Forbidden Rotations (Griffiths)

This is an explanation of why rotation along the long axis of molecule is forbidden by quantum mechanics.

a) What is the definition of moment of inertia $I$ for rotational motion?

b) For atomic Hydrogen estimate the electron moment of inertia in its ground state. Compare this with the moment of inertia of nucleus.

c) What is the effective radius in the moment of inertia for molecular Hydrogen?

d) What is the effective radius in the moment of inertia for spinning along the long axis?

e) What is the ratio of the two previous moments of inertia? What is the temperature required to for the Hydrogen to spin along  its long axis?

a)Definition of momentum of inertia is:

$I=∑_i m_i*R_i^2=∫ R*d m$    the integral is done over the whole existent mass (volume)
In the International System I is measured in $kg*m^2.$

b) Bohr radius is $a=0.53A=0.53*10^{-10} m$. The moment of inertia of electron of Hydrogen in ground state is

$I_0=ma^2=9.1*10^{-31}*(0.53*10^{-10})^2=2.556*10^{-51} (kg*m^2)$

c) If the molecule rotates about axis c) (short axis) the effective radius in the moment of Inertia is about the radius of a hydrogen atom (Bohr radius). In fact the distance between the two atoms of hydrogen is $d=74 pm=0.74 A$

d) If the molecule rotates along the long axis d), it rotates in fact both its nucleus (protons) about their axis. Therefore the effective radius for moment of inertia is exactly proton radius. The radius of proton is $R_p=0.8775 fm=0.8775*10^{-15} m$

e)

$I_c=2*m_p*(d/2)^2=m_p*d^2/2=1.66*10^{-27}*(0.74*10^{-10} )^2/2=4.545*10^{-48} (kg*m^2 )$

$I_d=2*2/5*m_p*R_p^2=4/5*1.66*10^{-27}*(0.8775*10^{-15} )^2=1.02*10^{-57} (kg*m^2)$

$I_c/I_d =4.44*10^9$

f) The rotation energies of a spinner are quantified as

$E_{rot}=(\hbar^2*m^2)/2I$   with $m=0,+or-1,+or-2,…$

First non zero energy is

$(\hbar^2)/(2I_d )=k T$   hence

$T=(\hbar^2)/(2kI_d )=(1.055*10^{-34} )^2/(2*1.38*10^{-23}*1.02*10^{-57} )=3.95*10^{11} K$