What is the role of spherical harmonics in quantum mechanics?

[KostasV]

Hello people !
I have been studying Zettili's book of quantum mechanics and found that spherical harmonics are written <θφ|L,M>.
Does this mean that |θφ> is a basis? What is more, is it complete and orthonormal basis in Hilbert?
More evidence that it is a basis, in the photo i uploaded , in (5.163) it seems that he "opens" a complete basis to orthonormalization condition ... Furthermore, why does it have integrals of φ and θ ?
I am confussed ... I have never heared about |θφ> basis ... (In expression |θφ> is it a product between θ and φ or should be a decimal point between them like in |L,M> ? )
I am asking because in potential wells we write the wave function like this <x|y> in position representation. So we use the |x> basis there.
I want to know if i can treat |θφ> Like i treat |x> as a base.
Thank you!

 

[BvU]

Hello Kostas, welcome to PF !

Seems you have seem so much detail that you have lost a little bit of the oversight.
Yes, $\theta$ and $\phi$ are coordinates. Spherical coordinates, very useful when studying situations with rotational symmetries such as with central forces. Spherical symmetry allows separation of variables and this section looks at angular dependences only.

And yes, you can treat $| r, \theta, \phi >$ and its subspace $| \theta, \phi >$ as bases in Hilbert space, in the same way as $| x, y, z >$

 

[KostasV]

Thank u very much for the welcome and the answer ! :)