Chapter 12. Quantum Spin

12.1 Discovery of electron spin

In 1922, German physicists Otto Stern and Walther Gerlach conducted an experiment to test the Bohr-Sommerfeld model of the atom[1] (discussed in Chapter 10 and Chapter 11). They passed a beam of silver atoms - which have a single electron in their outer shell - through a magnetic field with positive and negative regions. They then measured how the atoms were affected by the field.

If the orbits of electrons can have any orientation, and they are distributed randomly, then they will be deflected by a continuous range of values. This is the classical prediction. If they only have a limited number of orientations, then they will only be deflected by a limited number of angles. The Bohr-Sommerfeld model predicted an odd number of deflections, 1 in this case.

Stern and Gerlach found that neither theory was correct; although the electron orbits were quantised, the electrons were deflected by two values. This means that outer electrons with the same m value were divided into two groups, defined by a new quantum number (s).

Diagram showing predicted and actual results of the Stern and Gerlach experiment. The classical prediction is that electrons will fill the screen, the Bohr-Sommerfeld model predicts that they will be deflected by one value, and the actual results show they are deflected by two.

Figure 12.1
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Predictions and results for the Stern and Gerlach experiment.

The maximum number of m values can be found using maximum number = 2+1, and so assuming this is also the case for s values:

Maximum number of spin values = 2s+1 (12.1)

If the maximum number is equal to 2, then s must equal 1/2. In 1925, Austrian physicist Wolfgang Pauli described the atom as having a “two-valuedness” that could not be described classically.[2]

German-American physicist Ralph Kronig and Dutch-American physicists George Uhlenbeck and Samuel Goudsmit all suggested that these two extra angular momentum values, designated +1/2 and -1/2, may be due to the electrons rotating as they orbit the nucleus, just as the Earth rotates as it orbits the Sun.[3] The electrons were considered to be rotating in two directions, either clockwise or anticlockwise, and so this quality was named ‘spin’.

This idea was criticised by Pauli because the electron would have to be moving faster than the speed of light in order for it to rotate quickly enough to explain their findings.[4] This would violate German-Swiss-American physicist Albert Einstein’s theory of special relativity, which had been published in 1905[5] (discussed in Book I).

12.2 The Pauli exclusion principle

Pauli devised the Pauli exclusion principle in 1925. This states that no two electrons can share the same quantum state at the same time.[6] This means that no two electrons in a single atom can have the same n, , m, and s numbers.

This was later extended to show that all particles or atoms with a total spin number that is fractional obey the Pauli exclusion principle, whereas all particles or atoms with a total spin number that is a whole number do not. The former were named fermions and the latter bosons.

Bosons include photons, and some atoms, such as carbon-12 and Helium-4, and fermions include electrons, and atoms such as carbon-13 and helium-3. Carbon-12 and Carbon-13 and helium-3 and helium-4 are carbon and helium atoms that were known to have slightly different masses, despite having the same number of protons and electrons. These are known as isotopes. It was later shown that this extra mass comes from neutrons (discussed in Chapter 14).

Bosons obey Bose-Einstein statistics, which were developed for photons by Indian physicist Satyendra Nath Bose in 1924,[7] and generalised by Einstein the following year.[8,9] Fermions obey Fermi-Dirac statistics, which were independently discovered by Italian physicist Enrico Fermi[10] and British physicist Paul Dirac[11] in 1926. In 1927, American physicist David Dennison found that protons also have a spin of 1/2, and are therefore subject to Fermi-Dirac statistics.[12]

The spin-statistics relation - which states that all particles with a whole spin number are bosons, while all particles with a spin of half are fermions - was first formulated by Swiss physicist Markus Fierz in 1939[13].

The fact that bosons do not obey the Pauli exclusion principle means that an unlimited number of bosons can occupy the same energy state at the same time. This gives rise to a state of matter known as a Bose Einstein condensate, or macroscopic quantum wave function, and quantum effects, such as superconductivity and superfluidity (discussed in Chapter 13), become apparent on a macroscopic scale.[14]

12.3 Spin as an intrinsic property

Dirac provided a theoretical foundation for the concept of spin in 1928,[15,16] following the work of French physicist Louis de Broglie[17] (discussed in Chapter 15), German physicist Werner Heisenberg[18] (discussed in Chapter 16), and Austrian physicist Erwin Schrödinger[19] (discussed in Chapter 17). Dirac did this by developing a wave equation for the electron that is consistent with special relativity.

