What Is the Partition Function of a Diatomic Gas Molecule at Low Temperatures?

[Nylex]

According to QM, a diatomic gas molecule possesses rotational energy levels given by En = (1/2I)(h^2)n(n + 1). h is meant to be h-bar, Planck's constant over 2π here and I = moment of inertia. Energy level n has a degeneracy of 2n + 1.

Find the partition function of the rotational motion of a single such molecule.

Z = Σ gn.exp(-En/kT) = Σ (2n + 1)exp[-(n(n + 1)h^2)/kT]

Suppose T is sufficently small so that only the first 2 energy levels need to be considered. Find an expression for the mean rotational energy.

Here's where I'm stuck as I don't know which equation to use. Some of my notes say E avg = (1/Z)Σ En.gn.exp(-En/kT). In some of my other notes, there's an example for finding out the mean vibrational energy. E avg = (1/Z)(kT^2)dZ/dT. Obviously for rotation, the Boltzmann factor will be different as there's a different expression for En, but can I use this equation anyway?

Edit: I worked out Z for the first 2 states as Z = 1 + 3exp[-(h^2)/IkT].

 

[dextercioby]

Yes,as long as u're using the quantum canonical ensemble,you should be using:
$<E>=\frac{1}{Z^*}\sum_{n=1}^{n=2} E_{n}\exp({-\frac{E_{n}}{kT}})$,where the canonical state sum $Z^*$ is:
$$Z^* =\sum_{n=1}^{n=2} \exp({-\frac{E_{n}}{kT}})$$.
The problem gives u all the data u need to perform the calculations.Be grateful it's a quantum ensemble;it's all about correct summations,no more tricky integrals.

As for the vibrational energy,the question would be the same and the calculation would use the formula above,but with the proper spectrum of the Hamiltonian.

PS.In the case of quantum virtual statistical ensembles,it would be fair if we spoke about Von Neumann's factor;yet,we usually call it state sum,or sum over states.

 

[Nylex]

I have no idea about most of the stuff you just said to be honest, but ok (I've not heard the terms "quantum canonical ensemble", "canonical state sum", etc before, although our lecturer did use "sum over states"). Thanks .

 

[dextercioby]

I have no idea about most of the stuff you just said to be honest, but ok (I've not heard the terms "quantum canonical ensemble", "canonical state sum", etc before, although our lecturer did use "sum over states"). Thanks .

Then what kind of physics do they teach u there? Statistical physics,at least in its equilibrium part,is founded on the concept of "virtual statistic ensemble".Statistical interpretation of QM is based upon "quantum virtual statistical ensembles".
Depending on the external conditions imposed on the systems from the ensembe,equilibrium statistical ensembles are:microcanonic,canonic,macrocanonic,isobar-isothermal,generalized ad so on.All these ensembles come up in 2 versions:classical and quantum.Gibbs and Von Neumann are responsible for this part of equilibrium statistical physics.
So the canonical state sum is nothing but the partition function in the case of quantum canonical ensembles.The "weighing" propababilities u use when calculating the average of an observable on a statistical ensemble are nothing but the eigenvalues of the density operator,and so on...
I would suggest you read other sources of SM;i can't come up with good books by Englishmen or Americans,but the names of Kerson Huang,Bernard Diu,Schwabl,F.Mandl,L.D.Landau+E.Lifschitz could one day tell u somethig.

 

[Nylex]

Then what kind of physics do they teach u there? Statistical physics,at least in its equilibrium part,is founded on the concept of "virtual statistic ensemble".Statistical interpretation of QM is based upon "quantum virtual statistical ensembles".
Depending on the external conditions imposed on the systems from the ensembe,equilibrium statistical ensembles are:microcanonic,canonic,macrocanonic,isobar-isothermal,generalized ad so on.All these ensembles come up in 2 versions:classical and quantum.Gibbs and Von Neumann are responsible for this part of equilibrium statistical physics.
So the canonical state sum is nothing but the partition function in the case of quantum canonical ensembles.The "weighing" propababilities u use when calculating the average of an observable on a statistical ensemble are nothing but the eigenvalues of the density operator,and so on...
I would suggest you read other sources of SM;i can't come up with good books by Englishmen or Americans,but the names of Kerson Huang,Bernard Diu,Schwabl,F.Mandl,L.D.Landau+E.Lifschitz could one day tell u somethig.

Our lecture notes can be found here if you want to see. Mandl's book was one of the ones the lecturer recommended for the course, but he also suggested Bowley & Sanchez, "Statistical Mechanics" and that's the one I have.

 

[dextercioby]

Our lecture notes can be found here if you want to see. Mandl's book was one of the ones the lecturer recommended for the course, but he also suggested Bowley & Sanchez, "Statistical Mechanics" and that's the one I have.

The course is cleraly made by a nontheorist for nontheorists.That means it is outstanding when it comes to simplicity of exposure and comprhending,but,being written by a nontheorist,it virtually lacks when explaining the basis behind it all.Einstein came up with the notion of "virtual statistical ensemble" as his was famous for his ability of imagining experiments ("Gedankenexperiment"),yet,when VonNeumann restated Born's statistical interpretation of QM,he was very puzzled and remained so till the end of his life,notwilling to accept the way a theory was developing,a theory for the basis of which he contributed himself.He was the first one to accept Planck's idea of quantization.
To resume,i think your physics course on SM is superficial and does not cover the essence.
Did u know Bolzmann's formula is a particular case of Gibbs-Von Neumann axiomatical definition of entropy...??I didn't find it in the notes,maybe i should take a look again...