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- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_(Kok)/07%3A_Orbital_Angular_Momentum_and_Spin/7.02%3A_Spin\[\sigma_{i} \sigma_{j}=\sigma_{i} \sigma_{j}+\sigma_{j} \sigma_{i}-\sigma_{j} \sigma_{i}=\left\{\sigma_{i}, \sigma_{j}\right\}-\sigma_{j} \sigma_{i}=2 \delta_{i j}^{\mathbb{I}}-\sigma_{j} \sigma_{i}....σiσj=σiσj+σjσi−σjσi={σi,σj}−σjσi=2δIij−σjσi. \(\frac{1}{2} \operatorname{Tr}\left(A \sigma_{v}\right)=\frac{1}{2} \operatorname{Tr}\left(\sum_{\mu} a_{\mu} \sigma_{\mu} \sigma_{v}\right)=\frac{1}{2} \sum_{\mu} a_{\mu} \operatorname{Tr}\left(\sigma_{\mu} \sigma_{v}\right)=\sum_{\mu} a_{\mu} \delta_{\mu v}=a_{v}\tag{7.44}\]
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/09%3A_Spin_Angular_MomentumBroadly 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 fix...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 cent
- https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/09%3A_Symmetry_and_Bound_States/9.03%3A_Confined_Matter_WavesConfinement of a wave to a limited spatial region results in rather peculiar behavior — the wave can only fit comfortably into the confined region if the wave frequency, and hence the associated parti...Confinement of a wave to a limited spatial region results in rather peculiar behavior — the wave can only fit comfortably into the confined region if the wave frequency, and hence the associated particle energy, takes on a limited set of possible values. This is the origin of the famous quantization of energy, from which the “quantum” in quantum mechanics comes.
