In the Wikipedia page for the Pauli matrices, there is a list of the eigenvectors of the Pauli matrices.

Notice at $\sigma_y$, it's eigenvectors are $\begin{pmatrix} 1 \\ \pm i \end{pmatrix}$ but not $\begin{pmatrix} \pm i\\ 1 \end{pmatrix}$, same for $\sigma_x$ and $\sigma_y$.

By which formalism or customs stated that Pauli matrices' eigenvectors are $\begin{pmatrix} 1 \\ \pm i \end{pmatrix}$ but not $\begin{pmatrix} \pm i\\ 1 \end{pmatrix}$?

  • 2
    $\begingroup$ They are equal, modulo an $i$ or two. People liked the 1 up top... $\endgroup$ – Jon Custer Sep 3 '18 at 13:58

The standard way to encode vectors on the Bloch sphere sets the first component as a positive real number, and this convention is influenced by that.

But that's about it: it is a convention, and you're welcome to change it if there's a specific reason why that would suit your purposes better.

However, if there isn't a well-defined reason why it's better for you in some measure, then I would advice against trying to buck this convention: the only thing that it's going to achieve is that a good fraction of your readers is going to waste time wondering why the phase convention on your eigenbases has been chosen in a non-standard way, and you want your readers' attention to be focused on what you want to say instead of on your notation. The positive-first-component is just a part of the shared language that makes it easier to communicate clearly about the content; there's no good reason to make one choice over others (unlike in other domains) but having a common choice that breaks that degeneracy takes that uncertainty out and leaves more room for the physics.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.