Quantum Mechanics Course

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

← Part II/Wave Functions

3. Wave Functions

Reading time: ~40 minutes | Pages: 8

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The wave function is the fundamental object in quantum mechanics, encoding all information about a system's state.

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Video Lecture

What is the Wave Function? - MIT OCW

Prof. Allan Adams explains the physical meaning of the wave function, its probabilistic interpretation, and why it's the central object in quantum mechanics.

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

Position Representation

In position space, the state vector becomes a function:

$$\psi(\vec{r},t) = \langle \vec{r}|\psi(t)\rangle$$

Probability Interpretation

$$P(\vec{r} \in dV, t) = |\psi(\vec{r},t)|^2 dV$$

Normalization: $\int|\psi|^2d^3r = 1$

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Visualizing Wave Function Time Evolution

Infinite Square Well

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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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Probability Density |ψ(x)|²

1D Probability Distribution

Interpretation:

  • Brighter regions = higher probability of finding the particle
  • Total integrated probability = 1 (normalization)
  • Quantum predictions differ dramatically from classical (uniform) distribution

Normalizing a Wave Function

BASIC

Problem: A particle in a 1D box has wave function ψ(x) = A sin(πx/L) for 0 < x < L. Find the normalization constant A.

Given:

  • ψ(x) = A sin(πx/L) for 0 < x < L
  • ψ(x) = 0 elsewhere
  • Normalization condition: ∫|ψ|² dx = 1

Find: Normalization constant A

Momentum Representation

Fourier transform relates position and momentum:

$$\tilde{\psi}(\vec{p},t) = \frac{1}{(2\pi\hbar)^{3/2}}\int e^{-i\vec{p}\cdot\vec{r}/\hbar}\psi(\vec{r},t)d^3r$$

Gaussian Wave Packet in Momentum Space

INTERMEDIATE

Problem: A particle has position-space wave function ψ(x) = A exp(-x²/2σ²). Find the momentum-space wave function φ(p).

Given:

  • ψ(x) = A exp(-x²/2σ²)
  • Fourier transform: φ(p) = (1/√(2πℏ)) ∫ exp(-ipx/ℏ)ψ(x)dx
  • Gaussian integral: ∫ exp(-ax² + bx)dx = √(π/a)exp(b²/4a)

Find: φ(p) in momentum space

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Video Lecture

Fourier Transforms and Momentum - MIT OCW

Prof. Barton Zwiebach discusses the momentum representation and the relationship between position and momentum wave functions via Fourier transforms.

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

Schrödinger Equation in Position Space

$$i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V(\vec{r})\psi$$

Probability Current & Continuity

Probability density: $\rho = |\psi|^2$

Probability current:

$$\vec{j} = \frac{\hbar}{2mi}(\psi^*\nabla\psi - \psi\nabla\psi^*) = \frac{\hbar}{m}\text{Im}(\psi^*\nabla\psi)$$

Continuity equation:

$$\frac{\partial\rho}{\partial t} + \nabla\cdot\vec{j} = 0$$

Properties of Wave Functions

  • ✓ Normalized: $\int|\psi|^2d^3r = 1$
  • ✓ Square-integrable: $\psi \in L^2(\mathbb{R}^3)$
  • ✓ Continuous (usually)
  • ✓ Single-valued
  • ✓ Global phase arbitrary: $\psi$ and $e^{i\theta}\psi$ physically equivalent
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Self-Check Question

What is the physical interpretation of |ψ(x,t)|²?

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Self-Check Question

Why must wave functions be square-integrable (∫|ψ|²dx < ∞)?

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Self-Check Question

If ψ(x) is narrow in position space, what happens to φ(p) in momentum space?

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Self-Check Question

What does the continuity equation ∂ρ/∂t + ∇·j = 0 represent physically?

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Video Lecture

Wave Packets and the Uncertainty Principle - MIT OCW

Prof. Allan Adams demonstrates how wave packets evolve, spread, and manifest the uncertainty principle through Fourier analysis.

