Orthonormality of Spherical Harmonics Y_1,1 and Y_2,1

[MxwllsPersuasns]

Homework Statement

Here is a copy of the pdf problem set {https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU} the problem in question is problem number 1 which asks you to prove the orthonormality of the spherical Harmonics Y_1,1 and Y_2,1.

Homework Equations

Y_1,1 = -sqrt(3/8pi)e^(i(/))sin(-)
Y_2,1 = -sqrt(15/8pi)e^(i(/))sin(-)cos(-)
<l,m|l',m'> = Integral from 0 to pi of {sin(-)d(-)}*Integral from 0 to 2pi of {Y*_l,m(Y_l',m')d(/)} = delta_ll'(delta_mm')
(-) = THETA
(/) = PHI
delta = dirac delta function

The Attempt at a Solution

Looking at the 3 equations above something is a little ambiguous. It shows two integrals in the calculation of <l,m|l',m'>, one with d(-) and one with d(/). Now I see that both spehrical harmonics in question have (-) terms, i.e., sin(-), so is that d(-) integral just there from pulling the sin(-) out of Y_1,1 and Y_2,1? That's what I would imagine but it's doesn't explicate a dependence only on (/) in the spherical harmonics (unless the notion of being paired with d(/) only (and not d(-)) is rationale enough to interpret as only being functions of (/))

Actually as I'm looking at the forumlas Y_2,1 couldn't be a function of (/) only after we pulled out the sines as it also has a cos(-) term.. I'm very confused can someone help me move forward here?

 

[Orodruin]

Just write down and perform the integrals after inserting the expressions for the spherical harmocs. You will have an integrand that is on product form, which is easy to solve.

 

[DrClaude]

(-) = THETA
(/) = PHI

At the top of the field where you type the text to be posted, you will notice a $\sum$. Click on it to have access to common symbols. Also, x₂ and x² allows to write sub- and superscripts.

 

[MxwllsPersuasns]

I see well what I'm getting tripped up on is the fact that in the forumla for Y_2,1 there's a sin(-) (which gets extracted out into the d(-) integral) and ALSO a cos(-) but it doesn't seem that the cosine gets extracted out of Y_2,1 and hence still exists within the d(/) integral. I'm not sure why I'm having so much trouble with this but I know that the e^-i(/) and e^i(/) will cancel and that will leave me with my two sqrt coefficients and also the cos(-). Do I just move the cos(-) then into the d(-) integral and make it sin(-)cos(-)d(-) from 0 to pi? Or would it become something like (/)cos(-) as cos(-) has no (/) dependence it would be treated as a constant within the interval

 

[MxwllsPersuasns]

Oh hey thanks DrClaude I completely missed that! Appreciate the help :)

 

[DrClaude]

$$
\int \int f(\theta) g(\phi) d\theta d\phi = \int \left[ \int f(\theta) d\theta \right] g(\phi) d\phi = \left[ \int f(\theta) d\theta \right] \left[ \int g(\phi) d\phi \right]
$$
and so on.

 

[Orodruin]

ALSO a cos(-) but it doesn't seem that the cosine gets extracted out of Y_2,1 and hence still exists within the d(/) integral.

What do you mean by this? Of course the cosθ is part of the θ integral. Nothing can depend on θ once you have performed that integral.

Do I just move the cos(-) then into the d(-) integral and make it sin(-)cos(-)d(-) from 0 to pi?

dont forget the sine from the other harmonic or that coning from the area element!

 

[MxwllsPersuasns]

Okay I think I'm getting tripped up because it shows the formula worked out where only the sin θ is in the dθ integral. That led me to believe that the cos θ in the Y_2,1 harmonic stayed in that harmonic and thus within the d∅ integral. What you guys are telling me to do is indeed to move that cosθ into the dθ integral so then rather than the formula above I'd have something like:

∫sinθcosθdθ∫√(...)e^-i∅√(...)e^i∅d∅??​

 

[DrClaude]

∫sinθcosθdθ∫√(...)e^-i∅√(...)e^i∅d∅??​

As @Orodruin mentioned, you are missing some terms in $\theta$ in there. Are why are you not taking out the square roots out of the integration?

Considering the problems you are having here and in the other thread on the radial problem, I think you should first take out your calculus textbook and revise multi-variable integration as well as ODEs.

 

[Orodruin]

Okay I think I'm getting tripped up because it shows the formula worked out where only the sin θ is in the dθ integral.

What is "it"? Please write out everything you refer to in the thread.

 

[MxwllsPersuasns]

Oh I'm sorry Orodruin I was being lazy with my typing. The "it" I'm referring to is the third equation I listed in my first post

<l,m|l',m'> = ∫ sinΘdΘ ∫ Y*_l,mY_l',m'd∅

I was referring to the fact that the first integral shown here is displayed as having a sinΘ term (presumably from both the Spherical Harmonic functions) but it didn't show the cosΘ (which is involved in Y_2,1) so I assumed it was still within the Y_2,1 which is in the second integral (the d∅ one). So my question was given that, do I just pull the cosΘ out of the second integral and put it into the first where it would be properly integrated? Is that something I can do?

 

[Orodruin]

presumably from both the Spherical Harmonic functions

No, it is from the surface element of the sphere expressed in spherical coordinates. There is nothing that has been "extracted" here.

So my question was given that, do I just pull the cosΘ out of the second integral and put it into the first where it would be properly integrated? Is that something I can do?

The integrand is inside both integrals. There is no "changing" of which integral the terms belong to.

 

[MxwllsPersuasns]

Ah okay so then I carry out the integrals with the full formulae for the spherical harmonic functions and the sinΘ is "extra" in that its not from the spherical harmonic functions. I got tripped up by the way the formula was presented I suppose. I assumed all of the functions with explicit dependence on theta would be shown within the theta integral and simultaneously that the first integral "closed off" or stopped being valid for all things to the right of the dΘ term but I see the thetas are implicit in the Y-functions and are to be integrated along with all other terms dependent on theta.

 

[MxwllsPersuasns]

I ended up finishing the problem pretty easily after your clarification Orodruin, just wanted to say thanks again for the help. I really appreciate it. Have a nice evening!