Biomedical Engineering Reference
In-Depth Information
ε
2
k
s
x
2
d
2
ψ
d
x
2
8π
2
μ
h
2
1
+
−
=
0
(11.21)
where
k
s
is the force constant. The energy ε and wavefunction ψ now describe the nuclear
vibrations. This is a more difficult differential equation that the ones discussed earlier in this
chapter, but fortunately it is well known in many areas of engineering, physics and applied
mathematics. Detailed solution of the vibrational Schrödinger equation is given in all the
advanced texts and to cut a long story short we interest ourselves in bound states and so
impose a condition ψ
→
→±∞
. This gives rise to energy quantization described by
a single quantumnumber that is written
v
. The general formula for the vibrational energies is
0as
x
k
s
μ
v
h
2π
1
2
ε
v
=
+
(11.22)
v
=
0, 1, 2, ...
The harmonic oscillator is not allowed to have zero vibrational energy. The smallest
allowed energy is when
v
0, and this is referred to as the
zero point energy
. This finding
is in agreement with Heisenberg's uncertainty principle, for if the oscillating particle had
zero energy it would have also zero momentum and would be located exactly at the position
of the minimum in the potential energy curve.
The normalized vibrational wavefunctions are given by the general expression
=
1
2
v
v
1/2
β
π
H
v
(ξ ) exp
ξ
2
2
ψ
v
(ξ )
=
−
(11.23)
!
where
2π
√
k
s
μ
h
β
x
β
=
and ξ
=
The polynomials
H
v
are called the
Hermite polynomials
; they are solutions of the second-
order differential equation
d
2
H
dξ
2
2ξ
d
H
−
dξ
+
=
2
vH
0
(11.24)
and are most easily found from the recursion formula
H
v
+
1
(ξ )
=
2ξ
H
v
(ξ )
−
2
vH
v
−
1
(ξ )
The first few are shown in Table 11.1.
Squares of the vibrational wavefunctions for
v
=
0 and
v
=
5 (with numerical values
appropriate to
14
N
2
) are shown as Figures 11.10 and 11.11.
Table 11.1
The first few Hermite polynomials H
v
(ξ )
v
H
v
(ξ )
0
1
1
2
ξ
2
4
ξ
2
−
2
−
3
8
ξ
3
12
ξ
−
+
4
16
ξ
4
48
ξ
2
12
−
+
5
32
ξ
5
160
ξ
3
120
ξ
