I'm reading Sakurai's "Advanced Quantum Mechanics" (which is different from his "Modern Quantum Mechanics"). In chapter 3, which is about the Relativistic Quantum Mechanics of spin 1/2 particles, after discussing the covariance of the Dirac equation, he goes on to give some examples to clarify the significance of the operator S (as in ## \psi'(x')=S \psi(x) ##). In the first example, he considers an infinitesimal rotation around the z-axis and so he considers the transformation:(adsbygoogle = window.adsbygoogle || []).push({});

## \psi'(x')= \left[ 1+i\Sigma_3 \frac{\delta \omega}{2} \right] \psi(x) \ \ \ \ (1)##

where ## \vec{x}'=\vec x+\delta \vec{x} ## and ## \delta \vec x=(x_2 \delta \omega,-x_1 \delta \omega,0)##.

He then notes that:

## \psi'(x')=\psi'(x)+\delta x_1 \frac{\partial \psi'}{\partial x_1}+\delta x_2 \frac{\partial \psi'}{\partial x_2} \ \ \ \ (2)##

Then from all the above, he concludes that:

## \psi'(x)=\left[ 1+ i \Sigma_3 \frac{\delta \omega}{2}-\left(x_2 \delta \omega \frac{\partial}{\partial x_1}-x_1 \delta \omega \frac{\partial}{\partial x_2} \right) \right] \psi(x) \ \ \ \ (3)##

It seems he rearranged equation 2 and then used equation 1 to substitute for ## \psi'(x') ##. But there still remain the derivative terms which are derivatives of ## \psi'(x) ## which should somehow be related to the derivatives of ## \psi(x) ##. But he seems to just assumes that ## \frac{\partial \psi'(x)}{\partial x_i}=\frac{\partial \psi(x)}{\partial x_i }## which doesn't seem justified at all! In fact using equation 1, you can easily show how far this is from truth! Am I missing something here?

Thanks

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# A Angular momentum of Dirac particle

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