Final answer:
The Killing spinor equation for a special case on S² is derived using the 2d-Clifford algebra. Spin in quantum mechanics represents a particle's intrinsic angular momentum and only permits certain quantized values. The Pauli Exclusion Principle dictates that no two electrons can have the same set of quantum numbers.
Step-by-step explanation:
A Killing spinor on a Riemannian spin manifold M satisfies the equation ∇_Xψ = λX ⋅ ψ for all tangent vectors X, where λ is a constant, the Killing number. In the special case where λ=0, the spinor is known as a parallel spinor. When considering the 2-sphere S² and the 2d-Clifford algebra, this equation can be reduced to ∇_μ ψ = ± 1 / 2 γ_μ ψ. To derive this, a particular representation of the Clifford algebra relevant to two dimensions is used, alongside properties specific to the geometry of S².
In the context of quantum mechanics, spin is a fundamental property of particles such as electrons. The magnitude of the intrinsic spin angular momentum S of an electron is √s(s+1)ħ, with the spin quantum number s being 1/2 for electrons. The spin projection quantum number ms can be either +1/2 or -1/2, corresponding to 'spin up' and 'spin down' states, respectively.
The Pauli Exclusion Principle states that no two electrons can have the same set of quantum numbers, thereby prohibiting them from occupying the same quantum state. This principle is crucial to understanding a multitude of physical phenomena, including the structure of the periodic table and the behavior of electrons in atoms.