A diatomic molecule rotating about it's axis of symetry

[quasar987]

In my thermo text, they consider a diatomic molecule that is rotating about the axis joining the two atoms (also the axis of symetry) and make a quantum argument involving the energy levels of a rigid rotator to conclude that kT<<$\Delta E$ for this particular degree of freedom.

But isn't this kind of rotation impossible to begin with in QM? One of my HW problem in QM last year was to show that the energy needed to put a totally smooth cylinder in rotation about it's axis of symetry is infinite. This follows a principle like "it is not possible to put in motion an object that you cannot tell is in motion". And this is precisely what we have here with the diatomic molecule so I find it peculiar that we are even considering such rotations.

 

[Einstein Mcfly]

In my thermo text, they consider a diatomic molecule that is rotating about the axis joining the two atoms (also the axis of symetry) and make a quantum argument involving the energy levels of a rigid rotator to conclude that kT<<$\Delta E$ for this particular degree of freedom.

But isn't this kind of rotation impossible to begin with in QM? One of my HW problem in QM last year was to show that the energy needed to put a totally smooth cylinder in rotation about it's axis of symetry is infinite. This follows a principle like "it is not possible to put in motion an object that you cannot tell is in motion". And this is precisely what we have here with the diatomic molecule so I find it peculiar that we are even considering such rotations.

I might be wrong to think about it this way, but the symmetry operator that would describe such a rotation would just be the identity. The identity always commutes with the Hamiltonian and the change in energy for such a "rotation" would be zero and hence not really a rotation.

I'll admit that I'm having some trouble understanding the second part of your post though. Perhaps I don't know what you mean by "totally smooth".

 

[vanesch]

In my thermo text, they consider a diatomic molecule that is rotating about the axis joining the two atoms (also the axis of symetry) and make a quantum argument involving the energy levels of a rigid rotator to conclude that kT<<$\Delta E$ for this particular degree of freedom.

I think you misunderstood the text (or otherwise you better put it in a dustbin). The rigid rotor of 2 atoms is rotating around an axis of rotation which is PERPENDICULAR to the axis of rotational symmetry. You know, like those halters weightlifters use: imagine the weightlifter holding the thing up, and now you make it spin around the vertical axis...

As you say, it would make no sense to talk about a rotation around the rotational axis of symmetry...

 

[quasar987]

I'll admit that I'm having some trouble understanding the second part of your post though. Perhaps I don't know what you mean by "totally smooth".

The exercice consisted in considering a cylinder (make that a ring if you want) made up of N atoms and showing that in the limit N-->infty, the energy needed to set it in rotation about its axis of symetry was infinite.

 

[cesiumfrog]

The exercice consisted in considering a cylinder (make that a ring if you want) made up of N atoms and showing that in the limit N-->infty, the energy needed to set it in rotation about its axis of symetry was infinite.

I assume each atom's mass was proportional to 1/N so that the total mass remains constant? Was the exercise using classical mechanics, statistical thermodynamics or QM?

The ramifications of things being in principle indistinguishable is somewhat unintuitive, so it would be good to have simple examples handy.

 

[quasar987]

I assume each atom's mass was proportional to 1/N so that the total mass remains constant? Was the exercise using classical mechanics, statistical thermodynamics or QM?

The ramifications of things being in principle indistinguishable is somewhat unintuitive, so it would be good to have simple examples handy.

It's Gasiorowicz Ch.7 #12:

Consider the following model of a perfectly smooth cylinder. it it a ring of equally spaced, identical particles, with mass M/N, so that the mass of the ring is M and its moment of inertia MR², with R the radius of the ring. Calculate the possible values of the angular momentum. Calculate the energy eigenvalues. What is the energy difference btw the ground state of zero angular momentum, and the first rotational state? Show that this approaches infinity as N-->oo. Constrast this with the comparable energy for a thick "nicked" cylinder, which lacks the symetry under the rotation through 2pi/N radians. This exemple implies that it is impossible to set a perfectly smooth cylinder in rotation, which is consistent with the fact that for a perfectly smooth cyinder such a rotation would be unobservable.