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9: Spin Angular Momentum

Broadly speaking, a classical extended object (e.g., the Earth) can possess two different types of angular momentum. The first type is due to the rotation of the object’s center of mass about some fixed external point (e.g., the Sun)—this is generally known as orbital angular momentum. The second type is due to the object’s internal motion—this is generally known as spin angular momentum (because, for a rigid object, the internal motion consists of spinning about an axis passing through the center of mass). By analogy, quantum particles can possess both orbital angular momentum due to their motion through space (see Chapter [sorb]), and spin angular momentum due to their internal motion. Actually, the analogy with classical extended objects is not entirely accurate, because electrons, for instance, are structureless point particles. In fact, in quantum mechanics, it is best to think of spin angular momentum as a kind of intrinsic angular momentum possessed by particles. It turns out that each type of elementary particle has a characteristic spin angular momentum, just as each type has a characteristic charge and mass.

• 9.1: Spin Operators
Because spin is a type of angular momentum, it is reasonable to suppose that it possesses similar properties to orbital angular momentum. Thus, by analogy, we would expect to be able to define three operators that represent the three Cartesian components of spin angular momentum. Moreover, it is plausible that these operators possess analogous commutation relations to the three corresponding orbital angular momentum operators.
• 9.2: Spin Space
Unlike regular wavefunctions, spin wavefunctions do not exist in real space. Likewise, the spin angular momentum operators cannot be represented as differential operators in real space. Instead, we need to think of spin wavefunctions as existing in an abstract (complex) vector space. The different members of this space correspond to the different internal configurations of the particle under investigation. Note that only the directions of our vectors have any physical significance.
• 9.3: Eigenstates of Sz and S²
Because the operators Sz and S² commute, they must possess simultaneous eigenstates.
• 9.4: Pauli Representation
Up to now, we have discussed spin space in rather abstract terms. In the following, we shall describe a particular representation of electron spin space due to Pauli . This so-called Pauli representation allows us to visualize spin space, and also facilitates calculations involving spin.
• 9.5: Spin Precession
The expectation value of the spin angular momentum vector subtends a constant angle α with the z -axis, and precesses about this axis. This behavior is actually equivalent to that predicted by classical physics.
• 9.E: Spin Angular Momentum (Exercises)

Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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