Exploring the Dispersion Relation for a Boson Gas Under an Harmonic Potential

[lokofer]

Let's suppose we have a Boson Non-interacting gas under an Harmonic potential so

$$V(x)= \omega (k) x^{2}$$


the question is if we know what the Partition function is $$Z= Z (\beta )$$ we could obtain the specific Heat, and other important Thermodinamical entities...but could we know what the "dispersion relation" w(k) for k real is? , i have looked several books about "Solid State" but i don't find any info about how to get dispersion relations using partition functions or similar..or if we can find an Integral or differential equation for the w(k)..thanks

 

[Epicurus]

We can really only know the partition function if we know w(k). For a given Hamiltonian for which we can find the eigenspectrum, the partition function for n non-interacting, once we know the partition function for a single particle in this potential we are able to formulate the many-body partition function. I this what you are asking is to calculate the response function for the system, which is different from w(k)

 

[lokofer]

Sorry "Epicurus" I'm not Brittish or American so my english sometimes sounds ambigous..my problem is..

-Let's suppose we know the TOTAL partition function for the system $$Z(\beta )$$

- If we have a Non-interacting Boson gas we have that: $$Z(\beta)= \prod _k Z_k (\beta)$$

- I wish to calculate fro this...the "dispersion relation" $$\omega (k)$$ using the functions i know (Total partition function and Specific Heat, Gibss function and similar that can be obtained from the Total partition function )... for example getting a differential equation or other type of equation for $$\omega (k)$$ so it can be solved by numerical methods to obtain the "frecuencies"..Hope it's clearer (my question) now...

 

[Epicurus]

1-The expression you have written down in the third line is incorrect.

 

[lokofer]

- Are you referring that for a Non-interacting gas the "total partition function" (Harmonic approach) isn't equal to the product of all the partition function for all the particles taking N=1 ?...

 

[Epicurus]

Yes that correct. You talking about the distinguishable case, not the bosonic case.

 

[lokofer]

- Well in any case...is there any form to obtain the "structure" (unit cell) of the gas or the dispersion relation, speed of sound $$c(k)= \frac{d \omega }{dk}$$ or any quantity related to the "frecuencies"...? I know that from the partition function you could calculate "Entropy" , "Energy" (U) and other Thermodinamical functions but not the "frecuencies"..perhaps you could use X-ray scattering or other method but if you don't know the "shape" (unit cell) of the gas i think you can't do anything.