2 Principles and Characteristics of NIR Spectroscopy
25
H
= −
h
2
8π 2 μ
d
2
dQ 2 +
1
2
k Q
2
(2.19)
Now, we got Hamiltonian. Substituting this in ˆ
HΨ = EΨ and processing the
formula, a Schrödinger equation on harmonic oscillator of a diatomic molecule is
obtained.
d
2
ψ
dQ 2 +
8π
2
μ
h 2
E −
1
2
k Q
2
ψ = 0
(2.20)
It is not easy to solve this differential equation, but one can do it rigorously. As
how to solve this equation is described in detail in a number of textbooks, we will
explain only results. Formula 2.20 yields a solution only to the following eigen value
E υ :
E v =
v +
1
2
hν
(2.21)
where ν is a quantum number of a vibration (ν= 0, 1, 2,…). It can be seen from
Eq. 2.21 that under the harmonic oscillator approximation, the energies take discreet
values and their spacings are equal. Figure 2.7 shows the energy levels of vibration
of a diatomic molecule. It is noted that the lowest vibrational energy is not 0 but E 0 =
1/2 hν. E 0 is called zero point energy. Energies have discrete values; E 1 = 3/2 hν, E 2
= 5/2 hν, E 3 = 7/2 hν,…, and an energy difference between adjacent energy levels
is always hν.
An eigen function to each value of E υ is expressed as:
Fig. 2.7 Potential energy curve for a harmonic oscillator and allowed energy levels. a wave
functions and b probability density functions of the harmonic oscillator
25
H
= −
h
2
8π 2 μ
d
2
dQ 2 +
1
2
k Q
2
(2.19)
Now, we got Hamiltonian. Substituting this in ˆ
HΨ = EΨ and processing the
formula, a Schrödinger equation on harmonic oscillator of a diatomic molecule is
obtained.
d
2
ψ
dQ 2 +
8π
2
μ
h 2
E −
1
2
k Q
2
ψ = 0
(2.20)
It is not easy to solve this differential equation, but one can do it rigorously. As
how to solve this equation is described in detail in a number of textbooks, we will
explain only results. Formula 2.20 yields a solution only to the following eigen value
E υ :
E v =
v +
1
2
hν
(2.21)
where ν is a quantum number of a vibration (ν= 0, 1, 2,…). It can be seen from
Eq. 2.21 that under the harmonic oscillator approximation, the energies take discreet
values and their spacings are equal. Figure 2.7 shows the energy levels of vibration
of a diatomic molecule. It is noted that the lowest vibrational energy is not 0 but E 0 =
1/2 hν. E 0 is called zero point energy. Energies have discrete values; E 1 = 3/2 hν, E 2
= 5/2 hν, E 3 = 7/2 hν,…, and an energy difference between adjacent energy levels
is always hν.
An eigen function to each value of E υ is expressed as:
Fig. 2.7 Potential energy curve for a harmonic oscillator and allowed energy levels. a wave
functions and b probability density functions of the harmonic oscillator