How Do You Calculate Spherical Harmonics for Given Quantum Numbers?

[noblegas]

Homework Statement

Find the speherical harmonics $$(Y_1)^1, (Y_1)^0, (Y_1)^-1$$ as functions of the polar angles $$\theta$$ and $$\psi$$ and as functions of the cartesian coordinates x, y , and z.

Homework Equations

$$\(phi_l)^l= sin^l(\theta)*e^il\psi$$

$$L_\(phi_l)^l=(d/(d\theta))*\phi_l^l-l cot(\theta)\phi_l^l$$

The Attempt at a Solution

The first thing I should do is normalized$$\(phi_l)^l$$ to get a value for the A constant

A^2*$$(sin^l(\theta)*exp(il\psi))^2$$=1; should I plug in the values for m and l before I normalized the function or after I normalized the function

once I get the value for $$\(phi_l)^l$$ I can plug in this value into $$L_\(phi_l)^l=(d/(d\theta))*(\phi_l)^l-(l cot(\theta))(\phi_l)^l$$ correct?Not sure why I am finding the value for the lower opperator. Please inform me if you have a reallly really hard time understanding the latex code.

 

[noblegas]

just let me know if my latex is unreadable

 

[Troels]

just let me know if my latex is unreadable

Pretty much - I'm not sure what it is you need to do; Do you need to derive the spherical harmonics directly from their defining differential equation or do you merely need to express them in the different coordinate systems?