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

Evaluate the integral

<Y_{(l,m+2)} | L_{x}^{2} | 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 ?