Find the Eigenfunctions of Lz in Spherical Coordinates
Spin One-Half Matrices
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Fermions and Bosons

In analogy with orbital angular momentum, you can assume that m (the z-axis component of spin) can take the values –s, –s + 1, ..., s – 1, and s, where s is the total spin quantum number. For electrons, physicists Otto Stern and Walther Gerlach observed two spots, so you have 2s + 1 = 2, which means that s = 1/2. And therefore, m can be +1/2 or –1/2. So here are the possible eigenstates for electrons in terms of spin:


So do all subatomic particles have s = 1/2? Nope. Here are their options:

  • Fermions. In physics, particles with half-integer spin are called fermions. They include electrons, protons, neutrons, and so on, even quarks. For example, electrons, protons, and neutrons have spin s = 1/2, and delta particles have s = 3/2.

  • Bosons. Particles with integer spin are called bosons. They include photons, pi mesons, and so on; even the postulated particles involved with the force of gravity, gravitons, are supposed to have integer spin. For example, pi mesons have spin s = 0, photons have s = 1, and so forth.

So for electrons, the spin eigenstates are


For photons, the eigenstates are |1, 1 >, |1, 0 >, and |1, –1 >.

Therefore, the possible eigenstates depend on the particle you’re working with.

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