# PARTITION FUNCTION AND SPECTRUM

# PARTITION FUNCTION AND SPECTRUM

A direct application of the calculation of the partition function is provided by determining the spectrum of a Hamiltonian (which is assumed to be discrete, for simplicity). In this chapter, only situations where the Hamiltonian eigenvalues are not degenerate are considered. The power of the path integral formalism is then illustrated beyond simple perturbation theory: it is used to derive a variational principle. It is evaluated, by applying the steepest descent method, in the case of *O(N)* symmetric Hamiltonians, in the large *N* limit. Steepest descent calculations of path integrals involve determinants of differential operators that are given a perturbative definition. This chapter shows how to calculate the spectrum of a Hamiltonian in the semi-classical limit or WKB approximation, starting from the semi-classical expansion of the partition function obtained earlier. It is demonstrated that the ground state of simple quantum systems is not degenerate.

*Keywords:*
partition function, spectrum, Hamiltonians, ground state, path integral, variational principle, steepest descent method, semi-classical expansion

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .