How Do You Solve the Schrodinger Equation with Spherical Coordinates?

[KingBigness]

Homework Statement

See attached photo

The Attempt at a Solution

So I have no idea if I have even started this problem correctly so any help would be nice.

My working is set out in one of the pictures.

Any help would be appreciated I really am not quite sure what to do. Can't figure out where the x^2-y^2 comes from.

Thank you!

 

[tiny-tim]

Hi KingBigness!

Can't figure out where the x^2-y^2 comes from.

I'm guessing really, since this isn't my field , but if you replace r²sin²θcos2φ by r²sin²θ(cos²φ - sin²φ), that looks like x² - y²

 

[vela]

Leave the r² alone for now and use the fact that$$\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$$to get rid of the complex exponentials.

 

[KingBigness]

Thank you both for that, I shall try that and let you know how I go

 

[KingBigness]

Ok I tried that and out came the x^2-y^2

Thank you for that tip.

This is the answer I have ended up with can you let me know if it is correct or if I need to simplify it more? not really sure when to stop =\

 

[KingBigness]

I lied above...I brought the sin theta squared in before I converted which got rid of the sin theta squared in the final answer.

Is this now correct?

 

[vela]

Yes, that's right, because in spherical coordinates x = r sin θ cos φ and y = r sin θ sin φ.

 

[KingBigness]

Yes, that's right, because in spherical coordinates x = r sin θ cos φ and y = r sin θ sin φ.

Sweet finally got this question complete!

Thank you. Will double check all my algebra later to make sure I haven't done a silly mistake.

Thanks again for your help