Standard Pauli spin matrices are:(adsbygoogle = window.adsbygoogle || []).push({});

S_{x}:

$$\left(\begin{array}{cc}0&1/2\\1/2&0\end{array}\right)$$

S_{z}:

$$\left(\begin{array}{cc}1/2&0\\0&-1/2\end{array}\right)$$

The S_{z}eigenvectors are Z_{+}= (x=1,y=0) and Z_{-}= (x=0,y=1). These yield eigenvalues 1/2 and -1/2 respectively. Similarly, eigenvectors of S_{x}are defined to guarantee eigenvalues 1/2 and -1/2 (e.g. by making X_{+}= (1/2(x=sqrt(2),y=sqrt(2))).

My question is this:

Do they need to be those values? What is stopping me from instead defining the operators as follows:

S_{x}:

$$\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$

S_{z}:

$$\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$$

Retain the S_{z}eigenvectors. That gives eigenvalues +1 and -1. Make the S_{x}eigenvectors such that eigenvalues are +1 and -1 (e.g. by making X_{+}= (x=1/sqrt(2),y=1/sqrt(2)).

Is there some reason why the spin matrices don't take this form instead?

A more general question from which this spawns: Hermitian operators are invoked so as to sure that eigenvalues are always real numbers. But why should they be real numbers? What's wrong with complex eigenvalues?

Interested in your thoughts!

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# Physical significance of eigenvalues?

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