The Rayleigh-Ritz theorem forms the basis for the variational method, in which its expectation value for a trial wave function gives an upper bound on the ground state energy.
For a system coupled to a heat bath, its expectation value is related to the Helmholtz free energy in the Gibbs-Bogoliubov inequality. Its mean value is stationary in neighborhoods of its discrete eigenvalues, according to the Ritz theorem. In classical mechanics, the Routhian is defined so that it behaves like this function on all cyclic coordinates. The Rayleigh-Ritz theorem forms the basis for the variational method, in which its expectation value for a trial wave function gives an upper bound on the ground state energy. Defined in classical mechanics as a function of generalized coordinates and conjugate momenta, FTP name this quantum mechanical operator which corresponds to total energy.
For a system coupled to a heat bath, its expectation value is related to the Helmholtz free energy in the Gibbs-Bogoliubov inequality.
{Hamiltonian}