- #1

- 99

- 0

## Main Question or Discussion Point

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!

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!