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

Evaluate the integral

i) <$Y_{l,m+2}$ | $L_{x}^{2}$ | $Y_{l,m}$ >

I evaluated it, found that the value is

$$\hbar^2/4 \sqrt{[(l(l+1)-(m)(m+1)][(l(l+1)-(m+2)(m+1)]}$$

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 function. What is actual significance of such operations ?

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  • $\begingroup$ Strictly speaking this is not an expectation value. I would just call it a matrix element. $\endgroup$ – Feodoran Nov 9 '18 at 10:17
  • $\begingroup$ @Feodoran Sorry for my primitive knowledge. $\endgroup$ – Aditya Shrivastava Nov 9 '18 at 10:44
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    $\begingroup$ I would have expected $\hbar^2$, did you expect the result to be complex for some $\ell$ and $m$ ? $\endgroup$ – porphyrin Nov 9 '18 at 16:41
  • $\begingroup$ @porphyrin Thanks for pointing out, I really messed it. Now I have corrected it. I was really sleepy when I solved it. $\endgroup$ – Aditya Shrivastava Nov 9 '18 at 21:44
  • $\begingroup$ These kind of elements appear when constructing the Hamiltonian of for the rotation of an asymmetric top molecule. This term appears Because the moments of inertia along the spatial axes are different you cannot express the rotational energy in terms of $L^2$ and $L_z^2$ alone as you would do for a linear or symmetric top molecule. $\endgroup$ – Paul Jan 11 '19 at 22:04

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