Wednesday, 25 October 2017

How do you determine the value of the degeneracy factor in the partition function?


In the partition function, expressed as Z=jgjeβEj I'm wondering what determines the gj factor. I've been trying to look around the internet for an explanation of it but I can't find one. I guess it is the number of degenerate states in a given energy level? How do you determine how many degenerate states there are? A simple example involving how to determine the gj factor would be great. Thanks for the help!



Answer




Consider a quantum system with state (Hilbert) space H. For simplicity, let the Hamiltonian H of the system have discrete spectrum so that there exists a basis |n with n=0,1,2, for the state space consisting of eigenvectors of the Hamiltonian. Let ϵn denote the energy corresponding to each eigenvector |n, namely H|n=ϵn|n Now, it may happen that one or more of the energy eigenvectors |n have the same energy. In this case, we say that their corresponding shared energy eigenvalue is degenerate. It is therefore often convenient to have a the concept of the energy levels Ej of the system which are simply defined as the sequence of distinct energy eigenvalues in the spectrum of the Hamiltonian. So, whereas one can have ϵn=ϵm if nm, one cannot have En=Em if nm. Moreover, it is often convenient to label the energy levels in increasing index order so that Em<En whenever $m

The degeneracy gn of the energy level En is defined as the number of distinct energy eigenvalues ϵm for which ϵm=En. For simplicity, we assume that none of the levels is infinitely degenerate so that gn1 is integer for all n.


The partition function of a system in the canonical ensemble is given by Z=neβϵn In other words, the sum is over the state labels, not over the energy levels. However, noting that whenever there is degeneracy, sum of the terms in the sum will be the same, we can rewrite the partition function as a sum over levels Z=ngneβEn The degeneracy factor is precisely what counts the number of terms in the sum that have the same energy.


As for a simple example, consider a system consisting of two, noninteracting one-dimensional quantum harmonic oscillators. The eigenstates of this system are |n1,n2 where n1,n2=0,1,2, and the corresponding energies are ϵn1,n2=(n1+n2+1)ω. The canonical partition function is given by Z=n1,n2=0eβϵn1,n2=eβωn1=0,n2=0+eβ(2ω)n1=1,n2=0+eβ(2ω)n1=0,n2=1+ If you think about it for a moment, you'll notice that, in fact, the energy levels of this composite system are En=nω,n1+n2=n and that the degeneracy of the nth energy level is gn=n so that the partition function can also be written in the form that uses energy levels and degeneracies as follows: Z=n=1gneβEn=eβωn=1+2eβ(2ω)n=2+3eβ(3ω)n=3+


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...