← Wave Mechanics

# De Broglie Waves

Thursday, February 10, 2022

## De Broglie's Hypothesis

Louis de Broglie hypothesized that, just like how light behaves as both a wave and a particle, all matter has some wave-like property. Knowing $E=hf$ and $p=h/\lambda$ for light, de Broglie predicted that any particle moving with momentum $p$ has an associated wave with wavelength $\lambda$ such that

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

This is called the particle's de Broglie wavelength. These wavelengths are often far too small to be observed in a laboratory, but some waves (whose wavelengths are on the scale of atomic or nuclear sizes) can be observed.

### Consequences of de Broglie's hypothesis

Considering the idea that all particles have wavelengths, it is worth asking what wave exactly has the particle's de Broglie wavelength. It can be helpful to think of this as a wave that only presents itself when a wave experiment is done on the particle.

Another question is why these wavelengths hadn't be observed before de Broglie. Using experimental setups like the double slit experiment, it is impossible to observe the wave from ordinary objects. Only experimental verification of de Broglie's hypothesis has been done on the atomic scale.

## Experimental Evidence for de Broglie Waves

Experiments like the double slit experiment were not possible for atomic or subatomic particles until long after de Broglie's hypothesis (and still is not possible for some particles). Therefore, other experiments were conducted to verify the hypothesis.

### Particle diffraction experiments

Diffraction of light is often observed using a single slit. For light waves with a wavelength of $\lambda$ incident on a slit with width $a$, the diffraction minima occur at

$a\sin{\theta}=n\lambda$

Where $n$ is an integer greater than $1$.

To study electron wave diffraction, experiments use "slits" between atoms in a crystal instead of an artificial slit. Due to the similarities between the patterns of electrons found and the patterns of light found with X-ray diffraction experiments strongly suggest that electrons are behaving as waves.

Electron waves are also observed by passing a beam of electrons through a small aperture to scatter on a crystal of nickel, then observing them at some angle $\phi$ with a detector. The maxima for a diffraction grating with spacing $d$ is described as integer multiples of the wavelength:

$d\sin{\phi}=n\lambda$

This matched perfectly with the experimental observations of electron waves, further supporting the de Broglie hypothesis for electrons.

### Double-slit experiments for particles

Due to the technical challenge of creating double-slit experiments for particles like electrons, experiments to observe the wave-like nature of particles were not conducted until the mid-twentieth century. In 1961, however, a double-slit experiment was conducted with electrons by accelerating them through a potential of 50kV, and the resulting pattern was extremely similar to that of light waves.

The same experiment, albeit with a different distance between the slits, was performed with neutrons and even atoms, and all experiments point to these particles behaving as waves.

### Monumental significance

These experiments showed that, in general, particles could also behave as waves. This property is used extensively in modern science, such as in the electron microscope, or in neutron diffraction.

## Double-Slit Uncertainty

Issues arise when trying to measure which of the two slits a particle passes through during a double-slit experiment. Any measurement to determine which slit the particles pass through, such as an electromagnetic loop, destroys the wave behavior and the particles simply pass through and appear on the screen as two lines—no interference.

By measuring particle-properties like which slit the particle passes through, we only investigate the particle aspects and thus cannot observe the wave aspects. Conversely, when studying the wave aspects, we cannot observe the particle aspects.

The electron behaves as a wave or a particle, but never both simultaneously.

### Principle of complementarity

The principle of complementarity is that a complete description of a photon or any other particle cannot be made in terms of only its particle properties or only its wave properties; both aspects must be considered. Additionally, both properties cannot be observed simultaneously. Different experiments suggest different identities.