2.6 First results from the calculation of the state sum
The canonical partition function for a Hamiltonian which completely separates into subspaces can be
written in the form
The equation on the right side must fulfill the restriction
The grand canonical partition function of a Hamiltonian which completely separates into subspaces
can be written in the form
The big advantage of this partition function is the identity
i.e. states can be occupied independently and within the partition function an independent
sum over independent energy subspaces is performed, leading to a product over subspace
Finally we get
Just a reminder:
- All quantum mechanical particles with half value spin are called Fermions (electrons, holes,
...). They obey the Pauli principle, have antisymmetric complete wave functions and each
state can only be occupied once.
- All quantum mechanical particles with integer spin are called Bosons (photons, phonons,
...). They obey not the Pauli principle, have symmetric complete wave functions and there
is no restriction for the number of particles within one state.
For the grand canonical potential we find
holds we get
In this case we find for Fermions as well as for Bosons
This is the so called Boltzmann statistics. As we will see later the above assumption is fulfilled for
extremely deluded systems and at high temperatures (classical particles).
- Without much effort we could calculate the Fermi-, Bose- and Boltzmann statistics for the
grand canonical ensemble.
- Almost all calculations for solids are performed for the grand canonical ensemble.
are just Lagrange parameter (from a mathematically point of view); they allow to calculate
the partition function without any restrictions due to the particle number. From a physical
point of view
is the energy which a particles has when added to the system. Consequently
is a force leading to a particle flow.
- ; since
is the only extensive
parameter in the potential,
must be proportional to .
© J. Carstensen (Stat. Meth.)