Editing Spin quantum number (section)

== Algebra ==
The algebraic theory of spin is a carbon copy of the [[Angular momentum#Angular momentum in quantum mechanics|angular momentum in quantum mechanics]] theory.<ref>[[David J. Griffiths]], ''[[Introduction to Quantum Mechanics (book)]]'', Oregon, Reed College, 2018, 166 p. {{ISBN|9781107189638}}.</ref>
First of all, spin satisfies the fundamental [[Canonical commutation relation|commutation relation]]:
<math display="block">\ [S_i, S_j ] = i\ \hbar\ \epsilon_{ijk}\ S_k\ ,</math>
<math display="block">\ \left[S_i, S^2 \right] = 0\ </math>
where <math>\ \epsilon_{ijk}\ </math> is the (antisymmetric) [[Levi-Civita symbol]]. This means that it is impossible to know two coordinates of the spin at the same time because of the restriction of the [[uncertainty principle]].

Next, the [[Eigenstate|eigenvectors]] of <math>\ S^2\ </math> and <math>\ S_z\ </math> satisfy:
<math display="block">\ S^2\ | s, m_s \rangle= {\hbar}^2\ s(s+1)\ | s, m_s \rangle\ </math>
<math display="block">\ S_z\ | s, m_s \rangle = \hbar\ m_s\ | s, m_s \rangle\ </math>
<math display="block">\ S_\pm\ | s, m_s \rangle = \hbar\ \sqrt{s(s+1) - m_s(m_s \pm 1)\ }\; | s, m_s \pm 1 \rangle\ </math>
where <math>\ S_\pm = S_x \pm i S_y\ </math> are the [[ladder operator|ladder]] (or "raising" and "lowering") operators.