Matter Waves

1. Compton scattering

German-Swiss-American physicist Albert Einstein's theory of special relativity was first published in 1905[1]. This suggested that light has a momentum - which is classically equal to an object's mass multiplied by its velocity - even if photons have no rest mass.

Special relativity shows that energy is related to mass via e2 = p2c2 + m02c4, where e refers to energy, p to momentum, c to the speed of light, and m0 to an object's rest mass. An object with no rest mass, like a photon, will have an energy of e2 = p2c2, or e = pc.

American physicist Arthur Compton proved that photons do have the momentum Einstein predicted in 1922[2]. Compton did this by firing X-rays at aluminium foil.

When the X-rays hit the electrons in the outermost shell of the aluminium atoms, they transferred some of their angular momentum. The electrons gained enough energy to leave the atom, and the X-ray photon lost the same amount of energy. This process is now known as Compton scattering.

Diagram of Compton scattering, where a photon changes energy after colliding with a charged particle.

Compton scattering. Image credit: NASA's Imagine the Universe/Public domain.

When a photon collides with an electron and gains energy, the process is known as inverse Compton scattering.

Compton scattering is now utilised in radiobiology[3], and both Compton scattering and inverse Compton scattering are important in X-ray astronomy[4].

2. Electron waves

In 1924, French physicist Louis de Broglie used Einstein's equations to show that electrons can act like waves, just as photons can act like particles[5].

2.1 The wavelength of a photon

The wavelength of a photon is calculated by combining Einstein's equation for determining a photon's energy, with the Planck relation[6].

Using, e = pc (which is the energy of photons according to special relativity), and e = hν (which is the energy of photons according to the Planck relation), pc = hν.

A wavelength, λ, is a distance, and so equals Velocity × Time.

Light travels at a velocity of c, and Time = 1/ν, and so λ = Velocity × Time becomes λ =
c/ν
.
This means pc = hν becomes pc =
hc/λ
, and so:
λ =
h/p
.

2.2 The wavelength of particles

De Broglie proposed that particles also have a wavelength, and that this can be calculated using the same equation, except here the particle's momentum is equal to mV, where m is the particles' mass, and V is its velocity:

λ =
h/mV
.

2.3 The wavelength of electrons

De Broglie realised that electrons must orbit the nucleus like a standing wave - a wave that is constrained at each end. This means that only a whole number of wavelengths fit exactly around each orbit, and the angular momentum of each electron is quantised, in agreement with Danish physicist Niels Bohr's theory of the atom[7].

Bohr had previously shown that the angular momentum of each electron, L, is equal to the quantum number n multiplied by a constant.

Specifically, L = n
h/
= rp, where r is the radius of the orbit, which is assumed to be circular in Bohr's model.

The angular momentum derived from the theory of standing waves is the same:

using nλ = r (which is the equation for standing waves, where r is the circumference of a circle), and λ =
h/mV
(de Broglie's equation), n
h/mV
= r, and so n
h/
= rmV = rp = L.
A standing wave

A standing wave.
Image credit: Wiki Commons/Public domain.

A standing wave in a circle

A standing wave in a circle.
Image credit: Yuta Aoki/CC-SA.

If an electron drops to a lower energy level, its orbit has a smaller radius. This means that less full wavelengths can fit, and so the frequency and energy are lower. The difference in energy between shells is always a quantised number, because E = hν. The lowest minimum energy is always more than zero because a full number of wavelengths must fit in each shell.

Diagram showing standing waves in five circles. Each has a different number of peaks.

Standing waves in a circle for n = 2, 3, 4, and 5. Image credit: Ulrich Mohrhoff; 2/CC-SA, 3/CC-SA, 4/CC-SA, and 5/CC-SA.

De Broglie's theory was greatly extended by German physicist Werner Heisenberg[8] and Austrian physicist Erwin Schrödinger[9], in 1925-1926.

American physicists Clinton Davisson and Lester Germer proved that electrons have wave-like properties in 1927[10].

Davisson and Germer measured the wavelength of electrons by firing a beam of electrons at a nickel crystal, which acts like a diffraction grating, and then measuring the angles they were deflected by.

