Wave-Particle Duality

Light and matter travel like waves but arrive like particles. Fire them through a double slit one at a time and watch both behaviours at once.

Beginner

Double slit, one particle at a time

Each electron lands as a single dot at a random spot. Where it can land is set by a wave that passes through both slits. Watch the fringes build up from the dots.

Side view: source, double slit, screen

Screen, face on. Bars count hits; the line is the predicted |ψ|².

0 electrons detected

With only a few electrons the dots look random. Keep going: after a few hundred, stripes appear where the waves from the two slits add up.

de Broglie wavelength calculator

Every moving object has a wavelength λ = h / (m v). Choose an object or type your own mass and speed.

Wavelength λ
7.274 × 10⁻¹⁰ m
Momentum p = m v
9.109 × 10⁻²⁵ kg·m/s
Kinetic energy ½ m v²
4.554 × 10⁻¹⁹ J

Comparable to the spacing of atoms in a crystal (about 10⁻¹⁰ m), so crystals diffract it. This is how electron microscopes and neutron scattering work.

Waves that arrive as particles

Send electrons through two slits so slowly that only one is in the apparatus at a time. Each electron makes a single sharp dot on the screen, just like a particle would. Yet after thousands of dots the screen shows bright and dark stripes, an interference pattern that only waves make.

Quantum mechanics describes each electron by a wavefunction ψ. The wave passes through both slits and interferes with itself. The probability of detecting the electron at a point is |ψ|², so each individual landing spot is random, but the overall pattern is fixed. Bright fringes are places where the two parts of ψ arrive in step; dark fringes are where they cancel.

Add a detector that records which slit the electron used and the fringes disappear, leaving two plain bands. Knowing the path means only one part of ψ reaches the screen, so there is nothing to interfere with. You can see a wave pattern or which-path information, never both.

This trade-off is tied to the Heisenberg uncertainty principle: position and momentum cannot both be known exactly, Δx · Δp ≥ ħ/2. Pinning down which slit (Δx small) disturbs the sideways momentum by enough to wash out the fringes.

Key formulas

de Broglie wavelength

λ = h / p = h / (m v)

h
Planck constant, 6.626 × 10⁻³⁴ J·s
p = m v
momentum of the particle

Photon energy and momentum

E = h f, p = h / λ = E / c

f
frequency; c = speed of light

Double-slit fringe spacing

Δy = λ L / d

L
distance from slits to screen
d
slit separation

Uncertainty principle

Δx · Δp ≥ ħ / 2

ħ
h / 2π = 1.055 × 10⁻³⁴ J·s

About the model

The simulation uses a simplified screen pattern: each slit alone gives a broad band, and with both slits open the two amplitudes interfere with fringe spacing proportional to λ / d. Real patterns have the same structure.

How to use this visualizer

  1. 1
    Watch the side view: without a detector each particle spreads out as a wave from the source and then from both slits.
  2. 2
    Look at the screen. Early on the dots look random; after a few hundred hits the stripes appear. Raise the rate to speed things up.
  3. 3
    Compare the bars (counted hits) with the pink line (predicted |ψ|²). They match better the more particles you collect.
  4. 4
    Change the wavelength or slit separation. Longer λ or smaller d spreads the fringes apart.
  5. 5
    Turn on the which-path detector. Each particle now takes one definite slit, a lamp flashes, and the pattern becomes two bands.
  6. 6
    Use the calculator to compare wavelengths of an electron, a neutron and a baseball.

How big are matter waves?

ObjectMassλ = h / (m v)What it means
Electron at 10⁶ m/s9.11 × 10⁻³¹ kg7.3 × 10⁻¹⁰ mAtom-sized, diffracts off crystals
Thermal neutron, 2200 m/s1.67 × 10⁻²⁷ kg1.8 × 10⁻¹⁰ mUsed to probe crystal structure
C60 molecule, 200 m/s1.20 × 10⁻²⁴ kg2.8 × 10⁻¹² mInterference seen in 1999
Baseball, 40 m/s0.145 kg1.1 × 10⁻³⁴ mFar too small to ever detect

The evidence

Light as a wave

Young’s double slit (1801): light interferes, producing bright and dark fringes.

Light as particles

Photoelectric effect (Einstein, 1905): light delivers energy in lumps of E = hf.

Matter as a wave

Davisson and Germer (1927): electrons diffract off a nickel crystal exactly as de Broglie predicted.

One at a time

Tonomura (1989): single electrons sent through a biprism build up fringes dot by dot, as in this simulation.

Practice problems

Problem 1: Electron wavelength

An electron (m = 9.11 × 10⁻³¹ kg) moves at 2.0 × 10⁶ m/s. What is its de Broglie wavelength, and could it be diffracted by a crystal with atomic spacing 0.3 nm?
Show solution

p = m v = 9.11 × 10⁻³¹ × 2.0 × 10⁶ = 1.82 × 10⁻²⁴ kg·m/s

λ = h / p = 6.626 × 10⁻³⁴ / 1.82 × 10⁻²⁴

λ = 3.6 × 10⁻¹⁰ m = 0.36 nm

λ ≈ 0.36 nm, comparable to 0.3 nm, so yes, it diffracts

Problem 2: Why baseballs do not interfere

A 0.145 kg baseball is thrown at 40 m/s through a 1 m wide doorway. Estimate its wavelength and the angle of the first diffraction minimum, sin θ ≈ λ / a.
Show solution

p = 0.145 × 40 = 5.8 kg·m/s

λ = 6.626 × 10⁻³⁴ / 5.8 = 1.14 × 10⁻³⁴ m

sin θ ≈ λ / a = 1.14 × 10⁻³⁴ / 1 m

θ ≈ 10⁻³⁴ rad, utterly undetectable

The wave nature is still there, but for everyday masses the wavelength is about 10¹⁹ times smaller than a proton.

Discussion

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