Dirac’s discovery marked the beginning of quantum field theory - the application of quantum mechanics to fields (discussed in Chapter 21). This explained the results of the Stern and Gerlach experiment by showing that although electrons do not physically rotate, they do have an intrinsic angular momentum - a contribution to the total angular momentum that is not due to the orbital motion of the particle - which we call spin. This accounts for why electrons interact with magnetic fields, explaining the anomalous Zeeman effect. Spin is now considered to be an intrinsic property, like mass and charge.

12.3.1 Spinors

Dirac’s wave equation makes use of mathematical objects known as spinors.[20] These can be thought of as the quantum analogue to vectors - a mathematical quantity that has both a value and a direction. Velocity, for example, is a vector composed of speed and direction.

Spinors rotate differently from vectors, rotating a spin-1/2 particle by 360°, for example, does not bring it back to the same quantum state, but to the opposite state. It needs to be rotated by 720° in order to get back to its original state.

A state with a spin of 0 looks the same whichever way it is rotated, like a circle. A state with a spin of 1 must be rotated by 360° before it goes back to its original state, like an Ace in a deck of playing cards, and a spin-2 particle needs to be rotated by 180°, like a Queen.

A circle.

Figure 12.2
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The circle looks the same however it’s rotated, like a spin 0 state.

An ace of spades playing card.

Figure 12.3
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The Ace needs to be rotated by 360° to return to the same state, like a spin 1 state.

A queen of spades playing card.

Figure 12.4
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The Queen needs to be rotated by 180° to return to the same state, like a spin 2 state.

12.4 References

  1. Gerlach, W., Stern, O., Zeitschrift für Physik A Hadrons and Nuclei 1922, 8, 110–111.

  2. Pauli, W., Zeitschrift für Physik A Hadrons and Nuclei 1925, 31, 373–385.

  3. Uhlenbeck, G. E., Goudsmit, S., Naturwissenschaften 1925, 13, 953–954.

  4. Enz, C. P., No Time to be Brief: A Scientific Biography of Wolfgang Pauli, Oxford University Press, 2010.

  5. Einstein, A. in The principle of relativity; original papers, The University of Calcutta, 1920 (1905).

  6. Pauli, W., Zeitschrift für Physik A Hadrons and Nuclei 1925, 31, 765–783.

  7. Bose, S. N., Zeitschrift für Physik 1924, 26, 178–181.

  8. Einstein, A., Sitzungsberichte der Preussischen Akademie der Wissenschaften Physikalischmathematische Klasse 1924, 261–267.

  9. Einstein, A., Sitzungsberichte der Preussischen Akademie der Wissenschaften Physikalischmathematische Klasse 1925, 3–14.

  10. Fermi, E., Rend. Lincei 1926, 3, 145–149.

  11. Dirac, P. A. M., Proceedings of the Royal Society Series A 1926, 112, 661–677.

  12. Dennison, D. M., Proceedings of the Royal Society of London Series A 1927, 115, 483–486.

  13. Fierz, M., Pauli, W., Proceedings of the Royal Society of London Series A 1939, 173, 211–232.

  14. Pitaevskii, L. P., Stringari, S., Bose-Einstein Condensation, Clarendon Press, 2003.

  15. Dirac, P. A. M., Proceedings of the Royal Society of London Series A 1928, 117, 610–624.

  16. Dirac, P. A. M., Proceedings of the Royal Society of London Series A 1928, 118, 351–361.

  17. De Broglie, L., PhD thesis, University of Paris, 1924.

  18. Heisenberg, W., Zeitschrift für Physik 1925, 33, 879–893.

  19. Schrödinger, E., Physical Review 1926, 28, 1049–1070.

  20. Steane, A. M., “An introduction to spinors”, arXiv preprint arXiv:1312.3824, 2013.

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