I'm trying to understand the rotations of rigid diatomic molecules such as HCl. My understanding of the orbital angular momentum is that it is quantized with a total value equal to(adsbygoogle = window.adsbygoogle || []).push({});

$$E=\frac{\hbar^2}{2I}J(J+1)$$

whereIis the rotational moment of inertia andJis the quantum number. Also,Jcan be 0, 1, 2, etc. According to this, the minimum rotational energy is 0, which is surprising but ok. Also, the total angular momentum is given by

$$L^2=\hbar^2 J(J+1)$$

This implies that the total angular momentum is 0 ifJis 0. This is consistent with the rotational energy being 0.

This implies that I have complete knowledge of the angular momentum ifJ= 0 (i.e. it must be exactly 0). However, the Heisenberg uncertainty principle says that complete knowledge of the angular momentum would require complete lack of knowledge of the orientation. To the contrary, I know that the orientation is between 0 and 2π, which is not complete lack of knowledge and thus seems to violate the Heisenberg uncertainty principle.

If there were a zero-point energy for rotations, then everything would make sense, but it doesn't seem that there is one.

What am I doing wrong? Thanks for any replies!

-Steve

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# I How can the total orbital angular momentum be zero?

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