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8.4: Electron Spin

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  • The two possible spin states of the electron are illustrated as vectors of equal length, one pointing up and right, representing vector S spin up, and the other pointing down and right, representing spin down. The two vectors are at the same angle to the horizontal. Spin up has a z component of plus h bar over two, and spin down has a z component of minus h bar over 2.
    Figure \(\PageIndex{4}\): The magnetic interaction between the electron and proton in the hydrogen atom is used to map the spiral arms of the Pinwheel Galaxy (NGC 5457). (a) The galaxy seen in visible light; (b) the galaxy seen in 21-cm hydrogen radiation; (c) the composite image of (a) and (b). Notice how the hydrogen emission penetrates dust in the galaxy to show the spiral arms very clearly, whereas the galactic nucleus shows up better in visible light (credit a: modification of work by ESA & NASA; credit b: modification of work by Fabian Walter).

    A complete specification of the state of an electron in a hydrogen atom requires five quantum numbers: n, l, m, s, and \(m_s\). The names, symbols, and allowed values of these quantum numbers are summarized in Table \(\PageIndex{4}\).

    Table \(\PageIndex{4}\): Summary of Quantum Numbers of an Electron in a Hydrogen Atom
    NameSymbolAllowed values
    Principal quantum numbern1, 2, 3, …
    Angular momentuml0, 1, 2, … n – 1
    Angular momentum projectionm\(0, \pm 1, \pm 2, . . . \pm l\)
    Spins1/2 (electrons)
    Spin projection\(m_s\)\(- 1/2, \, +1/2\)

    Note that the intrinsic quantum numbers introduced in this section (\(s\) and \(m_s\)) are valid for many particles, not just electrons. For example, quarks within an atomic nucleus are also spin-half particles. As we will see later, quantum numbers help to classify subatomic particles and enter into scientific models that attempt to explain how the universe works.


    Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).