frac{1}{2}m\omega^2 x^2$$

Hamiltonian:

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2$$

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Harmonic Oscillator Potential & Energy Levels

Harmonic Oscillator

Selected Energy Level: n = 1

Number of Levels: 5

Show All Energy LevelsShow Wave Function

Legend:

Blue curve - Potential energy V(x)
Cyan line - Selected energy level
Green curve - Wave function ψ(x) at that energy
Gray dashed - Other energy levels

Algebraic Method: Ladder Operators

Define raising (†) and lowering operators:

$$\hat{a}^\dagger = \frac{1}{\sqrt{2m\hbar\omega}}(m\omega\hat{x} - i\hat{p})$$

$$\hat{a} = \frac{1}{\sqrt{2m\hbar\omega}}(m\omega\hat{x} + i\hat{p})$$

Commutation relation:

$$[\hat{a}, \hat{a}^\dagger] = 1$$

Deriving the Ladder Operator Algebra

Assumptions:

Canonical commutation: [x̂,p̂] = iℏ
Ladder operators: â and â† as defined above
Both operators are adjoints: (â)† = â†

Starting with:

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2$$

Energy Eigenvalues & Eigenstates

Energy spectrum (equally spaced!):

$$E_n = \hbar\omega\left(n + \frac{1}{2}\right), \quad n = 0,1,2,\ldots$$

Energy eigenstates constructed by ladder:

$$|n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}}|0\rangle$$

where |0⟩ is the ground state: â|0⟩ = 0

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Harmonic Oscillator Wave Function Evolution

Quantum Harmonic Oscillator

Quantum Number (n = 1)

Animation Speed: 1.0x

Show Probability Density |ψ|²

Note: Time evolution shows the oscillating phase of the quantum state.

Wave function ψ(x,t) oscillates with frequency ω = E/ℏ
Probability density |ψ|² remains constant in time (stationary state)
Higher quantum numbers have higher energies and faster oscillations

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Harmonic Oscillator Energy Calculator

Calculate energy levels for molecular vibrations

Formula:

$$E_n = \hbar\omega\left(n + \frac{1}{2}\right)$$

Angular frequency ω (rad/s)

Quantum number n

Notes:

Ground state (n=0) has zero-point energy E₀ = ℏω/2
Energy spacing ΔE = ℏω is constant between levels
Typical molecular vibrations: ω ~ 10¹³-10¹⁴ rad/s

Zero-Point Energy of a Diatomic Molecule

BASIC

Problem: Calculate the zero-point energy of an H₂ molecule vibrating at ω = 8.28 × 10¹⁴ rad/s.

Given:

Angular frequency: ω = 8.28 × 10¹⁴ rad/s
Ground state quantum number: n = 0
ℏ = 1.055 × 10⁻³⁴ J·s

Find: Zero-point energy E₀ in eV

Ladder Operator Action

INTERMEDIATE

Problem: If a harmonic oscillator is in state |n⟩, what state results from applying â†|n⟩?

Given:

Initial state: |n⟩
Raising operator: â†
Ladder operator property: â†|n⟩ = √(n+1)|n+1⟩

Find: The resulting state and its normalization

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Video Lecture

Quantum Harmonic Oscillator Applications - Stanford

Prof. Leonard Susskind discusses why the harmonic oscillator appears everywhere in physics - from atoms to quantum fields.

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Self-Check Question

Why does the quantum harmonic oscillator have a non-zero ground state energy (zero-point energy)?

Self-Check Question

What is special about the energy level spacing in the harmonic oscillator?

Self-Check Question

What does the lowering operator â do to the ground state |0⟩?

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Why the Harmonic Oscillator is Everywhere

1. Molecular Vibrations

Spectroscopy

Chemical bonds behave as quantum harmonic oscillators, enabling infrared and Raman spectroscopy for molecular identification.

Examples:

FTIR spectroscopy - identifying chemical compounds
Raman spectroscopy - non-destructive material analysis
Atmospheric monitoring - greenhouse gas detection
Medical diagnostics - cancer tissue identification

Impact: Fundamental tool in chemistry, materials science, and medicine

2. Phonons in Solids

Condensed Matter

Quantized lattice vibrations (phonons) explain thermal and acoustic properties of materials.

Examples:

Superconductivity - phonon-mediated Cooper pairing
Thermal management - heat sinks, thermoelectrics
Acousto-optic modulators - laser control systems
Brillouin scattering - material characterization

Impact: Essential for understanding solid-state physics and devices

3. Quantum Field Theory

Fundamental Physics

Harmonic oscillator provides the foundation for quantum field theory - each field mode is a quantum harmonic oscillator.

Examples:

Photon creation/annihilation in QED
Particle physics - quantum chromodynamics
Casimir effect - vacuum energy manifestation
Hawking radiation - black hole thermodynamics

Impact: Cornerstone of modern theoretical physics

💡 Understanding real-world applications helps connect abstract quantum concepts to tangible technology and motivates further study.

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Video Lecture

Harmonic Oscillator and Molecular Vibrations - Yale

Prof. Ramamurti Shankar connects the quantum harmonic oscillator to spectroscopy and real molecular systems.

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

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📝 Chapter Summary

Key Equations

Potential: V(x) = ½mω²x²

Energy: Eₙ = ℏω(n + ½)

Ladder: [â,â†] = 1

Hamiltonian: Ĥ = ℏω(â†â + ½)

States: |n⟩ = (â†)ⁿ/√n!|0⟩

Key Concepts

Equally-spaced energy levels (ΔE = ℏω)
Zero-point energy E₀ = ℏω/2 (quantum fluctuations)
Ladder operators create/destroy quanta
Algebraic solution (no differential equations!)
Foundation for quantum field theory
Appears in all small oscillations
Coherent states minimize uncertainty

The harmonic oscillator is the most important model system in all of physics. Its algebraic solution via ladder operators is elegant and powerful, serving as the foundation for quantum field theory where every field mode is a harmonic oscillator. Understanding this system deeply unlocks advanced quantum mechanics and beyond.

