26
Y. Ozaki and Y. Morisawa
ψ υ = N υ H υ (
√
α Q) exp
−
α Q
2
2
(2.22)
where N υ denotes a normalization constant, H υ is a Hermite polynomial, and α =
2π
√ μk
h
Wave functions to υ = 0, 1, and 2 are as follows.
ψ 0 = (α/π)
1/4 exp
−α Q
2
/2
ψ 1 = (α/π)
1/4
(2α)
1/2 Q exp
−α Q
2
/2
ψ 2 = (α/π)
1/4
1/
√
2
2α Q
2
− 1
exp
−α Q
2
/2
(2.23)
These formulas clearly show that a wave function of harmonic oscillator is an
even function when a quantum number is an even number but is an odd function
when a quantum number is an odd number. Figure 2.7 shows a potential energy, (a)
wave functions, Ψ υ , (b) probability density function, Ψ
2
υ and energy eigen values,
E υ , of the harmonic oscillator.
2.2.2.3 Vibrations of Polyatomic Molecules
As examples of vibrations of polyatomic molecules let us consider normal vibrations
of carbon oxide and water; these molecules are examples of linear and non-linear
triatomic molecules, respectively. CO 2 has 3 × 3−5 = 4 normal vibrations. Figure 2.8
exhibits its four normal modes in CO 2 , 1, 2, 3a and 3b. The normal vibrations 1
and 2 are vibrations where two CO bonds stretch and contract in phase (1) and
out of phase (2), respectively, called symmetric and anti-symmetric stretching
vibrations. Meanwhile, the vibrations 3a and 3b are both vibrations that the angle
of OCO changes and called bending vibrations. While the vibrations 3a and 3b are
independent of each other, energies required for the vibrations are principally equal
Fig. 2.8 Normal modes in
CO 2 . (+ and – denote
vibrations going upward and
downward, respectively, in
the direction perpendicular
to the paper plane). 1,
symmetric stretching
vibration. 2, antisymmetric
stretching vibration. 3a, 3b,
degenerate bending
vibrations