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On one hand, since the angular momentum is:

$$L^{ij} = r^i p^j - r^j p^i$$

so it makes sense for the angular momentum operator to be:

$$\hat{L}^{ij}= -i\hbar (r^i \partial^j - r^j \partial^i)$$

On the other hand, since the momentum operator is:

$$\hat{P_i} = -i\hbar \frac{d}{dx^i}$$

Wouldn't the angular momentum follow the same steps? As in:

$$\hat{L}=-i\hbar \frac{d}{d\theta}$$

I believe my confusion revolves around the generators of rotation.

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1 Answer 1

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Use the chain rule for partial derivatives to show that with $$ x=r \cos\theta\\ y= r \sin\theta $$ you have $$ -i \hbar (x\partial_y - y\partial_x)= -i \hbar \partial_ \theta. $$ Your two formulae are therefore the same.

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