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Visualisation Guide

This comprehensive guide explains all visualisations generated by the quantum solvers.

Table of Contents

Overview

Each solver generates multiple animated GIF files saved in the gifs/ folder. All animations are designed to illustrate fundamental quantum mechanical concepts through visual dynamics.

Animation Types

Energy Level Diagrams:

  • Progressive fade-in of quantum states
  • Shows how wavefunctions relate to energy levels
  • Combines spatial structure with energy quantisation

Single Eigenstate Evolution:

  • Time evolution of pure quantum states
  • Demonstrates quantum phase dynamics
  • Shows why eigenstates are "stationary"

Superposition Evolution:

  • Quantum interference in action
  • Probability density that moves and oscillates
  • Demonstrates how motion emerges from superposition

Wave Packet Dynamics:

  • Real-time propagation and spreading
  • Quantum tunneling phenomena
  • Classical-quantum correspondence

Infinite Square Well Visualisations

1. Energy Level Diagram Animation

Description: Progressive animation showing quantum states appearing one by one from n=1 to n=4. The left panel shows wavefunctions between the walls, while the right panel shows probability densities.

What You See:

  • n=1 (Ground State): Half sine wave, single hump in probability
  • n=2 (First Excited): Full sine wave with node at center, two probability peaks
  • n=3 (Second Excited): 1.5 sine waves with 2 nodes, three probability peaks
  • n=4 (Third Excited): 2 sine waves with 3 nodes, four probability peaks

Key Physics:

  • Energy quantisation: E_n = (n²π²ℏ²)/(2mL²)
  • Node theorem: State n has (n-1) internal nodes
  • Boundary conditions: ψ(0) = ψ(L) = 0
  • Orthogonality of quantum states

Educational Use:

  • Teaching energy quantisation
  • Illustrating wave-particle duality
  • Showing eigenstate structure
  • Demonstrating node patterns

2. Ground State Time Evolution

Description: Time evolution of the n=1 ground state showing quantum phase oscillation.

Top Plot - Wavefunction:

  • Blue line (Re(ψ)): Real part oscillates like a sine wave
  • Red line (Im(ψ)): Imaginary part oscillates 90° out of phase
  • Black lines: Infinite potential walls at boundaries

Bottom Plot - Probability Density:

  • Green line (|ψ|²): Remains completely static
  • Single peak at center of well
  • Zero at both walls

Key Physics:

  • Stationary state: |ψ(x,t)|² = |ψ(x)|²
  • Phase evolution: ψ(x,t) = ψ(x)e^(-iE₁t/ℏ)
  • Global phase has no observable effect
  • Lowest possible energy state

What to Notice:

  • Real and imaginary parts oscillate smoothly
  • Probability density NEVER changes
  • No actual particle motion occurs
  • Pure quantum phase rotation

3. Second State Time Evolution

Description: Time evolution of the n=2 first excited state.

Top Plot:

  • Wavefunction with one node at x=L/2
  • Two lobes oscillating with opposite phases
  • Node remains fixed at center

Bottom Plot:

  • Two probability peaks (symmetric)
  • Static in time
  • Particle more likely near x=L/4 and x=3L/4

Key Physics:

  • Higher energy: E₂ = 4E₁
  • Antisymmetric about center node
  • Demonstrates excited state structure
  • Still a stationary state

Comparison with n=1:

  • Faster oscillation (higher frequency ∝ E₂)
  • More complex spatial structure
  • Same stationary behavior

4. Superposition Animation

Description: Superposition of n=1 and n=2 states: ψ = (ψ₁ + ψ₂)/√2

Top Plot:

  • Complex interference patterns
  • Wavefunction shape changes over time
  • Asymmetric distributions appear

Bottom Plot:

  • Probability density oscillates!
  • Wave packet "sloshes" between walls
  • Moves left-right-left-right
  • Period: T = 2πℏ/(E₂ - E₁)

Key Physics:

  • Quantum beats at frequency ω = (E₂ - E₁)/ℏ
  • Interference creates dynamics
  • Non-stationary state
  • Demonstrates quantum superposition principle

What Makes This Special:

  • THIS IS WHERE MOTION APPEARS!
  • Probability density moves (unlike eigenstates)
  • Shows quantum-classical correspondence
  • Demonstrates that motion comes from superposition

Harmonic Oscillator Visualisations

1. Energy Level Diagram Animation

Description: Progressive animation of quantum states in parabolic potential V(x) = ½mω²x².

