Can Schrodinger's Equation be Transformed into Spherical Polar Coordinates?

[asdf1]

how do you change the Schrödinger's equation into the spherical polar coordinates?

 

[jtbell]

Look up the chain rule for partial derivatives, and the equations that give you $x, y, z$ in terms of $r, \theta, \phi$ for spherical coordinates. Use these to re-write the derivatives $\partial^2 \psi / \partial x^2$ etc. into the derivatives $\partial^2 \psi / \partial r^2$ etc. There's a lot of algebra. The final result (which you should be able to see in your textbook) contains both first- and second-order derivatives.

 

[Galileo]

The coordinate-free form of the S.E. is:

$$i\hbar\frac{\partial \Psi(\vec r,t)}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2 \Psi(\vec r,t)+V\Psi(\vec r,t)$$

You can (should) look up the laplacian operator $\nabla^2$ in various coordinate systems in your textbook. There's not much physics to be learned by deriving it.

 

[asdf1]

thank you very much! :)