Wave-particle duality (4.5.3)
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In 1924, de Broglie proposed that matter exhibits both wave-like and particle-like properties.
de Broglie’s hypothesis is known as wave–particle duality: particles can behave as waves and waves can behave as particles under different circumstances. Wave-particle duality applies to all matter.
The wavelength of a particle is known as the de Broglie wavelength.
Electromagnetic waves exhibit wave–particle duality.
The photoelectric effect provides evidence for light behaving as a particle. When light of high enough frequency is incident on a metal’s surface, electrons are emitted. This can only be explained by the light behaving as a stream of particles (photons).
Photons can also demonstrate wave properties. Light will diffract around obstacles and slits; it can also interfere with itself constructively and destructively, producing interference patterns.
Electrons also exhibit wave–particle duality, showing both particle-like and wave-like behaviour in different circumstances.
When behaving as particles, electrons can be accelerated by electric and magnetic fields due to their charges, and their motion can be described using classical mechanics.
Electrons can also display wave properties. When a beam of electrons is fired through very narrow openings, they can diffract and produce interference patterns.
To observe the wave nature of particles, the particles must exhibit wave behaviour, specifically producing an interference pattern. Such patterns are caused by diffraction and interference, both of which are properties of waves.
Diffraction effects are most pronounced when the de Broglie wavelength of the particles is of the same order of magnitude as the openings they pass through.
Graphite consists of layers of carbon atoms with a separation of This structure acts as an extremely fine diffraction grating, allowing the electrons to pass through and diffract.
The following experimental setup demonstrates electron diffraction.
- Electrons are fired from an electron gun in the direction of a thin piece of graphite and accelerated by an electric field so that their de Broglie wavelength is the same order of magnitude as the carbon atom spacing.
- The electrons diffract as they pass through the graphite in the same way as waves diffract when passing through a diffraction grating.
- A fluorescent screen is placed behind the piece of graphite to detect the electrons that emerge from it.
- An interference pattern consisting of concentric rings is observed due to the interference of the electrons.
Unlike a normal diffraction grating, where all openings are aligned, graphite has openings in random orientations, so overlapping diffraction from all the openings produces rings instead of discrete spots.
The setup is contained within a vacuum tube, ensuring that the electrons are not obstructed by air particles.
The wavelength of a particle is related to its momentum by the de Broglie equation, which states that:
where:
- is the momentum of the particle in
- is the Planck constant
- is the wavelength of the particle in
The de Broglie equation shows that the momentum of a particle is inversely proportional to its wavelength:
For particles travelling at the same speed, a greater mass results in a shorter wavelength.
The equation for the momentum of a particle in is:
Where:
- is the mass of the particle in and
- is the speed of the particle in
The de Broglie wavelength of a particle is inversely proportional to its momentum, and this can be observed in the electron diffraction experiment.
Slower electron acceleration by the electric field leads to lower momentum upon reaching the graphite grating. Consequently, the electrons exhibit a larger de Broglie wavelength, which is a closer match to the atomic spacing in the graphite. This results in increased diffraction and a broader interference pattern.
A greater momentum, which corresponds to faster electrons, leads to a smaller de Broglie wavelength. Consequently, the electrons diffract less, producing a narrower interference pattern.
Macroscopic objects, such as a football, have a large mass and therefore a large momentum. Since the de Broglie wavelength is inversely proportional to momentum, macroscopic objects have a negligible de Broglie wavelength:
- Diffraction is observed when the wavelength of the wave passing through an aperture is similar to the aperture width.
- Wave-like properties are not observed for macroscopic objects due to their small de Broglie wavelength.
An example of small de Broglie wavelengths is a tennis ball of mass travelling at a speed of , which has a momentum of:
Therefore, it has a de Broglie wavelength of:
This means it is physically impossible to observe the diffraction of a tennis ball as it is unable to fit through the opening that would cause diffraction.
Electrons are ideal for demonstrating the wave nature of particles because they can be accelerated to have de Broglie wavelengths comparable to atomic spacings, allowing for the observation of diffraction and interference patterns.