1cWave–Particle Duality
Discover how matter and light exhibit both wave and particle properties, and explore the revolutionary de Broglie hypothesis and double-slit experiment.
Wave–particle duality is the cornerstone of quantum mechanics — it explains electron microscopes, quantum tunneling, and the behavior of every particle in the universe at the fundamental level.
Lesson Overview
Wave–particle duality is one of the most profound ideas in physics: matter and light exhibit both wave-like and particle-like behavior depending on how they are observed. In 1924, Louis de Broglie proposed that all matter has an associated wavelength λ = h/p = h/(mv). This was confirmed by electron diffraction experiments. The double-slit experiment with electrons dramatically demonstrates wave behavior of matter, while the photoelectric effect shows particle behavior of light. Niels Bohr's complementarity principle states that wave and particle aspects are mutually exclusive but both necessary for a complete description.
Key Concepts
de Broglie Wavelength
λ = h/p = h/(mv) — every particle has an associated wavelength
Wave–Particle Duality
Light and matter exhibit both wave and particle properties
Double-Slit Experiment
Electrons fired one at a time create an interference pattern — wave behavior
Complementarity Principle
Wave and particle aspects cannot be observed simultaneously (Bohr)
Wave Function (ψ)
Mathematical description of a quantum particle; |ψ|² gives probability density
Davisson-Germer Experiment
First experimental confirmation of electron wave behavior (1927)
Calculate the de Broglie wavelength of an electron (m = 9.11×10⁻³¹ kg) moving at 2.0×10⁶ m/s.
A baseball (mass 0.145 kg) is thrown at 40 m/s. Calculate its de Broglie wavelength and explain why we never observe wave behavior for baseballs.
A photon has wavelength 500 nm. Calculate its momentum using the de Broglie relation.
Describe what happens in the double-slit experiment when electrons are fired one at a time through two slits. What pattern forms and what does it imply?
An electron is accelerated through a potential difference of 100 V. Find its de Broglie wavelength. (m_e = 9.11×10⁻³¹ kg, e = 1.6×10⁻¹⁹ C)
A proton (m = 1.67×10⁻²⁷ kg) moves at 3.0×10⁵ m/s. Find its de Broglie wavelength.
Hint: Use λ = h/(mv). Substitute h = 6.626×10⁻³⁴ J·s and calculate.
Why do we not observe wave-like behavior (diffraction, interference) for everyday objects like cars or people?
Hint: Calculate the de Broglie wavelength for a large mass — compare it to the size of any physical opening or obstacle.
In the double-slit experiment with electrons, what happens to the interference pattern if you place a detector at one slit to determine which slit each electron passes through?
Hint: Think about the complementarity principle — measuring particle position destroys wave information.
A particle has de Broglie wavelength 0.25 nm and mass 1.67×10⁻²⁷ kg. What is its speed?
Hint: Rearrange λ = h/(mv) to solve for v: v = h/(mλ).
How does the de Broglie wavelength change if the momentum of a particle is doubled?
Hint: λ = h/p — if p doubles, what happens to λ?
Key Vocabulary
de Broglie Wavelength
The wavelength associated with a moving particle: λ = h/p = h/(mv). All matter has wave properties at the quantum scale.
Example: An electron moving at 10⁶ m/s has λ ≈ 0.7 nm — comparable to X-ray wavelengths, enabling electron microscopy.
Wave–Particle Duality
The concept that quantum objects (photons, electrons, etc.) exhibit both wave-like and particle-like properties depending on the experiment.
Example: Electrons create interference patterns (wave) but are detected as point particles (particle) in the double-slit experiment.
Complementarity Principle
Bohr's principle that wave and particle aspects of quantum objects are mutually exclusive — measuring one property destroys information about the other.
Example: Determining which slit an electron passes through (particle info) destroys the interference pattern (wave info).
Wave Function (ψ)
A mathematical function describing the quantum state of a particle. The square of its magnitude |ψ|² gives the probability of finding the particle at a given location.
Example: Before measurement, an electron's wave function spreads through space; measurement collapses it to a definite position.
Interactive Practice — 5 Questions
The de Broglie wavelength of a particle is given by:
The double-slit experiment with electrons demonstrates:
According to the complementarity principle, wave and particle properties:
Why do macroscopic objects not show observable wave behavior?
The Davisson-Germer experiment (1927) confirmed:
Independent Practice
Calculate the de Broglie wavelength of a neutron (m = 1.675×10⁻²⁷ kg) moving at 1.5×10³ m/s.
An electron has de Broglie wavelength 0.10 nm. Find its kinetic energy in eV.
Describe the double-slit experiment with electrons. What result is observed and what does it prove about the nature of matter?
Compare the de Broglie wavelengths of an electron and a proton moving at the same speed. Which has the longer wavelength and why?
★ Electron microscopes use electrons instead of light to image objects. Explain why electrons can resolve features that visible light cannot, and calculate the minimum electron speed needed to resolve features of size 0.05 nm.
ChallengeCommon Mistakes
Thinking wave–particle duality means an object is "half wave and half particle" at all times.
A quantum object is neither purely a wave nor purely a particle. Which aspect it shows depends entirely on what measurement is performed.
Confusing de Broglie wavelength with photon wavelength — thinking only photons have wavelengths.
All matter has a de Broglie wavelength λ = h/p. Electrons, protons, and even whole atoms have been shown to diffract.
Math Tips
To find de Broglie wavelength after acceleration through voltage V: first find KE = eV, then v = √(2KE/m), then λ = h/(mv). Alternatively use λ = h/√(2meV) directly.