E
1
2
h
ϖ
3
2
h
ϖ
5
2
h
ϖ
7
2
h
ϖ
x
9
2
h
ϖ
11
2
h
ϖ
The
energy
levels of a
harmonic
oscillator
are evenly
spaced
with
separation
·
ω
, with
ω
= (k/m)
1/2
.
Even in its
lowest
state, an
oscillator
has an
Harmonic oscillator . . . Quantum mechanically . Energy levels
v=0
v = 1
v =2
v =3
v = 4
v =5
v=6
E=hv 1
2
v
Harmonic oscillator Quantum mechanically Vibration Spectroscopy
V R=V R
e
dV
dR
dR
e
1
2
d
2
V
dR
2
dR
e
2
1
8
d
3
V
dR
3
dR
e
3
...
Taylor expansion
0
small
0
V R= 1
2
d
2
V
dR
2
dR
e
2
= 1
2
k⋅dR
e
2
d
2
V
dR
2
=
k
Harmonic oscillator . . . Quantum mechanically
We note relation between bond energy D ;
bond order and force constant k
Harmonic oscillator . . . Quantum mechanically
The three
normal
modes of
H
2
O. The
mode v
2
is
predomina
ntly
bending,
and occurs
at lower
wavenumb
er than the
other two.