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1. Spin Quantum Mechanics

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   Historical Context and Emergence
   The necessity for an intrinsic angular momentum quantum number arose in the late 1920s following spectroscopic observations that could not be fully explained by the existing Schrödinger equation formalism. Specifically, the fine structure splitting in atomic spectra, particularly the anomalous Zeeman effect, suggested an internal degree of freedom. Uhlenbeck and Goudsmit proposed in 1925 that [electrons](/entries/…

2. Zeeman Effect

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   Anomalous Zeeman Effect
   The Anomalous Zeeman effect occurs when the total spin is non-zero ($S \neq 0$), as is typical for most atoms. Because both $\mathbf{L}$ and $\mathbf{S}$ contribute to the magnetic moment, and because spin-orbit coupling is often significant, the energy splitting is governed by the Landé g-factor ($g_J$).
   The resulting spectral line structure is often more complex than a simple triplet, potentially yielding five, seven, or more distinct lines, hence the term …