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Electrons possess intrinsic angular momentum (spin) with magnitude S = ℏ√(s(s+1)) where s = 1/2. The spin z-component is m_s = ±ℏ/2 (spin up or spin down). Spin produces a magnetic moment μ = −g_s(e/2m_e)S, where g_s ≈ 2 (anomalous in comparison to orbital angular momentum). This magnetic moment interacts with magnetic fields and causes level splitting.
Study spin-1/2 systems using Pauli matrices. Calculate expectation values of spin components. Understand spin as an intrinsic two-state system; appreciate that quantum spin has no classical analog.
Spin does not mean the electron is literally spinning (quantum spin is intrinsic, not due to rotation). The g-factor ≈ 2 is not exactly 2 due to quantum electrodynamic corrections. Spin-orbit coupling is not due to the spinning electron's magnetic moment interacting with the orbital motion magnetic field (it's a relativistic effect).
You've studied spin angular momentum as an abstract two-state quantum system. Now the physical stakes become clearer: spin isn't just a mathematical curiosity — it generates a real magnetic dipole moment that interacts measurably with external magnetic fields and with the electron's own orbital motion. The connection between spin and magnetism is what makes the electron a tiny magnet, and it drives much of the structure of atomic spectra.
The intrinsic magnetic moment of the electron is μ = −g_s (e/2m_e) S, where S is the spin angular momentum vector. The factor e/2m_e is the same that appears for orbital angular momentum (the gyromagnetic ratio), but the factor g_s ≈ 2 is not — it's the anomalous g-factor. For orbital angular momentum, the magnetic moment and angular momentum have g_L = 1; for spin, g_s ≈ 2.002. This factor of 2 was a complete mystery classically and was correctly predicted only by Dirac's relativistic quantum mechanics. The tiny departure from exactly 2 (the 0.002) is a quantum electrodynamic correction — one of the most precisely measured quantities in physics, tested to 12 significant figures.
In a magnetic field B, the interaction energy is U = −μ·B = g_s (e/2m_e) S·B. Since the z-component of spin is quantized as m_s = ±ℏ/2, the energy levels split into two: E = ±g_s (eℏ/2m_e) B/2 = ±μ_B g_s B/2, where μ_B = eℏ/2m_e is the Bohr magneton (≈ 9.27 × 10⁻²⁴ J/T). This splitting is the magnetic energy scale for electrons in atoms. The Stern-Gerlach experiment demonstrated exactly this splitting: silver atoms passed through an inhomogeneous magnetic field split into two beams, corresponding to m_s = +1/2 and m_s = −1/2.
Why does spin have g_s ≈ 2 and not 1, like orbital angular momentum? The short answer is that spin is an intrinsic property of the relativistic electron — it emerges naturally from Dirac's relativistic wave equation as a consequence of special relativity combined with quantum mechanics. There is no classical model: attempts to picture the electron as a spinning charged sphere fail because the equatorial velocity would exceed c for any reasonable electron radius. Spin is genuinely quantum mechanical with no classical analog. This is why the language is "intrinsic angular momentum" — it's a property the electron carries independent of any spatial motion, as fundamental to what an electron is as its charge or mass.