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I feel like I just fell off the short bus. I found a problem with a setup like this:

"2X2 matrix defined by: $$U=(a_0+i(\sigma \cdot a))/(a_0-i(\sigma\cdot a)) $$ where $a$ is a three-dimensional vector with real components and $a_0$ is a real number."

This looks like a scalar to me. Dot products are always scalar (to my knowledge) and addition, subtraction, and multiplication of two scalars is scalar... where is the 2X2 matrix coming from? How do I define this matrix?

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    $\begingroup$ To follow up on the previous comment: make sure to distinguish between a vector in space (i.e. a vector that points in some direction) and a vector in Hilbert space, which is the abstract space of quantum states. In this case, the Pauli operators, denoted $\sigma_x$, $\sigma_y$, and $\sigma_z$ are operators that act on a 2-dimensional Hilbert space (quantum state space; for example, a spin-1/2 system), but they form the components of the Pauli vector, which is a vector in real space: when you take the expectation value of it, you get the average direction in which the spin points. $\endgroup$ – march Dec 1 '15 at 22:59
  • $\begingroup$ Thank you, that is correct. After looking that up, this makes much more sense. $\endgroup$ – SillyInventor Dec 1 '15 at 23:12
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The $\sigma$ is a vector consist of three $2\times 2$ matrices called the pauli matrices, namely $$\sigma=\left(\sigma^1,\sigma^2,\sigma^3\right).$$ And the unit matrix should be always understood properly, for example, the $a_0$ is actually $a_0I$. The division should be understood as the inverse of a matrix. Hence you have a $2\times 2$ matrix.

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