医学物理学chapter 10b_13x

10.5 Line spectra and Energy quantization in atoms
10.5.1 Line spectra of Hydrogen atoms
Hydrogen always gives a set of line spectra in the same position.
• Using relativistic theory;
• Energy conservation; • Momentum conservation ;
• The key points of derivation procedure: (1) Relativistic theory:
E h
n 2,3,4, n 4,5,6,
(9.5.3)
Brackett series:
n 5,6,7, n 6,7,8,
Pfund series:
frequencies
The Lymann series is in the ultraviolet, and the Paschen, Brackett, and Pfund series are in the infrared. All these formulas can be generalized into one formula which is called the general Balmer series.
• Quantum explanation
The quantum theory, by contrast, provides a beautifully simple explanation.
p 0 , p0
·
P
φ
·
Fig. 10.5 Schematic diagram of Compton scattering.
•
Classical explanation
Based on the classical principles, the scattering mechanism is induced by the motion of electrons in the material, caused by the incident radiation. This motion must have the same frequency as that of incident wave because of forced vibration, and so the scattered wave radiated by the oscillating charges should have the same frequency. There is no reason that the frequency can be shifted by this mechanism.
2
where m0
h c 0.00243 nm mc
called Compton wavelength.
10.4 The duality of light
light shows the property of waves in its interference and diffraction and performances the particle property in blackbody radiation, photoelectric effect and Compton effect. Till now we say that the light has duality property.
mV

h h h h m V 2 cos 0 0 (3)
So the (mV)2 can be deleted from (2) and (3), so we have
2
2 2
(1)
2 2
For electron, we have
2 2
E (cp) (m0c )
2
(mV) (mc) (m0c)
Where p = mV
(2)
(3) The momentum conservation
0
2
h h n0 n mV
h/ h/0
•
434.1 656.3 486.1 410.2 364.6
(nm)
H
H H H H
Fig. 9.5 the Balmer series of atomic hydrogen.
• No one can explain it!
• Finally,
We can say that light is wave when it is involved in its propagation only like interference and diffraction.
P
S0 d

O
=d sin = y/L
Wave property: light interacts with itself. photon property: when it interact with other materials.
E p c m c
2 2 2
2 4 0
when m0 = 0
E pc mc hv
2
∴ the mass of photon is
E h mp 2 2 c c
and the momentum of photon should be
E h h p mc c c
1 1 R 2 2 , 2 n 1
(9.5.1)
where is the wavelength, R is a constant called the Rydberg constant, and n may have the integral values 3, 4, 5, etc.* if is in meters,
2 2
2
(4) - (5)
2 h h h 0 2m0c 2 ( 1 cos ) 0 0
The above equation can be rewritten as
h 1 cos 0 m0c
h 1 cos 0 m0 c h 2 2 sin m0 c 2 2c sin 2 is the electron mass.
• Theoretical derivation
• Suppose that 0 and are the wavelength of the incident and scattered radiation, respectively. The Compton effect could be described by the following procedures:
simple formula which gave the frequencies of a group lines emitted by atomic hydrogen.
Balmer found that the wavelength of these lines were given accurately by the simple formula
2
2 h h h 2 2 m c m0c 2 cos (4) 0 0
2
2
Squaring (1), we have
h h h h m c m0c 2 m c 0 (5) 0 0
Lymann series: Paschen series:
1 1 R 2 2 , 1 n 1 1 1 R 2 2 , 3 n 1 1 1 R 2 2 , 4 n 1 1 1 R 2 2 , 5 n 1
Other series spectra for hydrogen have since been discovered. These are known, after their discoveries, as Lymann, Paschen, Brackett and Pfund series. The formulas for these are
2
hc

2
E mc
2 2
2
E (cp) (m0c )
When m0 = 0, the momentum is equal to
E h h p c c
(2) Energy conservation
h 0 m0c h m c
2
2
h
0
m0c
h

mc
• Scattered radiation has smaller frequency and longer wavelength than the incident radiation. • The wavelength of scattered radiation depends on the scattered angle φ .
• Quantum explanation
We imagine the scattering process as a collision of two particles, the incident photon and an electron at rest as shown in Fig. 9.5. The photon gives up some of its energy and momentum to the electron, which recoils as a result of this impact; and the final photon has less energy, smaller frequency and longer wavelength than the initial one.
10.3 Compton effect
• X-rays impinges (冲击,撞击) on matter, some of the radiation is scattered (散射), just as the visible light falling on a rough surface undergoes diffuse reflection (漫射，diffusion).
Chapter 10
Introduction to Quantum Mechanics (2)
10.2.2 The mass and momentum of photon
According to relativity, the particles with zero rest mass are possibly existent. From the relativistic equation of energy-momentum,
R 1.09710 m
7
1
(9.5.2)
Substituting R and n = 3 into the above formula, one obtains the wavelength of the H-line:
656 .3nm
For n = 4, one obtains the wavelength of the H-line, etc. for n= , one obtains the limit of the series, at = 364.6nm – shortest wavelength in the series.