🎨 New Course Features Demo

This page demonstrates all the new interactive features added to the Quantum Mechanics course. These components are now available for use throughout the course to enhance learning.

1. Interactive Wave Function Visualizer

Visualize wave functions in real-time! Adjust quantum numbers, toggle between ψ(x) and |ψ|², and watch time evolution animations.

🎨 Interactive Wave Function Visualizer

Quantum Number (n = 1)

Green: Wave function ψ(x) |Cyan: Probability density |ψ|²

Red walls: Infinite potential barriers (ψ = 0 at boundaries)

Harmonic Oscillator Visualizer

Explore the quantum harmonic oscillator with its characteristic Gaussian envelope and Hermite polynomials:

🌊Quantum Harmonic Oscillator

Quantum Number n = 0

0 (Ground)5 (Excited)

What you're seeing:

• Gray curve: Parabolic potential V(x) = ½mω²x²
• Orange dashed line: Energy level E_n = (n + ½)ℏω
• Cyan/Green curve: Wave function ψ_n(x) or probability |ψ_n|²
• Orange markers: Classical turning points
• Note the zero-point energy: E₀ = ½ℏω even for n=0!
• Higher n states show more nodes (zeros) in the wave function

Hydrogen Atom Orbital Visualizer

Visualize the electron orbitals in hydrogen (1s, 2s, 2p, 3s, 3p, 3d) with proper quantum numbers:

⚛️Hydrogen Atom Orbitals

Select Orbital

Rotation: 0°

Quantum Numbers:

• n (principal): 1 — determines energy E = -13.6/n² eV
• l (angular momentum): 0 — determines orbital shape (s, p, d, f)
• m (magnetic): 0 — determines orientation in space

Color Legend:

• Blue/Cyan: Positive wave function
• Red/Orange: Negative wave function
• White dot: Nucleus (proton)
• Brightness indicates magnitude of ψ or |ψ|²

2. Physical Intuition Boxes

💡Why Can't Energy Be Zero?

Imagine trying to confine a particle to a box. Classically, you could have it sit perfectly still at the bottom with zero energy. But quantum mechanically, this violates the uncertainty principle!

If the particle were at rest (p = 0, Δp = 0), it would have definite momentum. But confining it to a box of width L means Δx ≤ L. The uncertainty principle Δx·Δp ≥ ℏ/2 would be violated!

Therefore, the particle must have some minimum kinetic energy (the zero-point energy E₁) just from being confined. This is a purely quantum effect with no classical analog.

3. Common Mistakes

Learn from typical errors students make when first encountering this material:

⚠️Common Mistakes to Avoid

❌

Mistake:

Thinking ψ(x) represents the probability of finding the particle at x

🤔

Why it's wrong:

ψ(x) is the wave function (probability amplitude), which can be complex and even negative. Probabilities must be real and non-negative.

✅

Correct approach:

|ψ(x)|² represents the probability density. The probability of finding the particle in a region [a,b] is ∫_a^b |ψ(x)|² dx.

❌

Mistake:

Assuming the particle is "bouncing back and forth" between the walls

🤔

Why it's wrong:

This classical picture is misleading. The particle doesn't have a definite trajectory. The standing wave is a superposition of left-moving and right-moving states.

✅

Correct approach:

The particle exists in a stationary state—the probability distribution |ψ(x)|² is time-independent for energy eigenstates.

❌

Mistake:

Setting boundary conditions as ψ(0) = ψ(L) = 0

🤔

Why it's wrong:

While ψ(0) = 0 is correct, the condition at the right boundary should be at the wall location, not necessarily at x=L if the well isn't positioned there.

✅

Correct approach:

For a well from 0 to L: ψ(0) = 0 and ψ(L) = 0. These boundary conditions lead to sin(nπ) = 0, giving the quantization condition.

4. Comparison Tables

Side-by-side comparisons help clarify key differences:

Classical vs. Quantum: Particle in a Box

How the quantum treatment differs fundamentally from classical mechanics

Aspect	Classical	Quantum
Energy	Any value E ≥ 0 allowed	Quantized: E_n = n²π²ℏ²/(2mL²)
Position	Particle has definite position x(t)	Position uncertain: described by \|ψ(x)\|²
Zero Energy	Particle at rest (E=0) allowed	Zero-point energy: E₁ ≠ 0 (uncertainty principle)
Probability	Uniform probability everywhere	Non-uniform: \|ψ(x)\|² with nodes at x=nL/k
Superposition	Not applicable	Particle can be in superposition of multiple states

5. Code Examples

Computational examples help build intuition and practical skills:

🐍

Infinite Square Well - Python Simulation

Plot wave functions and probability densities for the first three energy levels

python

import numpy as np
import matplotlib.pyplot as plt

# Parameters
L = 1.0  # Well width
N = 1000  # Number of points
x = np.linspace(0, L, N)

# Wave function for n=1,2,3
def psi_n(x, n, L):
    return np.sqrt(2/L) * np.sin(n * np.pi * x / L)

# Energy levels
def E_n(n, L, m=1, hbar=1):
    return (n**2 * np.pi**2 * hbar**2) / (2 * m * L**2)

# Plot wave functions
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

for n in [1, 2, 3]:
    psi = psi_n(x, n, L)
    prob = psi**2

    ax1.plot(x, psi, label=f'n={n}')
    ax2.plot(x, prob, label=f'n={n}')

ax1.set_xlabel('Position (x/L)')
ax1.set_ylabel('ψ(x)')
ax1.set_title('Wave Functions')
ax1.legend()
ax1.grid(True, alpha=0.3)

ax2.set_xlabel('Position (x/L)')
ax2.set_ylabel('|ψ(x)|²')
ax2.set_title('Probability Densities')
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# Print energy levels
print("Energy Levels (in units of π²ℏ²/2mL²):")
for n in range(1, 6):
    print(f"E_{n} = {n**2}E_1")

Output:

Energy Levels (in units of π²ℏ²/2mL²):
E_1 = 1E_1
E_2 = 4E_1
E_3 = 9E_1
E_4 = 16E_1
E_5 = 25E_1

💡 Run with: python script.py

6. Practice Problems with Solutions

Work through problems at your own pace. Click "Show Hint" for guidance, or "Show Solution" for the complete worked answer. Your progress is automatically saved!

