- #1
Ackbach
Gold Member
MHB
- 4,155
- 91
Here is this week's POTW:
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Let
\begin{align*}
\sigma_1&=\begin{bmatrix}0&1\\1&0\end{bmatrix} \\
\sigma_2&=\begin{bmatrix}0&-i\\i&0\end{bmatrix} \\
\sigma_3&=\begin{bmatrix}1&0\\0&-1\end{bmatrix}
\end{align*}
be the three Pauli spin matrices. Let $\vec{v}$ be a real, three-dimensional unit vector, and let $\theta$ be a real number. Compute $\exp(i\theta \, \vec{v}\cdot\vec{\sigma}),$ where
$$\vec{v}\cdot\vec{\sigma}=\sum_{j=1}^3 v_j \, \sigma_j.$$
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
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Let
\begin{align*}
\sigma_1&=\begin{bmatrix}0&1\\1&0\end{bmatrix} \\
\sigma_2&=\begin{bmatrix}0&-i\\i&0\end{bmatrix} \\
\sigma_3&=\begin{bmatrix}1&0\\0&-1\end{bmatrix}
\end{align*}
be the three Pauli spin matrices. Let $\vec{v}$ be a real, three-dimensional unit vector, and let $\theta$ be a real number. Compute $\exp(i\theta \, \vec{v}\cdot\vec{\sigma}),$ where
$$\vec{v}\cdot\vec{\sigma}=\sum_{j=1}^3 v_j \, \sigma_j.$$
-----
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!