We showed that the 3d rotational energy levels depend only on J and 2d energy le

Question

We showed that the 3d rotational energy levels depend only on J and 2d energy levels only on m (or m_J). Explain why this is true. The degeneracy of each 3d energy level is 2j + 1 and each 2d level is 2, except for m = 0. Explain why this is true. Vibrational energy levels have "zero point energy" as does the particle in a box. Rotational energy levels do not have zero point energy. Explain all of these results. For a 2d rotor the angular momentum vector lies along the z-axis (for rotation in the xy plane). Can the angular momentum vector for a 3d rotor lie along the z-axis? Why or why not? In 2d only one quantum number is needed to define the eigenfunctions but in 3d two quantum numbers are needed. Why?

Explanation / Answer

The degeneracy of 3d orbital is 2J+1.

J is made of orbital angular momentum L and spin angular momentum S.

Orbital quantum number ml varies from 0, 1, 2, ....., so on

and spin angular momentum ms varies from -l to +l

So, if you consider the ms value 0, you will not get the degeneracy which you are supposed to get.

If you draw energy levels for each orbital you will see that for 3d orbital the degeneracy occurs, however if you plug ms = 0, you will not formulate that level as denerate. That;s why m =0 is not allowed for the degeracy of 3d orbitals.

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