Spin Operator

#Physics

Topics

$\displaystyle \hat{S}^{2}\ket{s,m_{s}}=\hbar ^{2}s(s+1)\ket{s,m_{s}}$

$\displaystyle \hat{S}^{2}={\left\langle{S_{x},S_{y},S_{z}}\right\rangle}$
$\displaystyle s$ is the spin of the particle
$\displaystyle m_{s}$ is the spin quantum number
$\displaystyle \ket{sm_{s}}$ represents a state with a certain $\displaystyle s$ and $\displaystyle m_{s}$
$\displaystyle \hbar$ is the [reduced Planck constant]

$\displaystyle \hat{S}{z}\ket{s,m{s}}=\hbar m_{s}\ket{s,m_{s}}$

$\displaystyle \hat{S}{\pm }\ket{s,m{s}}=\hbar\sqrt{ s(s+1) -m_{s}(m_{s} \pm 1)}\ket{s,(m_{s} \pm 1)}$

$\displaystyle \hat{S}_{\pm }$ is a ladder operator

$\displaystyle \hat{S}^{2}{s}=\hbar ^{2}s(s+1)I{2s+1}$

The operator for a particle with spin $\displaystyle s$ in matrix form