Quantum Spin

1. Quantum number ms

In 1922, German physicists Otto Stern and Walther Gerlach conducted an experiment to test the Bohr-Sommerfeld model of the atom[1]. 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.

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.

Predictions and results for the Stern and Gerlach experiment. Image credit: Helen Klus/CC-NC-SA.

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 ml value were divided into two groups, defined by a new quantum number: ms.

The maximum number of ml values can be found using the formula Maximum number = 2l + 1, and so assuming this is also the case for ms values, then for the maximum number to equal 2, ms must equal 1/2.

In 1925, Austrian physicist Wolfgang Pauli described the atom as having a "two-valuedness not describable 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].

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, l, ml, and ms 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.

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, become apparent on a macroscopic scale[14].

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], German physicist Werner Heisenberg[18], and Austrian physicist Erwin Schrödinger[19]. 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.

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.

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 magnitude and 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 circle.

The circle looks the same however it is rotated, like a spin 0 state. Image credit: Helen Klus/Public domain.

A queen of spades playing card.

The Queen needs to be rotated by 180° to return to the same state, like a spin 2 state. Image credit: Xiong/CC-SA.

An ace of spades playing card.

The Ace needs to be rotated by 360° to return to the same state, like a spin 1 state. Image credit: Byron Knoll/Public domain.

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.

4. References

  1. Gerlach, W. and Stern, O., 1922, 'Der experimentelle Nachweis des magnetischen Moments des Silberatoms' ('The experimental proof of the magnetic moment of the silver atom'), Zeitschrift für Physik A Hadrons and Nuclei, 8, pp.110-111.

  2. Pauli, W., 1925, 'Über den Einfluß der Geschwindigkeitsabhängigkeit der Elektronenmasse auf den Zeemaneffekt' ('About the influence of the velocity dependence of the electron mass to the Zeeman effect'), Zeitschrift für Physik A Hadrons and Nuclei, 31, pp.373-385, see also Pauli, W., 1946, 'Exclusion Principle and Quantum Mechanics', Nobel Lecture.

  3. Uhlenbeck, G. E. and Goudsmit, S., 1925, 'Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons' ('Replacement of the hypothesis of unmechanical forced by a claim relating to the internal behaviour of each individual electron'), Naturwissenschaften, 13, pp.953-954.

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

  5. Einstein, A., 1920 (1905), 'On the electrodynamics of moving bodies', in 'The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity', Courier Corporation.

  6. Pauli, W., 1925, 'Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren' ('About the context of the conclusion of the electrons in the atom groups with the complex structure of the spectra'), Zeitschrift für Physik A Hadrons and Nuclei, 31, pp.765-783.

  7. Bose, S. N., 1924, 'Plancks gesetz und lichtquantenhypothese' ('Planck's law and light-quantum hypothesis'), Zeitschrift für Physik, 26, pp.178–181.

  8. Einstein, A., 1924, 'Quantentheorie des einatomigen idealen Gases, 1 Abhandlung' ('Quantum Theory of the Monatomic Ideal Gas, Part I'), Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pp.261–267.

  9. Einstein, A., 1925, 'Quantentheorie des einatomigen idealen Gases, 2 Abhandlung' ('Quantum Theory of the Monatomic Ideal Gas, Part II'), Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pp.3–14.

  10. Fermi, E., 1926, 'On the Quantization of the Monoatomic Ideal Gas', Rend. Lincei, 3, pp.145-149.

  11. Dirac, P. A. M., 1926, 'On the Theory of Quantum Mechanics', Proceedings of the Royal Society, Series A, 112, pp.661–677.

  12. Dennison, D. M., 1927, 'A note on the specific heat of the hydrogen molecule', Proceedings of the Royal Society of London, Series A, 115, pp.483-486.

  13. Fierz, M. and Pauli, W., 1939, 'On relativistic wave equations for particles of arbitrary spin in an electromagnetic field', Proceedings of the Royal Society of London, Series A, 173, pp.211-232.

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

  15. Dirac, P. A. M., 1928, 'The quantum theory of the electron, Part I', Proceedings of the Royal Society of London, Series A, 117, pp.610-624.

  16. Dirac, P. A. M., 1928, 'The quantum theory of the electron, Part II', Proceedings of the Royal Society of London, Series A, 118, pp.351-361.

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

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

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

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

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