Section 02
The Idea, Step by Step
Start simple: is it a tiny ball, or a ripple?
Light can act like a stream of tiny bullets — photons that arrive one click at a time on a detector — and also like ripples spreading on a pond, bending around corners and overlapping into bright-and-dark stripes. The strange discovery of the 1920s is that matter does the same thing. Fire electrons one at a time at a wall with two narrow slits and each one lands as a single dot, like a bullet. But let thousands pile up and the dots slowly paint a striped interference pattern — the unmistakable fingerprint of a wave. Each electron somehow goes through both slits and overlaps with itself. That double life is wave–particle duality.
Build it up: everything moving carries a wavelength
Louis de Broglie proposed that any object with momentum $p$ (mass times velocity) has a hidden wavelength. The faster or heavier it moves, the shorter its wavelength:
de Broglie wavelength (simple form)
$$\lambda=\frac{h}{p}$$
Here $h=6.63\times10^{-34}\,$J·s is Planck's constant — a fantastically small number, which is why waviness only shows up for fantastically small things. A worked number: an electron given $100\,$eV of kinetic energy has $p=\sqrt{2mK}$, giving $\lambda\approx0.12\,$nm. That is about the spacing between atoms in a crystal — which is exactly why electron beams diffract off crystals and why electron microscopes can resolve atoms. A thrown baseball, by contrast, has $\lambda\sim10^{-34}\,$m: far too small to ever notice.
Go deeper (AP / intro-college): waves of probability
Non-relativistically $p=\sqrt{2mK}$, so $\lambda=h/\sqrt{2mK}$. Send that wave through two slits a distance $d$ apart and the bright fringes on a screen a distance $L$ away are spaced by $w=\lambda L/d$ — smaller wavelength means tighter stripes. The wave itself is the wavefunction $\psi$, and Born's rule says $|\psi|^2$ is the probability of finding the particle at each spot: the stripes are where landings are likely. Because position and momentum are Fourier partners, pinning down location ($\Delta x$ small) forces a broad spread of momentum ($\Delta p$ large) — the uncertainty principle $\Delta x\,\Delta p\ge\hbar/2$. In the sim, the KE and mass sliders set $\lambda$, while slit separation d, slit width a and screen distance L reshape the pattern.
Try this in the simulation above
In de Broglie mode, push the KE slider up and watch $\lambda$ shrink — high-energy electrons probe finer detail. Switch to Double Slit and increase the slit separation $d$: the fringes crowd together, just as $w=\lambda L/d$ predicts. Then open the Uncertainty view and notice that squeezing $\Delta x$ toward zero sends the minimum $\Delta p$ soaring — you can never sit at the bottom-left corner.
Section 03
Equations & Derivation
de Broglie Wavelength
$$\lambda=\frac{h}{p}=\frac{h}{\sqrt{2mK}},\quad p=\sqrt{2mK}\quad\text{(non-relativistic)}$$
Heisenberg Uncertainty Principle
$$\Delta x\,\Delta p\geq\frac{\hbar}{2},\quad\Delta E\,\Delta t\geq\frac{\hbar}{2},\quad\hbar=h/(2\pi)=1.055\times10^{-34}\;\text{J·s}$$
Double-Slit Interference (Electrons/Photons)
$$y_n=\frac{n\lambda L}{d}\;\text{(bright fringes)},\quad w=\frac{\lambda L}{d}\;\text{(fringe width)}$$
Born Rule
$$P(x)=|\psi(x)|^2,\quad\int_{-\infty}^{\infty}|\psi|^2\,dx=1\;\text{(probability density)}$$
Key Constants
| Symbol | Quantity | Unit/Value |
|---|
| $h=6.626\times10^{-34}$ | Planck constant | J·s |
| $\hbar=h/2\pi$ | Reduced Planck constant | J·s = 1.055×10⁻³⁴ J·s |
| $m_e=9.109\times10^{-31}$ | Electron rest mass | kg |
| $\lambda=h/p$ | de Broglie wavelength | m |
1
Every particle has a wavelength. de Broglie (1924): $\lambda=h/p$. Electrons (KE=100 eV): $\lambda\approx0.12$ nm — comparable to atomic spacing, enabling electron diffraction. Proton (100 eV): $\lambda\approx2.9$ pm. A 1 kg object at 1 m/s: $\lambda=6.6\times10^{-34}$ m — unobservably small. Macroscopic quantum effects vanish because $\lambda\ll$ atomic scales.
2
Double slit with electrons. Each electron passes through both slits and interferes with itself. The interference pattern builds up one electron at a time. Demonstrated by Tonomura et al. (1989) sending electrons one at a time through a biprism — interference pattern emerges gradually.
3
Uncertainty principle. Not a measurement limitation — a fundamental property of waves. A well-localised wavepacket requires many wavelengths (broad $k$-spectrum, broad $p$-spectrum). $\Delta x\Delta p\geq\hbar/2$. Confined electrons (atoms) have large $\Delta p$ and hence kinetic energy — this is why atoms don't collapse.
4
Wavefunction $\psi$. Complete quantum description. $|\psi(x)|^2$ is probability density. $\psi$ satisfies the Schrödinger equation. Measurement collapses $\psi$ to a definite value — the Copenhagen interpretation. Many-worlds: $\psi$ never collapses, all outcomes occur in branching universes.
Ref: Halliday, Resnick & Walker 10th Ed., Ch. 38; Griffiths — Introduction to Quantum Mechanics (2nd Ed.), Ch. 1–2; Feynman Lectures Vol. III, Ch. 1.
Section 05
Common Misconceptions
❌ Electrons are either particles or waves, depending on the experiment.
✅ Modern quantum mechanics says electrons are neither classical particles nor classical waves — they are quantum objects described by wavefunctions. In some experiments, particle-like properties (localisation, discrete detection) dominate; in others, wave-like properties (interference, diffraction) dominate. Both behaviours stem from the same wavefunction.
📖 Feynman Lectures on Physics Vol. III, Ch. 1.
❌ The uncertainty principle just reflects measurement limitations.
✅ The uncertainty principle is a fundamental property of the quantum state, not a result of clumsy measurement. Even a perfectly isolated particle in a quantum state has $\Delta x\Delta p\geq\hbar/2$. It arises because position and momentum are Fourier conjugates: localising a wavepacket in position requires a broad momentum spectrum.
📖 HRW 10th Ed., §38-5; Griffiths — Introduction to Quantum Mechanics, §1.6.
❌ Large objects also exhibit quantum effects if we look carefully enough.
✅ Decoherence makes macroscopic quantum effects effectively unobservable. Any interaction with the environment — even a single photon or air molecule — measures the particle's position and collapses the superposition. Decoherence time scales as $\sim(\lambda/\Delta x)^2\times(m/m_e)$, becoming negligible for macroscopic objects within femtoseconds.
📖 Zurek — Decoherence and the Quantum-to-Classical Transition, Rev. Mod. Phys. (2003).
Misconception research: Driver et al. — Making Sense of Secondary Science; Muller & Weissman (2002) Am. J. Phys.