166
3 Ensembles: Systems of Particles
nondegenerate and the excited level has degeneracy 3. Discuss any differences
between the behaviours of these two systems.
16. For a microcanonical ensemble the number of microstates associated with a
given configuration {n 0 , n 1 , n 2 , . . .} with a total of N particles is given by the
expression
(n 0 , n 1 , n 2 , . . . , ) =
N !
n 0 !n 1 !n 2 ! · · ·
,
with it being understood that the total number of particles is N and the
total energy for the configuration is E. Provided that the basic postulate of
statistical mechanics holds, we may determine the most probable configuration
by finding the maximal value for via setting the differential dln to zero,
while recognizing that any changes to the numbers n i remain subject to the
constraints placed upon the total number of particles and the total energy for
the configuration. This requirement can be represented as
d ln =
i
∂ ln
∂n i
dn i = 0 ,
taken together with
i
dn i = 0 and
i
i dn i = 0 required in order that N and
E have fixed values: these three separate statements can be combined using the
method of Lagrange multipliers as
d ln =
i
∂ ln
∂n i
+ λ − ββ i
dn i = 0 .
This expression involves two Lagrange multipliers, λ and β, one for each
constraint that has been imposed. Show that the Boltzmann distribution follows
from this expression.
17. Consider a spin1
2 particle, such as an atom with no net electronic angular
momentum, but whose nucleus has an intrinsic (nuclear) spin angular momentum I and an associated magnetic moment μ = g I μ N I, with a constant of
proportionality that is the product of g I , known as the nuclear g-factor (specific
to the nucleus) and the nuclear magneton μ N that determines the order of
magnitude of nuclear magnetic moments. The interaction between a magnetic
moment and an applied (i.e., external) magnetic field is given by the Zeeman
Hamiltonian H = −μ · B, which determines the energy eigenstates that the
spin particle may access when such a field is applied. The direction of the
applied magnetic field B is typically employed to define the z-axis direction.
The nuclear angular momentum energy eigenstates are designated (in Dirac
notation) |I m I and have associated (eigen)energies E spin = −M I g I μ N B.
3 Ensembles: Systems of Particles
nondegenerate and the excited level has degeneracy 3. Discuss any differences
between the behaviours of these two systems.
16. For a microcanonical ensemble the number of microstates associated with a
given configuration {n 0 , n 1 , n 2 , . . .} with a total of N particles is given by the
expression
(n 0 , n 1 , n 2 , . . . , ) =
N !
n 0 !n 1 !n 2 ! · · ·
,
with it being understood that the total number of particles is N and the
total energy for the configuration is E. Provided that the basic postulate of
statistical mechanics holds, we may determine the most probable configuration
by finding the maximal value for via setting the differential dln to zero,
while recognizing that any changes to the numbers n i remain subject to the
constraints placed upon the total number of particles and the total energy for
the configuration. This requirement can be represented as
d ln =
i
∂ ln
∂n i
dn i = 0 ,
taken together with
i
dn i = 0 and
i
i dn i = 0 required in order that N and
E have fixed values: these three separate statements can be combined using the
method of Lagrange multipliers as
d ln =
i
∂ ln
∂n i
+ λ − ββ i
dn i = 0 .
This expression involves two Lagrange multipliers, λ and β, one for each
constraint that has been imposed. Show that the Boltzmann distribution follows
from this expression.
17. Consider a spin1
2 particle, such as an atom with no net electronic angular
momentum, but whose nucleus has an intrinsic (nuclear) spin angular momentum I and an associated magnetic moment μ = g I μ N I, with a constant of
proportionality that is the product of g I , known as the nuclear g-factor (specific
to the nucleus) and the nuclear magneton μ N that determines the order of
magnitude of nuclear magnetic moments. The interaction between a magnetic
moment and an applied (i.e., external) magnetic field is given by the Zeeman
Hamiltonian H = −μ · B, which determines the energy eigenstates that the
spin particle may access when such a field is applied. The direction of the
applied magnetic field B is typically employed to define the z-axis direction.
The nuclear angular momentum energy eigenstates are designated (in Dirac
notation) |I m I and have associated (eigen)energies E spin = −M I g I μ N B.