Boson gas...

lokofer

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

[tex] V(x)= \omega (k) x^{2} [/tex]


the question is if we know what the Partition function is [tex] Z= Z (\beta ) [/tex] 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 [tex] Z(\beta ) [/tex]

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

- I wish to calculate fro this...the "dispersion relation" [tex] \omega (k) [/tex] 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 [tex] \omega (k) [/tex] 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 refering 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 [tex] c(k)= \frac{d \omega }{dk} [/tex] 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.