where we used the relations ω ¼ ck and ω
0
¼ ck
0 with the third equality. Therefore,
we get
p
2 c
2
¼ ħ
2
ω
2
þ ω
0 2 À 2ωω
0 cos θ
:
ð1:13Þ
From (1.11) and (1.13), we have
2m e c
2
ħ ω À ω
0
ð
Þþħ
2
ω À ω
0
ð
Þ
2 ¼ ħ
2
ω
2
þ ω
0 2 À 2ωω
0 cos θ
:
ð1:14Þ
Equation (1.14) is simplified to the following:
2m e c
2
ħ ω À ω
0
ð
ÞÀ2ħ
2
ωω
0
¼ À2ħ
2
ωω
0 cos θ:
That is,
m e c
2
ω À ω
0
ð
Þ¼ħωω
0 1 À cos θ
ð
Þ :
ð1:15Þ
Thus, we get
ω À ω
0
ωω 0 ¼
1
ω 0 À
1
ω
¼
1
2πc
λ
0
À λ
ð
Þ¼
ħ
m e c 2 1 À cos θ
ð
Þ ,
ð1:16Þ
where λ and λ
0 are wavelengths of the initial and final X-ray beams, respectively.
Since λ
0
À λ ¼ Δλ, we have (1.4) from (1.16) accordingly.
We have to mention another important person, Louis-Victor de Broglie (1924) in
the development of quantum mechanics. Encouraged by the success of Einstein and
Compton, he propounded the concept of matter wave, which was referred to as the
de Broglie wave afterward. Namely, de Broglie reversed the relationship of (1.1) and
(1.2) such that
ω ¼ E=ħ,
ð1:17Þ
and
k ¼
p
ħ
or
λ ¼ h=p,
ð1:18Þ
where p equals jpj and λ is a wavelength of a corpuscular beam. This is said to be the
de Broglie wavelength. In (1.18), de Broglie thought that a particle carrying an
energy E and momentum p is accompanied by a wave that is characterized by an
angular frequency ω and wavenumber k (or a wavelength λ ¼ 2π/k). Equation (1.18)
implies that if we are able to determine the wavelength of the corpuscular beam
experimentally, we can decide a magnitude of momentum accordingly.
6
1 Schrödinger Equation and Its Application