For the spherical coordinates;

$$ x=r\sin{ \theta} \cos{\phi} $$

$$ y=r \sin{\theta} \sin{\phi} $$

$$ z=r\cos{\theta} $$

Where $ \theta $ is the polar angle and $ \phi $ is the azimutal angle, Is it possible to uae the chain rule for multivariable functions in order to derive an expression for the partial derivatives $ \frac{\partial}{\partial x}, \frac{\partial}{\partial y}$ and $ \frac{\partial}{\partial z}$ in terms of $ \frac{\partial}{\partial r}, \frac{\partial}{\partial \theta}$ and $ \frac{\partial}{\partial \phi}$ ?

More specifically, I have an operator in cartesian coordinates that I want to write in the spherical coordinates (without using results from the internet, I want to understand how to derive the expression). The operator is;

$$ L_x=-i\hbar \left(y \frac{\partial}{\partial z} - z\frac{\partial}{\partial y}\right)$$

That is the angular momentum operator for the $x$ coordinate in physics (up to the fact that Im not sure how to write h_bar properly)

Im not sure what method can I use to replace the partial derivatives in cartesian coordinates in the operator, with the spherical partial derivativea.

Thanks in advance

  • 2
    $\begingroup$ \hbar = $\hbar$ $\endgroup$ – user170231 Feb 23 at 14:47
  • $\begingroup$ What you described works perfectly. Just use chain rule. You will have six total derivative terms. $\endgroup$ – Ninad Munshi Feb 23 at 15:21
  • $\begingroup$ @Ninad Munshi Im not sure how exactly to use it, i juat thought that it might be the right way (but I do not feel comfortable enough with derivatives of multivariable functions) $\endgroup$ – FreeZe Feb 23 at 22:43

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