4.3(b)
The present subsection was developed using the partition function of the canonical formalism, because it was easier for me to integrate the exponential functions of the Gaussian bell, immediately afterwards I obtained the internal energy with the expressions that we obtained in class and substituted that relation in my partition function applying the Stirling approximation (thermodynamic limit), this way you can obtain entropy in terms of energy, and obtain thermodynamic variables. Finally, for the thermodynamic limit we understand that the participation of each oscillator and each dimension of the oscillator becomes less and less significant as the number of oscillators increases, that can further simplify the expression when considering all the equal frequencies, and therefore we can simplify the expression.
You partition function is not correct. The Boltzman factor is "N!", not "(dN)!". This is since the different degrees of freedom of a particle are distinguishable. only the identities of the particles is not! However, most importantly, you are using the canonical ensemble, but that ensemble does not yield the N(E), the number of microstates with energy less than E.
ResponderBorrarYou then do another mistake, in calculating the entropy as if the partition function was the one of the microcanonical ensemble.
Your last comment, "Finally, for the thermodynamic limit we understand that the participation of each oscillator and each dimension of the oscillator becomes less and less significant as the number of oscillators increases, that can further simplify the expression when considering all the equal frequencies, and therefore we can simplify the expression." I assume it was meant as an answer to the question about the scaling of the trap frequencies in the thermodynamic limit. But I don't understand it.