What are the functions X^m_1 that are eigenfunctions of L^2 and L_x?

[Brewer]

Homework Statement

The spherical harmonics Y^m_l with l=1 are given by
Y^{-1}_1 = \sqrt{\frac{3}{8\pi}}\frac{x-iy}{r}, Y^0_1 = \sqrt{\frac{3}{4\pi}}\frac{z}{r}, Y^1_1 = -\sqrt{\frac{3}{8\pi}}\frac{x+iy}{r}

and they are functions of L^2 and L_z where L is the angular momentum.

i) From these functions find a new set of three functions X^m_1 which are now eigenfunctions of L^2 and L_x.

Homework Equations

The Attempt at a Solution

I'm not 100% sure about this question. Is it asking me to give the spherical harmonics in terms of \theta, \phi? I think I can do that, but if that's not the question, could someone please explain to me what is being asked of me.

Thanks

Brewer

 

[stunner5000pt]

Homework Statement

The spherical harmonics Y^m_l with l=1 are given by
Y^{-1}_1 = \sqrt{\frac{3}{8\pi}}\frac{x-iy}{r}, Y^0_1 = \sqrt{\frac{3}{4\pi}}\frac{z}{r}, Y^1_1 = -\sqrt{\frac{3}{8\pi}}\frac{x+iy}{r}

and they are functions of L^2 and L_z where L is the angular momentum.

i) From these functions find a new set of three functions X^m_1 which are now eigenfunctions of L^2 and L_x.

Homework Equations

The Attempt at a Solution

I'm not 100% sure about this question. Is it asking me to give the spherical harmonics in terms of \theta, \phi? I think I can do that, but if that's not the question, could someone please explain to me what is being asked of me.

Thanks

Brewer

i think it would be better if you wrote your spherical harmonics in spherical coords ..

but in any case
if X_{lm} is an eignefunction of L^2 and Lz then
\hat{L_{z}} X_{lm} = mX_{lm}
and
\hat{L^2} X_{lm} = l(l+1) X_{lm}
and i think Xlm would b acquired by finding a superposition of the three given eigenfunctions.

 

[Brewer]

Thats what the question gives, cartesian coords.