Particle–Wave Duality & Uncertainty Principle

Overall Course Structure

• Full 30-step roadmap of the “Fundamentals of Electrical Properties of Materials and Exercises” series
  • 1 Orientation & Introduction
  • 2 (Lecture 1) Solid-state electronics & quantum theory
  • 3 (Lecture 2) Particle–wave duality ⇐ current lecture
  • 4 (Exercise 1) Particle–wave duality
  • 5 – 30 … (Schrödinger eq., potential wells, tunnelling, quantum devices, free-electron model, statistical distributions, band theory, electrical/optical properties, metals, semiconductors, final comprehensive practice)
• Position of today’s content: supplies conceptual backbone for every later quantum topic

Today’s Learning Targets

• Deepen conceptual grasp of particle–wave duality
• Comprehend the essence of quantum mechanics via the Uncertainty Principle
• Lay foundation for Schrödinger equation (next lecture)
• Recommended supplementary text for the latter half of course (from Lecture 13 onward): “Introduction to Electronic Properties” by Tomohiro Hamaguchi, Maruzen

Historical Prelude: Atomic Spectra ⇒ Quantization Clues

• Planck’s quantum hypothesis (1900) implied integers are fundamental
• 1885 Balmer et al. discovered that inverse wavelengths of hydrogen emission lines obey an integer formula $\frac{1}{\lambda}=R_H\left(\frac{1}{m^2}-\frac{1}{n^2}\right),\;(m,n=1,2,3,\dots)$
  • $R_H=1.09677691\times10^{7}\;\text{m}^{-1}$ (Rydberg constant)
• Series classification (by final quantum number $m$)
  • $m=1$ → Lyman (UV)
  • $m=2$ → Balmer (visible)
  • $m=3$ → Paschen (IR)
• Integer-ruled wavelengths hinted discrete atomic states

Rutherford’s $\,\alpha$-Scattering Experiment (1909–11)

• Gold-foil scattering of $\alpha$ particles revealed
  • Tiny, massive, positively charged nucleus
  • Electrons orbiting around it
• Set stage for a planetary model yet inconsistent with classical electrodynamics (radiative collapse issue)

Bohr’s Quantum Postulates (1913)

• Introduced two ad-hoc conditions to rescue classical collapse problem
  1. Stationary Orbits: Electron can move only on specific, stable orbits; no radiation emitted while on them
  2. Frequency Condition: Photons emitted/absorbed only when electron jumps between stationary orbits
     $h\nu = |E<em>n-E</em>m|$
• Angular-momentum quantization
  $2\pi L = 2\pi m_e v a = n h \quad (n=1,2,3,\dots)$

Radius & Energy Quantization in Hydrogen

• Coulomb force balancing centripetal force:
  $\frac{m<em>e v^2}{a}=\frac{e^2}{4\pi\varepsilon</em>0 a^2}$
• Eliminating $v$ and using $L=m<em>e v a$ gives orbit radius $a</em>n=\varepsilon<em>0\frac{h^2}{\pi m</em>e e^2}n^2=a<em>0 n^2$$a</em>0=0.529\times10^{-10}\,\text{m}=0.529\,\text{Å}$ (Bohr radius)
• Total energy per level $E<em>n=\frac12 m</em>e v^2-\frac{e^2}{4\pi\varepsilon<em>0 a</em>n}=-\frac{m<em>e e^4}{8\varepsilon</em>0^2 h^2}\frac{1}{n^2}$
  • Ground-state energy $E_1=-13.6\,\text{eV}$
• Transition wavelength formula recovered
  $\frac{1}{\lambda}=\frac{m<em>e e^4}{8\varepsilon</em>0^2 h^3 c}\left(\frac{1}{m^2}-\frac{1}{n^2}\right)=R_H\left(\frac{1}{m^2}-\frac{1}{n^2}\right)$
• Orbit radius $\propto n^2$; Level energy $\propto 1/n^2$

Visual Keys

• Diagram of energy ladder: $n=1$ (ground) at $-13.6\,\text{eV}$, $n=2$ at $-3.39\,\text{eV}$, $n=3$ at $-1.51\,\text{eV}$ …
• Vertical arrows depict photon absorption/emission aligning with Lyman, Balmer, Paschen wavelengths (121.6 nm, 656.3 nm, etc.)

de Broglie Hypothesis: Matter Waves (1924)

• Attempted unified viewpoint: light shows wave–particle duality, so matter should too
• Proposed wavelength associated with particle of momentum $p$:
  $\lambda=\frac{h}{p}=\frac{h}{m v}$
• Momentum equivalence for photon: $p=\frac{h\nu}{c}$ (consistency check)
• Purely theoretical until diffraction proof

Electron Diffraction: Experimental Verification (Davisson–Germer, Thomson, etc.)

