Wave-particle duality

Wave-particle duality is the idea that quantum entities (such as light) can exhibit both wave and particle like properties.

Light as a wave: In the early 1900s, the idea of light as a wave had been an accepted concept for over one hundred years. However, it had been challenged by Planck’s 1901 models of blackbody radiation and Einstein’s 1905 interpretation of the photoelectric effect.

Electrons as particle: Similarly, it was broadly accepted that electrons only had particle properties. It wasn’t until 1924 that it was suggested electron could also have wave properties.

Einstein and the photoelectric effect

The photoelectric effect describes the process whereby electrons are released from a material when electromagnetic radiation, such as light, hits the material. When the photoelectric effect was first observed by Heinrich Rudolf Hertz in 1887, a number of surprising aspects were noted:

When light hit the surface of the metal, electrons were emitted immediately with no time lag.
Increased light intensity led to an increased number of electron emitted. Increased intensity did not increase the maximum kinetic energy of the emitted electrons.
Red light (long wavelengths) did not result in electrons being emitted, no matter how intense.
Weak violet (low wavelengths) light resulted in only a few electrons being emitted but their maximum kinetic energies were greater than those for intense light of longer wavelengths.

These results were in direct contradiction with classical physics. Classical physics predicts that continuous light waves transfer energy to electrons and once they have accumulated enough energy, they would be emitted or released from the metal. Instead, the experimental results indicate that electrons are emitted only when light exceed a certain frequency, it is not dependent on intensity of duration.

Extending Planck’s proposal that energy is discrete or quantised, in 1905, Albert Einstein proposed a theory to explain these experimental results, the idea that light consisted of tiny packets of energy known as photons or quanta. Photons have energy $E = hv$ where $h$ is Planck’s constant and $v$ corresponds to frequency. Then, the maximum kinetic energy of an electron released from a material can be given by:

$E = hv - W$

$W$ is the work function, the is the minimum energy required to remove an electron from the surface of the material.

Emission of electron from a metal plate caused by photons [Ref: mmerevise.co.uk]

Maximum kinetic energy as a function of frequency [Ref: wikipedia]

de Broglie Wavelength

The idea that matter (such as electrons) could behave like a wave, was first proposed by Louis de Broglie in 1924. De Broglie considered both the theory of relativity and the photoelectric effect:

Relativity: $E = mc^2 = \sqrt{p^2c^2 + m_0c^4}$ → $p = \dfrac{E}{c}$

Photoelectric effect: $E = hf = \dfrac{hc}{\lambda}$ → $\dfrac{h}{\lambda} = \dfrac{E}{c}$

Giving:

$\lambda = \dfrac{h}{p}$

where $\lambda$ is the de Broglie wavelength, $h$ is Planck’s constant and $p$ is momentum

This hypothesis was confirmed through the Davisson-Germer experiment.

Davisson and Germer and the Davisson-Germer Experiment

Davisson-Germer Experiment was an experiment carried out primarily between 1923 to 1927 at Western Electric (now known as Bell Labs). In this experiment, electrons were directed at the surface of a nickel metal crystal, causing them to scatter and form distinctive diffraction patterns. The results of experiments not only confirmed the long-debated hypothesis of wave-particle duality but it contributed significantly to the advancement of wave mechanics, particularly in the context of the Schrödinger equation. It marked a pivotal moment in the history of quantum mechanics.

The Davisson-Germer experiment provided compelling evidence of particles exhibiting wave-like behaviour. While the Bragg Law had previously been applied to the diffraction of x-rays, this experiment was the first application to particle waves.

The experimental setup involved the design of an apparatus capable of measuring the energies of electrons scattered from the surface of a metal. A heated filament generated and released electrons, which were then accelerated by an applied voltage. These accelerated electrons were precisely directed at the nickel metal surface, where their scattering behaviour was observed. To detect and record the scattered electrons, an electron detector, also known as a Faraday cup, was used. Notably, this detector was mounted on an arc, allowing it to flexibly rotate and capture electron data at varying angles.

What was expected: Based on classical physics and considering electrons solely as particles, the expected result of the Davisson-Germer experiment would have been straightforward scattering of electrons from the nickel crystal surface with no diffraction pattern observed. In classical physics, particles such as electrons are expected to behave as discrete, localised entities without wave-like characteristics. Classical physics would have predicted that electrons directed at the crystal surface would follow simple trajectories, possibly bouncing off atoms or the crystal lattice in a manner similar to billiard balls. The electrons would have been expected to be scattered in various directions in no particular pattern.

What actually happened: The particles exhibited wave-like behaviour when interacting with the crystal lattice - intensity peaks were measured at specific angles. This phenomenon, reminiscent of wave interference patterns, played a pivotal role in reinforcing the concept of wave-particle duality. Notably, when the angle between the incident electron beam and the scattered electron beam was precisely 50 degrees, their electron detector registered a significant peak in the number of scattered electrons observed.

It had been shown in 1912 that the periodic crystal structure of a lattice serves as a type of three-dimensional diffraction grating. In order to determine the wavelength at the angle of maximum reflection, we can use Bragg’s law:

\begin{aligned} 2d sin (90 - \dfrac{\theta}{2}) &= n \lambda \\ \lambda &= 2 \times 0.91 \times sin(65)\\ \lambda &= 1.65 \end{aligned}

$\theta$ is the angle between the incident beam and the surface, in this case 50 degrees. $d$ corresponds to the lattice spacing in the crystal, which is 0.91 Angstroms and $n$ is the diffraction grating. In this case, $n = 1$ .

Applying the de Broglie wavelength formula, with $E = 54 eV$ and $m$ being the electron mass:

\begin{aligned} \lambda &= \dfrac{h}{p} = \dfrac{h}{mv} = \dfrac{h}{\sqrt{2mE}} = \dfrac{6.6 \times 10^{-34}}{\sqrt{2 \times 9.1\times10^{-31}\times54\times1.6\times 10^{19}}} \\ \lambda &= 1.66 Å \end{aligned}

With both the classical application of Bragg’s Law and the use of the de Broglie wavelength formula, we arrive at similar solutions for the electron wavelength. The Davisson-Germer experiment provided compelling evidence of the wave-particle duality of particles by demonstrating that electrons, traditionally considered as particles, exhibited wave-like behaviour when interacting with the crystal lattice. This wave-like behaviour was manifested in the form of diffraction patterns, confirming the dual nature of electrons.

References and further reading

Photoelectric effect wiki

Davisson-Germer experiment wiki

Matter wave (de Broglie wavelength) wiki

Hyperphysics - Davisson-Germer Experiment