deriving the pauli matrices from general su(2) matrix

jeudi 1 août 2013

Hi, I think I need a sanity check, because I've been working on this for a while and I can't see what I'm doing wrong!

According to several authors, including Sakurai (Modern QM eq 3.3.21), a general way to write an operator from SU(2) is

$$\left(\begin{array}{cc}e^{i\alpha}\cos\gamma&-e^{-i\beta}\sin\gamma\\e^{i\beta}\sin\gamma&e^{-i\alpha}\cos\gamma\end{array}\right)$$

This clearly gives a determinant equal to one and has three parameters, as it should. It's basically identical to what Sakurai gives in his book, I've just redefined some variables.

But when I try to find the generators from this matrix by taking derivatives with respect to the parameters and then setting the parameters to zero, I get the following three matrices:

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

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

and

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

I can see how the first two are similar to a couple of the Pauli matrices (divide by i for the first and multiply by i for the second, which seems inconsistent), but the third has me completely stumped. Can anyone see what I'm doing wrong?

Thanks!