 # Problem I.10.2 is either wrong or wolfram and I are missing something!

Wolfram alpha disagrees with the answer in question (b).

Wolfram alpha link to the solution

I have also worked out the example and got similar results to the answer in the second link above.

Am I missing something?

Hey @Nicolas_Hadjittoouli! Our answer is correct, but so is the one on Wolfram (sort of…)! Let me show you:

Wolfram’s eigenvectors are

\begin{align*} \vert \tilde{v}_1 \rangle & = -0.780776 i \vert 0 \rangle + \vert 1 \rangle \\ \vert \tilde{v}_2 \rangle & = 1.28078 i \vert 0 \rangle + \vert 1 \rangle \ \end{align*}.

I put the “tilde” (~) sign over those vectors because they aren’t normalized! After normalizing them, we get

\begin{align*} \vert v_1 \rangle & = -0.61541221 i \vert 0 \rangle + 0.780776 \vert 1 \rangle \\ \vert v_2 \rangle & = 0.780776 i \vert 0 \rangle + 0.61541221 \vert 1 \rangle \end{align*}.

So, at this stage, \vert v_1 \rangle matches what we have in the codebook. But \vert v_2 \rangle still looks a little different. Here’s what we can do to make it “look” like ours:

\begin{align*} \vert v_2 \rangle & = 0.780776 i \vert 0 \rangle + 0.61541221 \vert 1 \rangle \\ \vert v_2 \rangle & = i \left( 0.780776 \vert 0 \rangle - 0.61541221 i \vert 1 \rangle \right) \\ \end{align*}

In the second line, I factored out the factor of i, which essentially acts like a global phase to the state. We can simply “do away” with the global phase and redefine \vert v_2 \rangle as

\vert v_2 \rangle =0.780776 \vert 0 \rangle - 0.61541221 i \vert 1 \rangle,

which is the answer that we have in the codebook.

So, two things to remember when finding eigenvectors:

1. Always normalize them
2. Global phases don’t matter
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Just to illustrate this with a more familiar-feeling example, the Pauli Y matrix

\hat{\sigma}^y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}

has eigenvalues / eigenvectors

1, \sqrt{\frac{1}{2}}\left(\vert 0 \rangle + i \vert 1 \rangle\right)

and

-1, \sqrt{\frac{1}{2}}\left(\vert 0 \rangle - i \vert 1 \rangle\right).

Try factoring out factors of i and playing the same game that we did with the codebook example and you will find that they don’t affect the eigenequations!

Oh, right! I couldn’t see, how I could get there. I totally forgot about the normalization. Thanks a lot for the help!  1 Like

Thank you for posting here! I forget about normalizing every once and a while too 