Infinite Square Well (Enhanced)
📋 Prerequisites
Required:
- Time-independent Schrödinger equation
- Boundary conditions
- Wave function normalization
- Basic calculus & trigonometry
Helpful:
- Fourier series
- Eigenvalue problems
- Linear algebra basics
The infinite square well (also called "particle in a box") is the simplest bound state problem in quantum mechanics. Despite its simplicity, it reveals fundamental quantum phenomena including energy quantization, wave-particle duality, and zero-point energy.
📜 Historical Context
Erwin Schrödinger solved the infinite square well in his groundbreaking 1926 paper "Quantisierung als Eigenwertproblem" (Quantization as an Eigenvalue Problem). This was one of the first exact solutions to his wave equation and demonstrated that quantization emerges naturally from boundary conditions - a revolutionary insight that particle energies aren't arbitrarily chosen but arise from wave confinement.
The solution validated de Broglie's wave hypothesis and provided the first mathematical framework for understanding atomic spectra, laying the foundation for modern quantum mechanics.
Video Lecture
Particle in a Box - Introduction
MIT OpenCourseWare lecture introducing the infinite square well and its fundamental importance in quantum mechanics. Excellent overview of the physics and mathematical framework.
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
The Setup
Consider a particle of mass $m$ confined to a one-dimensional region $0 \leq x \leq L$. The potential is:
Interactive Visualization
n = 1: Energy E_1 = 1 × E₁
Wave function has 1 half-wavelengths fitting in the well
Physical interpretation: Infinite potential walls mean the particle cannot exist outside [0, L]. The particle is perfectly confined - a mathematical idealization useful for understanding real quantum wells.
Solving the Schrödinger Equation
Complete Derivation of Energy Eigenstates
Assumptions:
- Infinite potential at x = 0 and x = L
- Particle confined to 0 < x < L
- Time-independent problem
- Non-relativistic (E << mc²)
Starting with:
Key Results
Video Lecture
Infinite Square Well - Energy Levels and Wave Functions
Clear explanation of quantized energy levels, wave functions, and the physical interpretation of the infinite square well solution
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Energy Eigenvalues
Energy Eigenfunctions
Orthonormal: $\langle \psi_n | \psi_m \rangle = \delta_{nm}$
Complete: Any function on [0,L] can be expanded as $f(x) = \sum_n c_n \psi_n(x)$
Real: Can choose eigenfunctions to be real (simplifies calculations)
Infinite Well Energy Calculator
Calculate energy levels and wavelengths for a quantum particle in a box
Formula:
Notes:
- Mass input is in units of electron mass (mₑ = 9.109×10⁻³¹ kg)
- For comparison: proton mass ≈ 1836 mₑ, neutron ≈ 1839 mₑ
- Typical quantum dots have L ~ 10-100 nm
- de Broglie wavelength: λ = h/p where p = √(2mE)
Video Lecture
Complete Solution of the Infinite Square Well
Step-by-step walkthrough of solving the Schrödinger equation with boundary conditions, deriving the quantized energy levels and normalized wave functions. Perfect complement to the derivation above.
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Important Physical Features
🔢 Energy Quantization
Energy can only take discrete values $E_n \propto n^2$. Not continuous!
Spacing increases with n: ΔE = E₂ - E₁ = 3E₁, but E₃ - E₂ = 5E₁
⚡ Zero-Point Energy
Minimum energy E₁ $\neq 0$. Particle cannot be at rest!
Consequence of uncertainty principle: Δx ~ L implies Δp ~ ℏ/L
🌊 Node Structure
nth state has (n-1) nodes inside the well
Higher n → more nodes → shorter wavelength → higher energy
📏 Size Dependence
Energy $\propto 1/L^2$. Smaller box → higher energy!
Explains why atoms are small but stable: confinement energy balances attraction
Worked Examples
Example 1: Ground State Energy of Electron in 1 nm Well
BASICProblem: An electron is confined to a one-dimensional infinite square well of width L = 1.0 nm. Calculate the ground state energy in electron volts.
Given:
- Well width: L = 1.0 nm = 1.0 × 10⁻⁹ m
- Particle: electron with mass mₑ = 9.109 × 10⁻³¹ kg
- Planck's constant: h = 6.626 × 10⁻³⁴ J·s
- Conversion: 1 eV = 1.602 × 10⁻¹⁹ J
Find: Ground state energy E₁ in eV
Example 2: Energy Absorption and Transitions
INTERMEDIATEProblem: An electron in a 1 nm infinite square well is initially in the ground state. A photon is absorbed, exciting the electron to the n = 3 state. What is the photon's wavelength?
