. . . )¹

If you go to $n$ much larger than 10, it might be wise to plot $\log F_n$ vs. $n$ .

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. . . WikipediA²

You can Google them, or go directly to http://en.wikipedia.org/wiki/Pauli_matrices or http://mathworld.wolfram.com/PauliMatrices.html to get a nice compact introduction to their mathematical properties. Be sure you understand what "Hermitian" and "unitary" mean.

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. . . particle.³

Actually, the Pauli matrices can be used to describe the quantum mechanics of any two-state system, which makes them useful not only in elementary particle physics but also in a wide variety of quantum computing topics.

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. . . matrices.⁴

I don't find MatLab to be as great as it claims, since I have not found a compact, elegant way to express the three Pauli matrices $\{\sigma_x, \sigma_y, \sigma_z\}$ as a vector $\Vec{\sigma}$ of matrices in MatLab, even though that is a "natural" description in Physics. This is possible with python; Google NumPy for Matlab Users.

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. . . thoroughly.⁵

You'll thank me later!

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. . . respectively.⁶

Thus $\vert \! \uparrow \rangle$ and $\vert \! \downarrow \rangle$ are the eigenvectors of $\sigma_z$ : operating on them with $\sigma_z$ is the same as multiplying them by a number which is their eigenvalue with respect to $\sigma_z$ . In the language of spin orientation, the eigenvalue is the projection of the particle's spin along the $\Vec{z}$ direction). This follows from the fact that $\sigma_z$ is diagonal.

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. . . vectors,⁷

Such an interaction is known as a contact interaction or a hyperfine interaction or a Heisenberg spin-spin interaction.

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. . . diagonal.⁸

If you need help with this, just ask!

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. . . . ⁹

Actually there are $\ell \ne 0$ states mixed into the "$1s$ " state of a hydrogen atom if the two spins are parallel; but this is a very small effect.

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