2. Calculate the number of accessible microscopic states of a system of two
localized and independent quantum oscillators, with fundamental frequencies
and
, respectively, and total energy
.
;
;
.
3. Consider a classical one-dimensional system of two noninteracting
particles of the same mass The motion of the particles is restricted to
a region of the axis between and . Let and
be the position coordinates of the particles, and and be the
canonically conjugated momenta. The total energy of the system is between
and
. Draw the projection of phase space in a plane defined by
position coordinates. Indicate the region of this plane that is accessible
to the system. Draw similar graphs in the plane defined by the momentum
coordinates.
4. The position of a one-dimensional harmonic oscillator is given by
5. Now consider the classical phase space of an ensemble of identical
one-dimensional oscillators with energy between and
. Given
the energy , we have an ellipse in phase space. So, the accessible region
in phase space is a thin elliptical shell bounded by the ellipses associated
with energies and
, respectively. Obtain an expression for
the small area of this elliptical shell between and .
Show that the probability may also be given by
, where is the total area of the elliptical shell. This is one of the few
examples where we can check the validity of the ergodic hypothesis and the
postulate of equal a priori probabilities.
6. Consider a classical system of localized and weakly interacting
one-dimensional harmonic oscillators, whose Hamiltonian is written as
7. The spin Hamiltonian of a system of localized magnetic ions is given
by
8. In a simplified model of a gas of particles, the system is divided into cells of unit volume. Find the number of ways to distribute
distinguishable particles (with
) within cells, such that
each cell may be either empty or filled up by only one particle. How would
your answer be modified for indistinguishable particles?
9. The atoms of a crystalline solid may occupy either a position of
equilibrium, with zero energy, or a displaced position, with energy
. To each equilibrium position, there corresponds a unique
displaced position. Given the number of atoms, and the total energy ,
calculate the number of accessible microscopic states of this system.