What You See:

  • Black dashed curve: Parabolic potential well
  • Horizontal lines: Energy levels at E_n = ℏω(n + ½)
  • Colored curves: Wavefunctions drawn at their energy heights
  • Right panel: Probability densities

Energy Levels:

  • n=0: E₀ = 0.5ℏω (zero-point energy)
  • n=1: E₁ = 1.5ℏω
  • n=2: E₂ = 2.5ℏω
  • n=3: E₃ = 3.5ℏω
  • n=4: E₄ = 4.5ℏω

Key Physics:

  • Equal energy spacing: ΔE = ℏω (unique property!)
  • Zero-point energy: Ground state has E > 0
  • Gaussian-like ground state
  • Hermite polynomial eigenfunctions

Symmetry Patterns:

  • Even n (0,2,4): Symmetric wavefunctions, even parity
  • Odd n (1,3,5): Antisymmetric wavefunctions, odd parity

2. Ground State Time Evolution

Description: Time evolution of n=0 ground state (Gaussian wavefunction).

Top Plot:

  • Single hump centered at x=0
  • Real and imaginary parts oscillate
  • Parabolic potential shown

Bottom Plot:

  • Gaussian probability distribution
  • Remains static (stationary state)
  • Maximum at equilibrium position

Key Physics:

  • Lowest energy state: E₀ = ½ℏω
  • Zero-point motion (cannot be at rest)
  • Minimum uncertainty state: Δx·Δp = ℏ/2
  • No nodes in wavefunction

Classical Analogy:

  • Like a vibrating spring at its lowest energy
  • But particle never stops moving (quantum)
  • Probability spread cannot be reduced further

3. First Excited State Time Evolution

Description: Time evolution of n=1 first excited state.

Top Plot:

  • Two lobes with node at x=0
  • Antisymmetric wavefunction
  • Oscillates with phase

Bottom Plot:

  • Two probability peaks (symmetric)
  • Zero probability at center
  • Static distribution

Key Physics:

  • Energy: E₁ = 3/2 ℏω
  • Odd parity: ψ(-x) = -ψ(x)
  • One node (n-1 rule holds)
  • Stationary state

Comparison with n=0:

  • Higher energy → faster phase oscillation
  • Node structure (0 vs 1 node)
  • Both have static |ψ|²

4. Superposition Animation

Description: Superposition of n=0 and n=1: ψ = (ψ₀ + ψ₁)/√2

Top Plot:

  • Wavefunction shifts position
  • Alternates between symmetric and asymmetric
  • Complex interference patterns

Bottom Plot:

  • Probability density oscillates!
  • Wave packet moves back and forth
  • "Breathing" motion (expands and contracts)
  • Period: T = 2π/ω (matches classical period!)

Key Physics:

  • Coherent state oscillation
  • Quantum beats at ω = ΔE/ℏ = ω
  • Demonstrates quantum harmonic motion
  • Classical-quantum correspondence

Special Property:

  • Oscillation period = classical period
  • Shows how quantum → classical for oscillators
  • Wave packet moves like classical particle
  • But maintains quantum superposition

Time Evolution Visualisations

1. Free Particle Evolution

Description: Gaussian wave packet in free space (no potential).

