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I try to solve problems from Problems in Quantum Computing.

I stuck with problem #3:

I do the following:

Because: $$ \sigma_2 = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}$$ Then: $$ -i \frac{\phi}{2}\sigma_2 = \begin{pmatrix} 0 & -\frac{\phi}{2}\\ \frac{\phi}{2} & 0 \end{pmatrix} $$

$$\exp\begin{pmatrix} 0 & -\frac{\phi}{2}\\ \frac{\phi}{2} & 0 \end{pmatrix} = \begin{pmatrix} 1 & \exp(-\frac{\phi}{2})\\ \exp(\frac{\phi}{2}) & 1 \end{pmatrix}$$

If I multiply the result of the last calculation with $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ or $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ I can't get $\psi_1(\phi)$ or $\psi_2(\phi)$. I get some unnormalised state like:

$$ \begin{pmatrix} 1 \\ \exp(\frac{\phi}{2}) \end{pmatrix} $$

Does it mean that the definition of the problem is not correct?

I haven't known about this "hack". Could you suggest a site where I can read about it? – Kenenbek Arzymatov – 2019-12-06T21:12:13.253

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This "hack" should be in any good QM or Quantum Information course; see wikipedia; I substituted $\hat{n}=(0,1,0)$

– kludg – 2019-12-06T21:19:32.137This is an application of a matrix function definition. See my answer below. – Martin Vesely – 2019-12-06T23:43:49.447