Wave-Particle Duality

The wave-particle duality of light

Light was considered to be propagating as waves until the early 20th century. Only 25 years before de Broglie found that particles had wave-like behaviour, Einstein had studied the photoelectric effect, assuming that light was composed of a small stream of particles with an energy equal to its frequency f and the Planck constant h. This revolutionised our understanding of light, which now could also be described as a particle.

The history of wave-particle duality

At the beginning of the 19th century, scientists believed light to be a particle. Even concepts, such as the vacuum and light being part of the electromagnetic spectrum, had not yet been established. The concepts of light having the properties of both a particle and a wave, and of small particles having the properties of light, led to several important developments:

• Thomas Young demonstrated how light is a wave phenomenon.
• Albert Einstein proposed that light is composed of small particles called quanta.
• Louis de Broglie developed a theory explaining that small particles have wave-like properties.
• Clinton Davisson, Paget Thomson, and Lester Germer conducted experiments on electron diffraction patterns.

The corpuscular theory

In the beginning, it was proposed that light is composed of small particles travelling in space. This theory tried to explain light as particles travelling through a medium that filled the universe and was named aether.

However, the corpuscular theory of light being small objects was not able to explain all the properties of light, such as how waves reduced their speed and changed their direction when entering water if they were not travelling through the water. An important argument against the corpuscular theory was this inability to explain the diffraction of light.

The experiments of Thomas Young

Experiments conducted by British scientist Thomas Young provided a new perception of light. The experiments were simple but also very smart. Passing a ray of light through a small aperture in a series of plates, he observed patterns of wave-like behaviour.

If light was a particle, it could simply pass through and would show over the open slits. If, however, light was a wave, it would spread after the slits, showing a pattern of interference. Young obtained an interference pattern, which confirmed that light behaved like a wave.

The contribution of Albert Einstein

Einstein proposed that light consists of small particles and that its energy depends on its frequency.

His ideas developed in connection with his work on the photoelectric effect. It was expected that more intense light would make the electrons jump more, but this did not happen. Only when the frequency of the light was increased did the electrons jump from the metallic plate.

Einstein, therefore, proposed that it was the energy of a particle called quantum that was impacting the metal plate and that it was this that was responsible for ejecting the electrons from the plate.

The contribution of Louis de Broglie

Having described how electrons disperse after impacting a crystal, de Broglie developed a theory in which he proposed that light behaves as both a wave and a particle. He found that the dispersion of the electrons presented a wave-like pattern and proposed a formula that connects the velocity and mass of particles with their wavelength.

Electron diffraction experiments

Clinton Davisson, Paget Thomson, and Lester Germer conducted experiments in which they fired electrons onto a crystal. The electrons did not collide against the crystal but rather passed through the material, showing a wave-like pattern after the impact.

These diffraction experiments conducted by Davisson and others were the final confirmation that electrons can behave like a wave.

What is the relationship between waves and particles?

De Broglie concluded that if electrons could behave like waves, particles had a wavelength. He linked the energy of the wavelength of the light particles to the energy of a particle moving with a certain kinetic energy. This tells us that the photon energy must be the energy given to the particle to put it into motion.

The energy of a photon or a wave

In the case of light that can be seen as an electromagnetic wave, its energy is inversely proportional to its wavelength, with smaller wavelengths having larger amounts of energy. In the calculation below, λ is the photon’s wavelength in metres, while h and c are the Planck constant and light’s velocity in a vacuum, with the following values:

\(h = 6.63 \cdot 10^{-34} m^2 kg/s = 3 \cdot 10^8 m/s\)

\[E_{photon} = \frac{h \cdot c}{\lambda}\]

The energy of a particle

Einstein derived a relationship between the energy of a particle and its mass ‘m’ given in kilograms. E is the energy given in joules, and c is the light’s velocity in a vacuum.

\[E_{particle} = m c^2\]

This says that the mass of a particle at rest has an energy equivalence.

The wavelength-particle energy relationship

We can simply say that the energies of both a particle and a photon are the same.

\[E_{particle} = E_{photon}\]

The mass is m, the particle velocity is v, the wavelength of the photon impacting the particle is λ, and h and c are the Planck constant and the velocity of light in a vacuum.

\[\frac{h \cdot c}{\lambda} = m c^2\]

To obtain the related wavelength of the particle, we equate both formulas and solve for the wavelength λ.

\[\frac{h \cdot c}{m c^2} = \lambda \]

Reducing this, we get:

\[\frac{h}{mc} = \lambda\]

Here, c can be exchanged with v, which is the proper velocity of the moving particle.

\[\frac{h}{mc} = \lambda\]

This wavelength is known as de Broglie’s wavelength of a particle.