Top Plot:

  • Wave packet moves to the right
  • Packet spreads out over time
  • No potential (flat line)

Bottom Plot:

  • Probability density moves with velocity v = ℏk/m
  • Width increases (uncertainty principle)
  • Peak height decreases (conservation of probability)

Key Physics:

  • Wave packet spreading: Δx(t) = Δx(0)√[1 + (ℏt/2mΔx²)²]
  • Group velocity: v_g = ℏk₀/m
  • Dispersion relation: ω = ℏk²/2m
  • Uncertainty principle: Δx·Δp ≥ ℏ/2

What to Notice:

  • Packet moves with constant velocity (initially)
  • Spreading is inevitable (quantum mechanics)
  • Momentum uncertainty causes spreading
  • No forces acting on particle

2. Harmonic Oscillator (Wave Packet)

Description: Gaussian wave packet oscillating in harmonic potential.

Top Plot:

  • Parabolic potential curve
  • Wave packet bounces back and forth
  • Breathing motion (width oscillates)

Bottom Plot:

  • Probability density oscillates
  • Period matches classical: T = 2π/ω
  • Packet width varies periodically

Key Physics:

  • Coherent state evolution
  • Classical-quantum correspondence
  • Revival phenomena
  • Quasi-classical behavior

Classical Connection:

  • Behaves like classical particle
  • Period identical to classical oscillator
  • But maintains quantum width
  • Demonstrates correspondence principle

3. Quantum Tunneling

Description: Wave packet encountering potential step barrier at x=0.

Top Plot:

  • Step potential (flat, then elevated)
  • Packet approaches from left
  • Splits into reflected and transmitted parts

Bottom Plot:

  • Partial reflection: Some probability bounces back
  • Partial transmission: Some probability penetrates barrier
  • Even when E < V (classically forbidden!)

Key Physics:

  • Tunneling probability: T ∝ exp(-2κL) where κ = √[2m(V-E)]/ℏ
  • Reflection coefficient + Transmission coefficient = 1
  • Purely quantum phenomenon
  • No classical analog

Three Phases:

  1. Approach: Packet moves toward barrier
  2. Interaction: Packet encounters barrier
  3. Aftermath: Split into reflected and transmitted parts

Observable Effects:

  • Reflected packet moves backward
  • Transmitted packet has reduced amplitude
  • Some probability penetrates classically forbidden region
  • Total probability conserved

Understanding the Animations

Animation Components

Top Plot (Wavefunction):

  • Blue line: Re(ψ) - Real part of wavefunction
  • Red line: Im(ψ) - Imaginary part of wavefunction
  • Black dashed line: Potential V(x) (if present)
  • Title: Shows quantum state and time

Bottom Plot (Probability):

  • Green line: |ψ|² - Probability density
  • Area under curve = 1 (normalization)
  • Shows where particle is likely to be found

Reading the Title

Single Eigenstates:

Quantum State n=1, E=4.935, t=2.145
  • n: quantum number
  • E: energy eigenvalue
  • t: current time

Superpositions:

Superposition: ψ_1+ψ_2, t=2.145
  • Lists which states are combined
  • Shows current time

Interpreting Results

Stationary vs Non-Stationary States

Stationary States (Single Eigenstates):

  • ✅ Probability density |ψ|² is constant in time
  • ✅ Only phase changes: ψ(t) = ψ(0)e^(-iEt/ℏ)
  • ✅ No observable motion
  • ✅ Definite energy

Non-Stationary States (Superpositions):

  • ✅ Probability density |ψ|² changes with time
  • ✅ Observable motion and dynamics
  • ✅ Quantum interference effects visible
  • ✅ No definite energy (spread of energies)

What "Static Probability" Means

For single eigenstates, the probability density doesn't move because:

|ψ(x,t)|² = |ψ(x)·e^(-iEt/ℏ)|²
          = |ψ(x)|² · |e^(-iEt/ℏ)|²
          = |ψ(x)|² · 1
          = |ψ(x)|²  ← Time independent!

The phase factor has unit magnitude, so it cancels in the probability!

