150
3 Ensembles: Systems of Particles
in a linear momentum interval dp: that is, (V , ,)d can be obtained by calculating
the number of momentum states associated with dp.
We shall consider translational motion as inherently quantum mechanical, and
thus note that the Heisenberg Uncertainty Principle, which may be expressed in
each dimension as ∼ h, then necessitates the introduction of a factor h −3
into the calculation of the number of momentum states contained in a momentum
interval dp. The expression thereby obtained for the energy density of states is thus
(V , ,)d =
1
h 3
V
d p dr
dp
=
1
h 3
V
2π
0
π
0
p
2 sin ϑdϑdϕdr
dp .
We shall employ Cartesian coordinates to represent the position, r, of a particle
in the cylindrical column and spherical polar coordinates to represent its momentum
p. As the energy of a particle in a gravitational field is given by
(r, p) =
p 2
2m
+ mgz ,
with z representing the vertical direction in the cylindrical gas column while g is
the acceleration due to gravity (acting downwards), the integrations over the x and
y coordinates can be carried out trivially, to give πR 2 ≡ A, the cross-sectional area
of the cylindrical column of gas, and the integration over the two angles ϑ and ϕ
associated with the particle momentum gives an additional factor 4π . This gives the
energy density of states as
(V , ,)d =
4π A
h 3
H
0
p
2 (z) dz
dp ,
with H the height of the gas column. From the expression for the particle energy,
we obtain the magnitude of the linear momentum and its differential as
p = [2m(( − mgz)]
1
2
and dp = [2m(( − mgz)]
−
1
2 m d
so that the energy density of states is thus given as
=
4π A(2m)
1
2 m
h 3
z u
z
(( − mgz)
1
2 dz
= −
4π A(2m)
3
2
3gh 3
(( − mgz)
3
2
u
.
3 Ensembles: Systems of Particles
in a linear momentum interval dp: that is, (V , ,)d can be obtained by calculating
the number of momentum states associated with dp.
We shall consider translational motion as inherently quantum mechanical, and
thus note that the Heisenberg Uncertainty Principle, which may be expressed in
each dimension as ∼ h, then necessitates the introduction of a factor h −3
into the calculation of the number of momentum states contained in a momentum
interval dp. The expression thereby obtained for the energy density of states is thus
(V , ,)d =
1
h 3
V
d p dr
dp
=
1
h 3
V
2π
0
π
0
p
2 sin ϑdϑdϕdr
dp .
We shall employ Cartesian coordinates to represent the position, r, of a particle
in the cylindrical column and spherical polar coordinates to represent its momentum
p. As the energy of a particle in a gravitational field is given by
(r, p) =
p 2
2m
+ mgz ,
with z representing the vertical direction in the cylindrical gas column while g is
the acceleration due to gravity (acting downwards), the integrations over the x and
y coordinates can be carried out trivially, to give πR 2 ≡ A, the cross-sectional area
of the cylindrical column of gas, and the integration over the two angles ϑ and ϕ
associated with the particle momentum gives an additional factor 4π . This gives the
energy density of states as
(V , ,)d =
4π A
h 3
H
0
p
2 (z) dz
dp ,
with H the height of the gas column. From the expression for the particle energy,
we obtain the magnitude of the linear momentum and its differential as
p = [2m(( − mgz)]
1
2
and dp = [2m(( − mgz)]
−
1
2 m d
so that the energy density of states is thus given as
=
4π A(2m)
1
2 m
h 3
z u
z
(( − mgz)
1
2 dz
= −
4π A(2m)
3
2
3gh 3
(( − mgz)
3
2
u
.