5.5 Simple Harmonic Oscillator Ensembles
241
1
2 mv 2
x +
1
2 mv 2
y +
1
2 mv 2
z . We note that the total energy E in this case contains three
terms, each of which is quadratic in a Cartesian velocity component.
The second simplest type of particle is a one-dimensional simple harmonic
oscillator of mass m, which may be represented by a point particle constrained
to move symmetrically along a Cartesian direction (say the x-direction) about a
fixed position (or origin) due to a constraining (or restoring) force described by
Hooke’s law F = −kx, with a corresponding potential energy V (x) =
1
2 kx 2 .
Such an oscillator will clearly possess a (translational) kinetic energy T =
1
2 mv 2
x
and an associated Newtonian force F T = ma = m ¨
x, with ¨
x being the second
time derivative of x. The total energy E for a SHO is then the sum of its kinetic
and potential energies, namely, E =
1
2 mv 2
x +
1
2 kx 2 . The total energy E of a onedimensional SHO is constant and is determined by the initial condition (set at time
t = 0) imposed either by releasing the particle at a distance x max from its origin or
by impulsively giving the point particle a speed v max at the origin. The total energy
E for each vibrational degree of freedom thus has two quadratic components (i.e.,
kinetic energy and potential energy), rather than one quadratic component, as for
translational motion.
We may determine when the classical equipartition theorem is applicable by
examining our statistical mechanical expression for the thermodynamic internal
energy per particle, u(T , V ), for these two types of motion. For one-dimensional
(1D) translational motion modelled as a particle in a 1D box, we may write the 1D
analogue of Eq. (3.2.21) for the canonical partition function for three-dimensional
(3D) translational motion as
z(T , L) =
L
(T )
,
in which L is the length of the 1D box and (T ) is the usual thermal de Broglie
wavelength. The thermodynamic internal energy per particle, u(T , L), is then given
by
u(T , L) = k B T
2
∂ ln z
∂T
L
=
1
2 k B T .
We could also have written expression (3.2.21) for 3D translational motion as
z(T , L x , L y , L z ) =
L x L y L z
3 (T )
,
with V = L x L y L z the volume of the (rectangular) box, from which it will be clear
that the thermodynamic internal energy per particle is given by
u(T , L x , L y , L z ) =
3
2 k B T .
241
1
2 mv 2
x +
1
2 mv 2
y +
1
2 mv 2
z . We note that the total energy E in this case contains three
terms, each of which is quadratic in a Cartesian velocity component.
The second simplest type of particle is a one-dimensional simple harmonic
oscillator of mass m, which may be represented by a point particle constrained
to move symmetrically along a Cartesian direction (say the x-direction) about a
fixed position (or origin) due to a constraining (or restoring) force described by
Hooke’s law F = −kx, with a corresponding potential energy V (x) =
1
2 kx 2 .
Such an oscillator will clearly possess a (translational) kinetic energy T =
1
2 mv 2
x
and an associated Newtonian force F T = ma = m ¨
x, with ¨
x being the second
time derivative of x. The total energy E for a SHO is then the sum of its kinetic
and potential energies, namely, E =
1
2 mv 2
x +
1
2 kx 2 . The total energy E of a onedimensional SHO is constant and is determined by the initial condition (set at time
t = 0) imposed either by releasing the particle at a distance x max from its origin or
by impulsively giving the point particle a speed v max at the origin. The total energy
E for each vibrational degree of freedom thus has two quadratic components (i.e.,
kinetic energy and potential energy), rather than one quadratic component, as for
translational motion.
We may determine when the classical equipartition theorem is applicable by
examining our statistical mechanical expression for the thermodynamic internal
energy per particle, u(T , V ), for these two types of motion. For one-dimensional
(1D) translational motion modelled as a particle in a 1D box, we may write the 1D
analogue of Eq. (3.2.21) for the canonical partition function for three-dimensional
(3D) translational motion as
z(T , L) =
L
(T )
,
in which L is the length of the 1D box and (T ) is the usual thermal de Broglie
wavelength. The thermodynamic internal energy per particle, u(T , L), is then given
by
u(T , L) = k B T
2
∂ ln z
∂T
L
=
1
2 k B T .
We could also have written expression (3.2.21) for 3D translational motion as
z(T , L x , L y , L z ) =
L x L y L z
3 (T )
,
with V = L x L y L z the volume of the (rectangular) box, from which it will be clear
that the thermodynamic internal energy per particle is given by
u(T , L x , L y , L z ) =
3
2 k B T .