📝 Practice Problems

Progress: 0/3 (0%)

⭐ Easy

A particle is confined in a 1D infinite square well of width L = 1 nm. Calculate the energy of the ground state (n=1) in eV.

Given: m_electron = 9.11 × 10^-31 kg, ℏ = 1.055 × 10^-34 J·s

⭐⭐ Medium

For an infinite square well, show that the wave functions for different quantum numbers (n ≠ m) are orthogonal.

⭐⭐⭐ Hard

A particle in an infinite well (width L) is initially in the state:

Ψ(x,0) = A[ψ₁(x) + 2ψ₂(x)]

where ψ_n are the energy eigenstates.

(a) Find the normalization constant A

(b) What is the probability of measuring E₁?

📊 Problem Set Statistics

Total Problems

Attempted

Completion

Notice the breadcrumb navigation at the top of the page and the table of contents in the sidebar. These help you track where you are in the course and navigate efficiently.

8. Concept Glossary with Tooltips

Hover over technical terms to see definitions! For example, try hovering over these terms:

In quantum mechanics, a Hermitian operator has real eigenvalues. An eigenstate satisfies the eigenvalue equation Â|ψ⟩ = a|ψ⟩.

The commutator [Â, B̂] = ÂB̂ - B̂Â determines whether two observables can be simultaneously measured. When operators commute, they share eigenstates.

A complete set of states allows any state to be expressed as asuperposition. The wave function collapse occurs upon measurement.

9. Downloadable Resources

Download course materials in various formats for offline study:

📥 Downloadable Resources

Study materials for offline learning and practice

📄

Summary Sheet

Key equations, concepts, and formulas in a 2-page reference card

pdf · 1.2 MB

📝

Problem Set

Practice problems with space for solutions (answers separate)

pdf · 850 KB

📚

Lecture Notes

Complete formatted notes with all derivations

pdf · 3.5 MB

📐

LaTeX Source

Editable LaTeX source code for notes and equations

tex · 45 KB

📓

Jupyter Notebook

Interactive Python notebook with visualizations and calculations

notebook · 120 KB

🐍

Python Examples

Complete code examples ready to run

code · 25 KB

💡 How to use these resources:

• Summary Sheets: Quick reference during problem-solving
• Problem Sets: Print and practice, check solutions online
• LaTeX Source: Modify equations for your own notes
• Notebooks: Run interactively with Jupyter or Google Colab
• Code Examples: Copy into your own projects

10. Reading Progress Tracking

Notice the progress bar at the top of the page and the floating progress indicator in the bottom-right corner. These track your reading progress and time spent, with estimates of time remaining!

Features:

Color-coded progress bar (red → orange → yellow → green)
Circular progress indicator with percentage
Actual time spent reading tracked automatically
Estimated time remaining based on your reading speed
Motivational messages as you progress
Completion celebration when you reach 99%+

11. Global Course Search

Press Cmd/Ctrl + K to open the global search. Search across all pages, equations, and concepts instantly!

Search capabilities:

📄 Search all course pages by title and content
📐 Find equations instantly (Schrödinger, commutator, etc.)
💡 Look up concepts and definitions
Filter results by type (pages, equations, concepts)
Keyboard navigation with arrow keys and Enter
Press ESC to close the search modal

🎯 What's Next?

These features are being rolled out across all pages in the course. Check out:

Your Progress Dashboard - Track your learning journey and earn achievements
Equation Reference Database - Searchable database of all important QM equations
Enhanced Infinite Well Page - See these features in action
Course Home - Return to the main course

Quantum Mechanics Course

🎨 New Course Features Demo

1. Interactive Wave Function Visualizer

🎨 Interactive Wave Function Visualizer

Harmonic Oscillator Visualizer

🌊Quantum Harmonic Oscillator

What you're seeing:

Hydrogen Atom Orbital Visualizer

⚛️Hydrogen Atom Orbitals

Quantum Numbers:

Color Legend:

2. Physical Intuition Boxes

💡Why Can't Energy Be Zero?

3. Common Mistakes

⚠️Common Mistakes to Avoid

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

4. Comparison Tables

Classical vs. Quantum: Particle in a Box

5. Code Examples

Infinite Square Well - Python Simulation

6. Practice Problems with Solutions

📝 Practice Problems

📊 Problem Set Statistics

7. Enhanced Navigation

8. Concept Glossary with Tooltips

9. Downloadable Resources

📥 Downloadable Resources

Summary Sheet

Problem Set

Lecture Notes

LaTeX Source

Jupyter Notebook

Python Examples

💡 How to use these resources:

10. Reading Progress Tracking

Features:

11. Global Course Search

Search capabilities:

🎯 What's Next?