[AdiDex]

Recently, I was solving some questions on Rigid Rotor. I found this question very amusing.

Evaluate the integral

<Y_(l,m+2) | L_x² | Y_(l,m+2)  >

I evaluated it , value is

ħ((l(l+1)-(m+2)(m+1))^(1/2)

But I didn't get the point of evaluating it. Is there any application of such kind of integrals?
Is there any physical significance of it?
I know similar kind of integral are evaluated in case of transition dipole integral,  where we take two different wave functions.
What is actual significance of such operations ?

[Irlanur]

Integrals like this might pop up everywhere in QM where rotations are involved. One important point is the matrix representation of operators. If you want to calculate anything in a computer in QM, you usually want to have everything as matrices. Matrices need some kind of basis. In this case, it is the spherical harmonics. The matrix elements are then always of this form.

[Corribus]

How did you evaluate this? The spherical harmonics aren't eigenfunctions of the x-oriented angular momentum operator. I.e., the angular momentum in the x direction is not quantized (at least, under the convention that the wavefunctions are chosen so that the angular momentum in the z-direction is).

In the case of the transition dipole, such integrals are usually evaluated to determine whether a spectroscopic transition is allowed or not. But here you have the initial and final states being the same, and the operator is wrong, so the problem isn't relevant to the transition dipole.

[AdiDex]

Sorry. It was -

<Y_(l,m+2) | L_x² | Y_(l,m)  >

And Even answer is

ħ²/4 [((l(l+1)-(m)(m+1))((l(l+1)-(m+1)(m+2))]^(1/2)

I was really sleepy  .

[AdiDex]

Here I am attaching the solution. There are some writing errors as I didn't have much battery in my phone so I did this in hurry.