Interactive Simulation

Run this Python code to compute the wave functions using Hermite polynomials and visualize the quantum harmonic oscillator energy levels. Try modifying the parameters to explore different systems!

Quantum Harmonic Oscillator Wave Functions

Python

Calculate and visualize energy eigenstates using Hermite polynomials

harmonic_oscillator.py100 lines

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import hermite
from scipy.special import factorial

# Physical constants
hbar = 1.055e-34  # J·s
m = 1.67e-27      # proton mass (kg)
omega = 1e14      # angular frequency (rad/s)

# Characteristic length scale
x0 = np.sqrt(hbar / (m * omega))

# Position array (in units of x0)
x = np.linspace(-5, 5, 1000) * x0

def psi_n(n, x_vals):
    """Wave function for quantum harmonic oscillator state n"""
    # Normalization constant
    N_n = 1 / np.sqrt(2**n * factorial(n)) * (m*omega/(np.pi*hbar))**(1/4)

# Hermite polynomial of order n
    H_n = hermite(n)

# Dimensionless position
    xi = x_vals / x0

# Wave function
    return N_n * H_n(xi) * np.exp(-xi**2 / 2)

# Calculate energies
def E_n(n):
    """Energy of nth level"""
    return hbar * omega * (n + 0.5)

# Display energies
print("=" * 60)
print("Quantum Harmonic Oscillator Wave Functions")
print("=" * 60)
print(f"\nParameters:")
print(f"  Mass m = {m:.2e} kg (proton mass)")
print(f"  Angular frequency ω = {omega:.2e} rad/s")
print(f"  Characteristic length x₀ = {x0*1e12:.3f} pm")
print("\nEnergy Levels:")
for n in range(5):
    E = E_n(n)
    E_eV = E / 1.602e-19
    print(f"  n = {n}: E = {E:.3e} J = {E_eV:.4f} eV")

# Plot wave functions and probability densities
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

# Plot wave functions
colors = ['blue', 'red', 'green', 'purple', 'orange']
for n in range(5):
    psi = psi_n(n, x)
    # Offset by energy level for clarity
    offset = E_n(n) / (hbar * omega)
    ax1.plot(x/x0, psi*x0*2e11 + offset, label=f'n={n}', color=colors[n], linewidth=1.5)
    # Draw energy level line
    ax1.axhline(y=offset, color=colors[n], linestyle='--', alpha=0.3, linewidth=1)

# Plot potential
x_pot = np.linspace(-5, 5, 100)
V = 0.5 * x_pot**2
ax1.plot(x_pot, V, 'k-', linewidth=2, label='V(x) = ½mω²x²')
ax1.fill_between(x_pot, 0, V, alpha=0.1, color='gray')
ax1.set_xlabel('Position (x/x₀)', fontsize=12)
ax1.set_ylabel('Energy (ℏω) / Wave function', fontsize=12)
ax1.set_title('Wave Functions with Energy Levels', fontsize=14)
ax1.legend(fontsize=9, loc='upper right')
ax1.grid(True, alpha=0.3)
ax1.set_ylim(-0.5, 6)
ax1.set_xlim(-5, 5)

# Plot probability densities
for n in range(5):
    psi = psi_n(n, x)
    prob = np.abs(psi)**2
    ax2.plot(x/x0, prob*x0, label=f'|ψ_{n}|²', color=colors[n], linewidth=1.5)
    ax2.fill_between(x/x0, 0, prob*x0, alpha=0.15, color=colors[n])

ax2.set_xlabel('Position (x/x₀)', fontsize=12)
ax2.set_ylabel('Probability density |ψ(x)|²', fontsize=12)
ax2.set_title('Probability Densities', fontsize=14)
ax2.legend(fontsize=9)
ax2.grid(True, alpha=0.3)
ax2.set_xlim(-5, 5)

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

print("\nKey Features:")
print("  • Ground state (n=0) has zero-point energy ℏω/2")
print("  • Energy spacing is constant: ΔE = ℏω")
print("  • Wave functions involve Hermite polynomials Hₙ(x/x₀)")
print("  • State n has exactly n nodes")
print("  • Probability extends into classically forbidden region")
print("\n✓ Plot saved as 'harmonic_oscillator.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Finite Well Scattering →

Quantum Mechanics Course

4. Quantum Harmonic Oscillator

Your Progress

Video Lecture

The Hamiltonian

Harmonic Oscillator Potential & Energy Levels

Algebraic Method: Ladder Operators

Deriving the Ladder Operator Algebra

Energy Eigenvalues & Eigenstates

Harmonic Oscillator Wave Function Evolution

Harmonic Oscillator Energy Calculator

Zero-Point Energy of a Diatomic Molecule

Ladder Operator Action

Video Lecture

Self-Check Question

Self-Check Question

Self-Check Question

Why the Harmonic Oscillator is Everywhere

1. Molecular Vibrations

2. Phonons in Solids

3. Quantum Field Theory

Video Lecture

Related Topics & Learning Path

Classical Harmonic Motion

Ladder Operators

Infinite Square Well

Coherent States

📝 Chapter Summary

Key Equations

Key Concepts

Interactive Simulation

Quantum Harmonic Oscillator Wave Functions