is an idealized model in classical physics. It can be expected
in a conservative system, when a point mass m is connected to an
infinite counter mass by an ideal spring.
An ideal spring is characterized by Hooke's law
F = k x ,
according to which an elongation from the equilibrium position
requires a force F that is proportional to the elongation
x.
"Elongation" in this context can mean both, compression or stretching.
If the mass is left free after an elongation, it will move back
to the equilibrium position and beyond that.
Since there is no friction in a conservative system,
and since compression and stretching are equivalent, an infinite
oscillation will occur.
The force, which is necessary for a certain elongation, and the
force for the movement back to the equilibrium position are equal in
magnitude, but of opposite sign.
F = kx = -m .
The oscillation therefore can be described by the differential equation
+ (k / m ) x
= 0 .
The ansatz
x = A cos t
yields
-A ² cos t + ( k / m )
A cos t = 0
and therefore
² = k / m
or
=
.
The period of a full vibration movement is the reciprokal value of its
frequency ;
after this period, the elongation must be equal to the initial value.
x(t=0) = x(t=1/)
= x(t=2/) = ...
cos 0 = cos /
= cos 2 / = ...
This is fulfilled for /
= 2 .
Hence, the frequency is given by
=
.
The frequency of a harmonic oscillation therefore depends only on the
force constant k and the
mass m, but not
on the elongation of the spring.
A more realistic case is s model, in which
an ideal spring connects two finite masses m1 and
m2.
It turns out that one can still use
the above results, if the single mass m is replaced by the
reduced mass
µ = m1m2 /
(m1+m2).
This operator reflects the idealized character of the harmonic
oscillator model: both, compression and stretching, cause
an increase of the potential energy.
yields the following harmonic oscillator energy eigenvalues:
E = ( + ) h .
is the vibration quantum number, which
may be 0, 1, 2, ....
A typical feature of the harmonic oscillator is the constant
difference between neighbouring energy levels.
Molecular vibrations can be treated
according to the model of a spring and attached masses, on the one hand,
because chemical bonds do have certain characteristics of a spring:
they have force constants that are independent of temperature and independent
of the attachmed masses. The reason for this is that
the chemical bond is an electronic effect, and
the electronic Hamiltonian (Theoretical Chemistry Tutorial)
contains only the nuclear charges, but not the nuclear masses.
On the other hand, the potential energy of a chemical bond is very different
from that of an ideal spring: compression of atoms results in a
steeper energy increase, and stretching a bond longer and longer
( = dissociation) results in slower and slower increase of energy.
Potentials with such a characteristic are used in the
anharmonic oscillator model.
However, the Schrödinger equation
of anharmonic oscillators can only be solved in special cases.
It is the exact solution of the
Schrödinger equation, which makes the
harmonic oscillator such an important model system, also for
molecules.
The selection rule
for the harmonic oscillator is
v = ±1.
Transitions that absorb or emit light
may occur between neighbouring energy levels only.
Since the energy difference between all neighbouring energy levels is
identical, the spectrum of a harmonic oscillator would contain only a
single line for each
vibration mode.
The lowest allowed value of the quantum number is 0,
which corresponds to the energy E =
h.
Each molecular vibration therefore has a minimum energy, even at
a temperature of 0 K. The vibrational ground state v=0 therefore
often is called zero point vibration. Its existence is the reason for
the thermodynamic quantities
U and
H, and it ensures the validity of the
Heisenberg uncertainty principle also at 0 K.
Molecular oscillations occur in near
IR; examples are the halogen hydrides
wave numbers:
HF =
3962 cm-1,HCl =
2886 cm-1,HBr =
2559 cm-1,HI =
2230 cm-1.
The associated force constants indicate the strength of the
recpective chemical bonds:
kHF = 880 N/m,kHCl = 478 N/m,kHBr = 382 N/m,kHI = 291 N/m.
Vibrations that do not involve hydrogen atoms occur at significantly
lower frequencies, because of the significantly larger reduced mass.
For example, the wave number of DCl is
2069 cm-1
(k = 478 N/m) and that of
Cl2
is 557 cm-1
(k = 320 N/m).
.
t
yields
.
;
after this period, the elongation must be equal to the initial value.
.
=
E
) h 
Theoretical Chemistry Tutorial)
contains only the nuclear charges, but not the nuclear masses.
v = ±1.
Transitions that absorb or emit light
may occur between neighbouring energy levels only.
Since the energy difference between all neighbouring energy levels is
identical, the spectrum of a harmonic oscillator would contain only a
single line for each
vibration mode.
![[idealized spectrum]](http://fptchlx02.tu-graz.ac.at/im_tutorial/spektrum_harmon.gif)
Postulates.
Rigid rotor.