Why Superpositions Show Motion

For superposition ψ = c₁ψ₁ + c₂ψ₂:

|ψ|² = |c₁ψ₁|² + |c₂ψ₂|² + 2Re[c₁c₂*ψ₁*ψ₂e^(i(E₁-E₂)t/ℏ)]
                              ↑
                    Interference term oscillates!

This interference term creates time-dependent dynamics.

Troubleshooting

Common Issues

Problem: Animation shows flat line

  • Cause: Y-axis scale is wrong
  • Solution: Check that solve() was called before animating
  • Fix: Verify wavefunction normalization

Problem: Probability density > 1

  • Cause: Normalization error
  • Solution: Increase grid resolution (larger N)
  • Check: np.trapz(|ψ|², x) should equal 1.0

Problem: Non-zero at walls (infinite well)

  • Cause: Boundary conditions not enforced
  • Solution: Check T[0,0] and T[-1,-1] are set to large values
  • Verify: ψ(0) and ψ(L) should be exactly zero

Problem: Superposition looks static

  • Cause: States have same energy (degenerate)
  • Solution: Use states with different n values
  • Check: E₁ ≠ E₂ for quantum beats to occur

Problem: GIF file too large (>50 MB)

  • Cause: Too many frames or high dpi
  • Solution: Reduce n_frames or dpi parameter
  • Optimize: Use n_frames=100, dpi=80

Problem: Animation is choppy

  • Cause: Not enough frames
  • Solution: Increase n_frames parameter
  • Recommend: Use n_frames=150 or higher

Advanced Observations

Quantum Nodes

Pattern Recognition:

  • State n has (n-1) internal nodes
  • Nodes are points where ψ(x) = 0
  • Higher energy → more nodes → more oscillation

Physical Meaning:

  • More nodes = higher curvature = higher kinetic energy
  • Nodes represent destructive quantum interference

Classical Turning Points

For harmonic oscillator, classical turning points where E = V(x):

x_classical = ±√(2E/mω²)

Observation:

  • Quantum wavefunction extends beyond these points
  • Exponential decay in classically forbidden region
  • Tunneling probability ∝ exp(-∫κdx)

Quantum-Classical Correspondence

When Quantum Looks Classical:

  1. Large quantum numbers (n >> 1)
  2. Superposition of many states
  3. Coherent states in harmonic oscillator
  4. Wave packets in free space

Correspondence Principle: As n → ∞, quantum mechanics → classical mechanics

Tips for Educational Use

In Presentations

Energy Diagrams:

  • Use to introduce quantum quantisation
  • Show progressive build-up of states
  • Pause at each state to discuss

Single Eigenstates:

  • Emphasize stationary nature
  • Contrast real and imaginary parts
  • Explain phase vs. probability

Superpositions:

  • Highlight moving probability
  • Compare with eigenstates
  • Demonstrate interference

Tunneling:

  • Show before and after barrier
  • Discuss classical impossibility
  • Connect to real applications (STM, α-decay)

For Students

Key Questions to Ask:

  1. Why doesn't |ψ|² change for eigenstates?
  2. Where does motion come from in superpositions?
  3. How many nodes does state n have?
  4. What is zero-point energy and why does it exist?
  5. How does quantum tunneling violate classical physics?

Exercises:

  • Predict |ψ|² from ψ(x)
  • Calculate beat frequency from energies
  • Identify quantum numbers from node count
  • Estimate tunneling probability

File Specifications

Technical Details

Parameter Value
Format GIF (Graphics Interchange Format)
Frame Rate 20 fps
Resolution 80-100 dpi
Duration 5-10 seconds
File Size 1-5 MB per animation
Color Depth 256 colors per frame

Viewing Requirements

Compatible With:

  • All modern web browsers
  • Image viewers (Photos, Preview, etc.)
  • Presentation software (PowerPoint, Google Slides)
  • Markdown documents
  • LaTeX documents (with graphicx package)

Playback:

  • Loops automatically
  • No additional software required
  • Works offline
  • Cross-platform compatible

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