San José State University
Thayer Watkins

Silicon Valley

& Tornado Alley

The So-Called Spin of Subatomic Particles

Particle spin is a standard part of quantum physics, but there are admonitions that the spin of particle spin is different from that of macroscopic spin. Landau and Lifshitz assert that there is nothing like quantum spin at the macrosopic level. This raises the question of how it came to be that a quantum phenomenon was given the name spin when it does not correspond to macroscopic spin.

The Nature of the
Evidence for Particle Spin

If a particle with a charge distribution has spin then that particle generates a magnetic field. If there is an external magnetic field then there will be an interaction of the field generated by the particle with that field. That interaction imposes a force upon the particle.

If some of the particles of a collection has spin in one direction (parallel to the axis of rotation) then there will be other particles in the collection with spin in the opposite direction which are subject to a force in the opposite direction.

The net result is that due to the spins of the particles the opposite directions the beam would split into two parts. When Otto Stern and Walter Gerlach created an experiment involving a beam of silver atoms passing through a magnetic field of varying intensity they did find that the beam was split into two parts. The effect was due to the valence electrons of the silver atoms so the beam was effectively a beam of heavy electrons. This was evidence for electron spin but it was not recognized as such by Stern and Gerlach.

The title of the article published in 1922 by Stern and Gerlach was, translated into English, "Experimental Proof of Space-Quantization in a Magnetic Field." By space-quantization they meant that the angle of the angular momentum vector with the gradient of the magnet field may take on only discrete values.

In 1926 Samuel A. Goudsmit and George E. Uhlenbeck published in the British science journal Nature an article entitled "Spinning Electrons and the Structure of Spectra." In that article Goudsmit and Uhlenbeck showed the value of the idea of electron spin in explaining the multiplicity of spectral lines, as in the Zeeman effect.

The Theory Supporting
the Two-Valuedness of Spins

Spins in the three principal directions are represented by three 2×2 complex matrices, known as the Pauli matrices after their discoverer Wolfgang Pauli.

The Pauli matrices are:

σ0 = | 10 |
    | 01 |

σ1 = | 01 |
    | 10 |

σ2 = | 0-i |
    | i 0 |

σ3 = | 1 0 |
    | 0-1 |

σ0 is the 2×2 identity matrix and the symbol I will be used to represent this matrix hereafter.

These matrices have the properties that

σj² = I
for j=1, 2, 3.


σ1σ2 = iσ3
σ2σ3 = iσ1
σ3σ1 = iσ2

It is also true that

σ2σ1 = −iσ3
σ3σ2 = −iσ1
σ1σ3 = −iσ2

From these relations it is seen that

σjσk = −σkσj
and hence σjσk + σkσj = 0
for j, k = 1, 2, 3

where 0 is a 2×2 matrix of zeroes.

The Eigenvalues of
the Pauli Matrices

The squares of all the Pauli matrices are equal to the 2×2 identity matrix. This means that the eigenvalues of the squares of all of the Pauli matrices are equal to 1. This in turn means that the eigenvalues of the Pauli matrics σj for j = 1, 2, 3 are +1 or −1.

The magnetic moment of a particle is equal to the spin multiplied by μ, the Bohr magnetron. Thus the projection of the magnetic moment in each of the three principal directions can only have the values μ and −μ. This may be generalized.

Let {x1, x2, x3} be real numbers such that x1² + x2² + x3² = 1.

Now consider

1x1 + σ2x2 + σ3x3

Because of the properties of the Pauli matrices this expression reduces to

(x1² + x2² + x3²)I = I

This means that the expression (x1σ1+x2σ2+x3σ3) have eigenvalues of +1 or −1 and the corresponding magnetic moment can only have the values μ and −μ.

Any direction can be represented by a triple {x1, x2, x3}. This means that no matter what the direction is the magnetic moment it can only have the values +μ or −μ. Thus a beam of electrons passing through a magnetic field will split into two parts.

For further details of the Stern-Gerlach experiment and a translation of their article see Stern-Gerlach.

(To be continued.)

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