The wave-like nature of electrons meant that electron microscopes could be built in the 1930s[11]. The fact that electrons are more massive than photons means they have a smaller wavelength, and this is why electron microscopes have a better resolution than microscopes that use light.

In 1961, German physicist Claus Jönsson performed British natural philosopher Thomas Young's double-slit experiment with electrons, and found that they behave the same way as photons[12]. This experiment has since been conducted on larger particles and molecules. In 2012, Thomas Juffmann and physicists from University of Vienna in Austria conducted the experiment on molecules containing over 100 atoms[13].

By 1993, electron microscopes could be used to create images of individual atoms on metallic surfaces, known as quantum corrals[14]. Electron waves could be seen in these images, and look like ripples on the metal's surface.

A quantum corral, showing that electron waves can be produced by arranging atoms in a circle.

A sculpture showing a quantum corral made from iron atoms (the raised points) on copper, the ripples on the surface are electron waves. Image credit: Julian Voss-Andreae/CC-SA.

3. What is the wavelength of a person?

De Broglie's theory can be extended to show that all matter exhibits the same wave-particle duality as light. This means that everything in the universe can act like a wave[15].

λ =
h/mV
, and h = 6.626×10-34 m2 kg s-1.

This means that an object's wavelength gets smaller the more massive it is, and the faster it's moving.

If a person has a mass of 75 kg, and is jogging at 8 km/h (which is about 2.2 m/s), then

λ =
6.626×10-34/75 × 2.2
= 4.016×10-36 m.

This is about 700 billion, billion times smaller than the classical electron radius, which is about 2.8×10-15 m.

Diffraction works best if the slit is about the same size as the wavelength, and so this explains why we do not notice wave-like behaviour in people.

4. References

  1. Einstein, A., 1905, 'On the electrodynamics of moving bodies', Annalen der Physik, 17, pp.891-921, reprinted in in 'The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity', 1920, Courier Corporation.

  2. Compton, A. H., 1923, 'A quantum theory of the scattering of X-rays by light elements', Physical review, 21, pp.483-502.

  3. Saha, G. B., 2013, 'Physics and Radiobiology of Nuclear Medicine', Springer Science & Business Media.

  4. Frank, J., King, A., Raine, D., 2002, 'Accretion Power in Astrophysics', Cambridge University Press.

  5. De Broglie, L., 1924, 'On the Theory of Quanta', PhD thesis.

  6. Planck, M., 1901, 'On the law of the energy distribution in the normal spectrum', Annalen der Physik, 4, pp.90-100.

  7. Bohr, N., 1913, 'On the constitution of atoms and molecules', The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26, pp.1-25.

  8. Heisenberg, W., 1925, 'Quantum-theoretical re-interpretation of kinematic and mechanical relations', Zeitschrift für Physik, 33, pp.879-893.

  9. Schrödinger, E., 1926, 'An undulatory theory of the mechanics of atoms and molecules', Physical Review, 28, pp.1049-1070.

  10. Davisson, C. and Germer, L. H., 1927, 'Diffraction of electrons by a crystal of nickel', Physical Review, 30, pp.705-741.

  11. Palucka, T., 'Overview of Electron Microscopy', Caltech, last accessed 15-02-16.

  12. Jönsson, C., 1961, 'Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten' ('Electron Diffraction at Multiple Slits'), Zeitschrift für Physik, 161, pp.454-474.

  13. Juffmann, T., et al, 2012, 'Real-time single-molecule imaging of quantum interference', Nature nanotechnology, 7, pp.297-300.

  14. Crommie, M. F., Lutz, C. P., and Eigler, D. M., 1993, 'Confinement of electrons to quantum corrals on a metal surface', Science, 262, pp.218-220.

  15. Katz, D. M., 2016, 'Physics for Scientists and Engineers: Foundations and Connections, Extended Version with Modern', Cengage Learning.

Blog | Space & Time | Light & Matter | Mind & Multiverse | Timeline

RSS Feed | Images | About | Copyright | Privacy | Comments