• Setup essentials
  • Electron gun → slits/crystal sample → fluorescent screen
  • Observed concentric rings/spots identical to X-ray diffraction, confirming wave nature
• Allowed measurement of electron de Broglie wavelength matching $\lambda=\frac{h}{p}$ within experimental error

Photon Diffraction & Single-Particle Interference

• Pinhole (Airy disk) experiment
  • Classical: circular diffraction rings when many photons pass (wave picture)
  • Quantum: send photons one-by-one → each exposes a single grain (particle picture); accumulation recreates classical pattern → probability distribution emerges from wavefunction
• Two-slit analogy for both photons and electrons: build-up of interference fringes from discrete impacts

Probabilistic Interpretation & Wavefunction (Born, 1926)

• Continuous electromagnetic field amplitude $E$ at point $P$ → intensity $I(P)\propto|E(P)|^2$
• Suggestion: $|E|^2$ equals probability density that a photon arrives at $P$
• Generalized to matter waves: define wavefunction $\Psi(x,y,z,t)$ such that
  $|\Psi(x,y,z,t)|^2\,dx\,dy\,dz$ gives probability of finding the particle in volume element $dx\,dy\,dz$ around $(x,y,z)$ at time $t$
• Unifies classical analog (continuous field) with quantum discrete detection events (photons/electrons)

Heisenberg Uncertainty Principle (1927)

• Measurement on microscopic scale unavoidably disturbs system
• Gedanken: locate electron with light of wavelength $\lambda$
  • Larger $\lambda$ ⇒ small photon momentum $p=\frac{h}{\lambda}$ ⇒ gentle probe but significant diffraction – position blurry $\Delta x\sim\lambda$
  • Smaller $\lambda$ ⇒ precise position but large photon momentum kicks electron $\Delta p_x\sim\frac{h}{\lambda}$
• Fundamental limit (independent of experimental skill) $\Delta x\,\Delta p_x\gtrsim h$
  • Often written $\Delta x\,\Delta p_x\ge\frac{\hbar}{2}$ with $\hbar=\frac{h}{2\pi}$ for rigorous derivations
• Denies simultaneous exact knowledge of conjugate variables (position & momentum, energy & time, etc.)
• Breaks down classical deterministic trajectory concept; introduces statistical causality

Complementarity Principle (Bohr)

• Mutually exclusive experimental arrangements (particle vs. wave, position vs. momentum) reveal complementary facets of quantum object
• No single description suffices; full understanding requires considering both views together
• Trajectory (defined sequence of positions & velocities) does not exist in quantum mechanics; only wavefunction evolves deterministically

Key Numerical & Symbolic Constants (for quick reference)

• $h=6.626\times10^{-34}\,\text{J·s}$, $\hbar=1.055\times10^{-34}\,\text{J·s}$
• $e=1.602\times10^{-19}\,\text{C}$
• $m_e=9.109\times10^{-31}\,\text{kg}$
• $\varepsilon_0=8.854\times10^{-12}\,\text{F·m}^{-1}$
• Bohr radius $a_0=0.529\,\text{Å}$
• Rydberg constant $R_H=1.097\times10^{7}\,\text{m}^{-1}$

Ethical / Philosophical Implications

• Classical determinism replaced by intrinsic probabilism
• Measurement plays constitutive role; reality not independent of observation at microscopic level
• Long–running debates (Einstein–Bohr) on completeness of quantum description

Connections to Later Lectures

• Schrödinger equation (Lecture 3) formalizes wavefunction dynamics
• Potential well, tunnelling, quantum devices rely on particle–wave duality and $\Psi$ interpretation
• Free-electron and band theory (Lectures 7-9) derive from de Broglie concept extended to solids

Illustrative Thought Experiments / Multimedia

• Photon two-slit video: https://www.youtube.com/watch?v=F75y3ZjHDyY
• Quantum eraser demonstration: https://www.youtube.com/watch?v=Le7S7B2a7iw

Self-Check / Study Prompts

• Explain in your own words particle–wave duality and give 3 experimental evidences
• Derive Bohr radius and ground-state energy from angular-momentum quantization and force balance
• List wavelengths for the first three lines of Lyman, Balmer, Paschen series using $R_H$ formula
• Compute de Broglie wavelength of 100 eV electron and compare with X-ray wavelengths
• Use uncertainty principle to estimate minimum kinetic energy of an electron confined within a nucleus radius $\sim10^{-15}\,\text{m}$
• Describe how complementarity resolves apparent contradictions between wave and particle pictures

Quick Mnemonics

• “BALmer → BALl-visible colors” (series in visible region)
• “Lyman starts with Light higher energy (UV)”

Wrap-Up

• Particle–wave duality, probabilistic interpretation, and uncertainty principle form triad that overturns classical physics and underpin all quantum-mechanical phenomena studied in subsequent lectures.