Given:
- Initial state: n₁ = 1 (ground state)
- Final state: n₂ = 3
- Well width: L = 1.0 nm
- From Example 1: E₁ = 0.376 eV
- Photon energy: E_photon = hc/λ
- Constants: hc = 1240 eV·nm
Find: Photon wavelength λ
Example 3: Expectation Value of Position
ADVANCEDProblem: Calculate ⟨x⟩ for an electron in the n = 2 state of an infinite square well of width L.
Given:
- State: n = 2
- Wave function: ψ₂(x) = √(2/L) sin(2πx/L)
- Domain: 0 ≤ x ≤ L
Find: Expectation value ⟨x⟩
Self-Check Questions
Self-Check Question
Why can't the quantum number n be zero in the infinite square well?
Self-Check Question
How does the ground state energy change if the well width L is doubled?
Self-Check Question
Which of the following is TRUE about the wave functions ψₙ(x)?
Self-Check Question
An electron transitions from n = 3 to n = 1, emitting a photon. How does this photon's energy compare to a transition from n = 2 to n = 1?
Self-Check Question
What happens to the probability density |ψ(x)|² for high quantum numbers (n → ∞)?
🔬 Real-World Applications
Video Lecture
Quantum Wells in Modern Technology
Exploring how the particle-in-a-box model applies to real nanotechnology: quantum dots, semiconductor lasers, and nanoscale devices. See quantum mechanics in action in everyday technology!
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
1. Quantum Dots
Semiconductor nanocrystals (2-10 nm) confine electrons in all three dimensions, behaving like 3D infinite wells. The energy quantization determines their optical properties.
Example: Quantum dots are used in:
- QLED displays (Samsung TVs) - size determines color
- Medical imaging - fluorescent biomarkers
- Solar cells - improved efficiency through size tuning
- Quantum computing - qubit implementations
2. Semiconductor Devices
Quantum wells in transistors and lasers confine charge carriers, leading to discrete energy levels that can be engineered.
Technologies:
- Quantum Well Lasers - used in optical communications
- High Electron Mobility Transistors (HEMTs)
- Quantum Cascade Lasers - infrared and THz sources
- LEDs with quantum well active regions
3. Nanowires and Carbon Nanotubes
1D confinement in nanowires creates quantum wire behavior, with applications in:
- Nanoelectronics - ultra-small transistors
- Thermoelectric devices - waste heat recovery
- Sensors - extremely sensitive chemical detection
- Energy storage - battery electrodes
4. Molecular Systems
Conjugated molecules (like polyenes) approximate 1D boxes:
Example: β-carotene
The molecule has ~11 conjugated double bonds forming a "wire" of length L ≈ 3 nm. Treating π electrons as particles in a box correctly predicts the orange color (absorption ~450 nm)!
⚠️ Common Mistakes to Avoid
1. Using n = 0
❌ Wrong: "The ground state has n = 0"
✅ Correct: The ground state is n = 1. When n = 0, ψ ≡ 0 everywhere (no particle).
2. Confusing Energy and Wave Function
❌ Wrong: "Higher energy means larger ψ(x)"
✅ Correct: All ψₙ are normalized to 1. Higher energy means more oscillations (nodes), not larger amplitude.
3. Forgetting Unit Conversions
❌ Wrong: Using L = 1 nm directly in E = n²h²/(8mL²) with h in J·s
✅ Correct: Convert nm → m first, or use consistent units (e.g., atomic units).
4. Misunderstanding Transitions
❌ Wrong: "A transition from n = 3 to n = 1 releases energy E₃"
✅ Correct: Released energy is ΔE = E₃ - E₁ = 8E₁ (the difference, not E₃ itself).
5. Ignoring Boundary Conditions
❌ Wrong: "The wave function can be any sine or cosine"
✅ Correct: Only ψ(x) = √(2/L) sin(nπx/L) satisfies ψ(0) = ψ(L) = 0. The cosine terms are eliminated.
📝 Chapter Summary
Key Equations
Key Concepts
- Energy is quantized due to boundary conditions
- Zero-point energy E₁ > 0 (uncertainty principle)
- Energy spacing increases with n (E ∝ n²)
- Energy inversely proportional to well size (E ∝ 1/L²)
- nth state has (n-1) nodes inside well
- Eigenfunctions form complete orthonormal basis
Why This Matters
The infinite square well is the foundation for understanding:
- Quantum dots and nanotechnology
- Semiconductor devices and lasers
- Molecular electronic structure
- The correspondence principle (classical limit)
- More complex potential wells and barriers