A diatomic molecule rotating about it's axis of symetry

In summary: It's Gasiorowicz Ch.7 #12:The rotational energy of a perfectly smooth cylinder is infinite, while the energy for a "nicked" cylinder is finite.
  • #1
quasar987
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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<<[itex]\Delta E[/itex] 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.
 
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  • #2
quasar987 said:
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<<[itex]\Delta E[/itex] 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".
 
  • #3
quasar987 said:
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<<[itex]\Delta E[/itex] 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...
 
  • #4
Einstein Mcfly said:
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.
 
  • #5
quasar987 said:
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.
 
  • #6
cesiumfrog said:
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.
 

Related to A diatomic molecule rotating about it's axis of symetry

1. What is a diatomic molecule?

A diatomic molecule is a molecule consisting of two atoms bonded together. Examples of diatomic molecules include oxygen (O2), nitrogen (N2), and hydrogen (H2).

2. What does it mean for a molecule to rotate about its axis of symmetry?

When a molecule rotates about its axis of symmetry, it means that the molecule is spinning around an imaginary line that runs through its center and divides it into two equal halves. This axis of symmetry is often referred to as the principal axis of rotation.

3. How does the rotation of a diatomic molecule affect its properties?

The rotation of a diatomic molecule can affect its properties in several ways. For example, it can change the molecule's moment of inertia, which in turn affects its rotational energy and vibrational energy. Additionally, the orientation of the molecule's axis of symmetry can affect its dipole moment, which is a measure of its polarity.

4. What factors influence the rotational behavior of a diatomic molecule?

The rotational behavior of a diatomic molecule is influenced by several factors, including its moment of inertia, the distance between the two atoms, and the strength of the bond between them. Additionally, external forces such as electric or magnetic fields can also affect the molecule's rotation.

5. How is the rotational motion of a diatomic molecule described in quantum mechanics?

In quantum mechanics, the rotational motion of a diatomic molecule is described using the Schrödinger equation, which takes into account the molecule's moment of inertia and the potential energy due to its rotation. The solutions to this equation provide information about the energy levels and wavefunctions associated with the molecule's rotational motion.

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