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10: Addition of Angular Momentum

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    Consider an electron in a hydrogen atom. As we have already seen, the electron’s motion through space is parameterized by the three quantum numbers \(n\), \(l\), and \(m\). (See Section [s10.4].) To these we must now add the two quantum numbers \(s\) and \(m_s\) that parameterize the electron’s internal motion. (See the previous chapter.) Now, the quantum numbers \(l\) and \(m\) specify the electron’s orbital angular momentum vector, \({\bf L}\), (as much as it can be specified) whereas the quantum numbers \(s\) and \(m_s\) specify its spin angular momentum vector, \({\bf S}\). But, if the electron possesses both orbital and spin angular momentum then what is its total angular momentum?

    • 10.1: General Principles of Angular Momentum
      This page covers angular momentum operators in quantum mechanics, particularly orbital (\(L\)) and spin (\(S\)) angular momentum. It explains their commutation relations, leading to total angular momentum (\(J = L + S\)), and notes that \(L\) and \(S\) commute. The page discusses measurement properties, allowing simultaneous measurements of squared magnitudes of \(L\) and \(S\) with \(J_z\), or the squared magnitude of \(J\) with \(J_z\).
    • 10.2: Angular Momentum in Hydrogen Atom
      This page explores the quantum mechanics of electrons in hydrogen atoms, highlighting the coupling of orbital and spin angular momentum. It discusses the wavefunction representation through spherical harmonics and spinors, and how to express higher composite states as linear combinations, utilizing coefficients. The text also focuses on the eigenstates of angular momentum, employing Clebsch-Gordon coefficients to combine states and deriving eigenstate relationships.
    • 10.3: Two Spin One-Half Particles
      This page explores the total spin angular momentum operator for a system of two spin-1/2 particles without orbital angular momentum. It details the compatible measurements of spin, the corresponding quantum numbers, and the outcomes of combining the particles, leading to triplet (s=1) and singlet (s=0) states. Additionally, the page introduces Clebsch-Gordon coefficients and explicitly presents the forms of these states.
    • 10.E: Addition of Angular Momentum (Exercises)
      This page covers electron spin states and measurements in quantum mechanics, focusing on angular momentum, spin, and total angular momentum. It discusses measurement outcomes, probabilities, and spatial probability densities related to hydrogen atoms. The implications of potential energy in neutron-proton systems, the behavior of entangled electron pairs, and correlations in measurements between two electrons in a singlet state are also examined.

    This page titled 10: Addition of Angular Momentum is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard Fitzpatrick via source content that was edited to the style and standards of the LibreTexts platform.