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

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Wave Functions in Modern Technology

1. Electron Microscopy

Materials Science

Transmission Electron Microscopes (TEM) use the wave nature of electrons (ψ(r) for electrons) to achieve sub-angstrom resolution. The electron wave function's wavelength λ = h/p is much smaller than visible light, enabling atomic-resolution imaging.

Examples:

  • STEM - scanning transmission electron microscopy (0.5 Å resolution)
  • Protein structure determination for drug design
  • Semiconductor defect analysis (Intel, TSMC chip development)
  • Graphene and 2D material characterization

Impact: Essential for nanotechnology, materials science, and structural biology

2. Bose-Einstein Condensates

Atomic Physics

Ultracold atoms (<1 μK) occupy the same macroscopic wave function ψ(r,t), creating a 'matter wave' visible to the naked eye. The collective wave function exhibits quantum interference on macroscopic scales.

Examples:

  • MIT BEC experiments (2001 Nobel Prize - Ketterle, Cornell, Wieman)
  • Atom interferometry for precision measurements
  • Quantum simulation of condensed matter systems
  • Tests of quantum mechanics at macroscopic scales

Impact: New state of matter enabling quantum technologies and fundamental tests

3. Molecular Wavefunctions in Chemistry

Computational Chemistry

Electronic wave functions ψ(r₁,r₂,...,rₙ) determine molecular structure, bonding, and reactivity. Quantum chemistry software solves Schrödinger equations for multi-electron systems to predict chemical properties.

Examples:

  • Drug discovery - predicting protein-ligand binding (Schrödinger Inc.)
  • Materials design - catalysts, batteries, solar cells
  • Reaction pathway calculations - transition state theory
  • Spectroscopy prediction - IR, Raman, UV-Vis spectra

Impact: Multi-billion dollar computational chemistry industry for pharmaceuticals and materials

💡 Understanding real-world applications helps connect abstract quantum concepts to tangible technology and motivates further study.
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Related Topics & Learning Path

From: Wave Functions

📘 PrerequisitePart II

Postulates of QM

Postulate I defines wave functions as state vectors in Hilbert space

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📘 PrerequisitePart I

Hilbert Spaces

Mathematical structure underlying wave functions (L² spaces)

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📙 RelatedPart II

Measurement Theory

How measurement collapses the wave function

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📙 RelatedPart II

Uncertainty Principle

Fourier duality of ψ(x) and φ(p) leads to ΔxΔp ≥ ℏ/2

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📙 RelatedPart III

Infinite Square Well

First concrete example: solving for ψₙ(x) with boundary conditions

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📙 RelatedPart III

Free Particle

Plane wave solutions and wave packet spreading

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📕 AdvancedPart IX

Density Matrix

Generalization beyond pure states for mixed ensembles

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💡 Learning Path: Blue (prerequisites) → Purple (related) → Red (advanced)

📝 Chapter Summary

Key Equations

Position rep: ψ(r,t) = ⟨r|ψ(t)⟩
Probability: P(r)dV = |ψ(r,t)|²dV
Normalization: ∫|ψ|²d³r = 1
Fourier: φ(p) = (2πℏ)⁻³/²∫e⁻ⁱᵖ·ʳ/ℏψ(r)d³r
Current: j = (ℏ/m)Im(ψ*∇ψ)
Continuity: ∂ρ/∂t + ∇·j = 0

Key Concepts

  • ψ encodes complete information about quantum state
  • |ψ|² is probability density (Born interpretation)
  • ψ must be normalized, square-integrable, continuous
  • Position ↔ momentum via Fourier transform
  • Narrow in x → broad in p (uncertainty principle)
  • Probability is conserved (continuity equation)
  • Global phase e^(iθ) is unobservable

The wave function is the heart of quantum mechanics. It contains all knowable information about a system, evolves deterministically via Schrödinger's equation, but yields only probabilistic predictions upon measurement. Mastering wave functions is essential for understanding all quantum phenomena.

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