Why does the rotational constant B decrease and transition spacings decrease as the mass of a particle increase?

I understand from a purely equation perspective that since

$$B = \frac{h} {8\pi ^2 cI}$$

that as $I$ increases the denominator increases and so $B$ decreases. But what is the physical reasoning behind this? Why or in what way is the rotational constant dependent on mass?

  • $\begingroup$ I is the moment of inertia of the molecule, which is given by $$I = \mu R^2$$ R is the distance between the two atoms and $\mu$ is the reduced mass of a bimolecular system, given by $$\mu = \frac {m_1 m_2} {m_1 + m_2}$$ If the mass of either particle increases, then the reduced mass increases, causing $I$ to increase, which then causes $B$ to decrease. $\endgroup$ Dec 5, 2019 at 8:41
  • $\begingroup$ thank for simply restating my question and not answering it at all. $\endgroup$ Dec 5, 2019 at 14:29
  • $\begingroup$ How is the energy related to B? $\endgroup$
    – Buck Thorn
    Dec 5, 2019 at 17:54
  • 3
    $\begingroup$ See, it is pretty much the same with any quantum system (think of PIB, think of HO). A heavier particle means more classic-like behavior, which means "less discrete" energy spectrum, which means smaller transition spacings. $\endgroup$ Dec 5, 2019 at 21:13

1 Answer 1


First, take a look at classical physics. The angular momentum of a particle rotating in a plane is defined as $$L = I \omega$$ and its kinetic energy is $$E = \frac{1}{2} I \omega^2 = \frac{L^2}{2I}.$$

So if you formulate your energy in terms of the angular momentum of your rotating particle, you arrive at the inverse relation.

In analogy to the classical picture, the eigenvalues of the rotational Schrödinger equation, $$ E = hcBJ(J+1),$$ likewise depend quadratically on the angular momentum quantum number, and thus have a similar inverse dependence on the moment of inertia through $B$.


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