Half Spin
Contents
Half Spin#
Eigenvalue Relations#
\[
\begin{gather*}
\hat S_n \ket{\uparrow_n} = \frac{\hbar}{2}\ket{\uparrow_n}, \qquad \hat S_n \ket{\downarrow_n} = -\frac{\hbar}{2}\ket{\downarrow_n}
\end{gather*}
\]
Spin States#
\[\begin{split}
\begin{gather*}
\ket{\uparrow_z} = \begin{pmatrix}
1 \\ 0
\end{pmatrix} \qquad
\ket{\downarrow_z} = \begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
\frac{1}{\sqrt{2}}\ket{\uparrow_x} = \begin{pmatrix}
1 \\ 1
\end{pmatrix} \qquad
\frac{1}{\sqrt{2}}\ket{\downarrow_x} = \begin{pmatrix}
1 \\ -1
\end{pmatrix}\\
\frac{1}{\sqrt{2}}\ket{\uparrow_y} = \begin{pmatrix}
1 \\ i
\end{pmatrix} \qquad
\frac{1}{\sqrt{2}}\ket{\downarrow_y} = \begin{pmatrix}
1 \\ -i
\end{pmatrix}
\end{gather*}
\end{split}\]
Spin Operators and Pauli Matrices#
\[\begin{split}
\begin{gather*}
\hat S_n = \frac{\hbar}{2}\sigma_n\\\\
\sigma_x = \begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}, \qquad
\sigma_y = \begin{pmatrix}
0 & -i \\ i & 0
\end{pmatrix}, \qquad
\sigma_z = \begin{pmatrix}
1 & 0 \\ 0 & 1
\end{pmatrix}
\end{gather*}
\end{split}\]
Arbritrary Spin States and Operators#
For an arbritrary direction \(\hat n = (\sin\theta \cos\phi, \sin\theta\sin\phi, \cos\theta)\),
\[\begin{split}
\begin{gather*}
\hat S_n = \frac{\hbar}{2}\begin{pmatrix}
\cos\theta & \sin\theta^{i\phi} \\ \sin e^{i\phi} & -\cos\theta\\
\end{pmatrix}\\\\
\ket{\uparrow_n} = \begin{pmatrix}
\cos\frac{\theta}{2} \\ \sin\frac{\theta}{2}e^{i\phi}
\end{pmatrix} \qquad \ket{\downarrow_n} = \begin{pmatrix}
-\sin \frac{\theta}{2}e^{i\phi} \\ \cos \frac{\theta}{2}
\end{pmatrix}
\end{gather*}
\end{split}\]