Chapter 9

Introduction to Quantum Mechanics (2)

(May. 25, 2005)

A brief summary to the last lecture

• Blackbody radiation and Planck’s hypothesis: (Stefan and Boltzmann’s law, Wein’s displacement law, Rayleigh-Jeans formula, Wein’s formula, Planck’s formula and his hypothesis)

• Photoelectric effect: (shows that the absorption of light is as light quantum, photoelectric effect equation)

What does the Stefan-Boltzmann’s Law explain?

4)( TTM

M(T) is radiation energy, T is the absolute temperature.

For the radiation power M

(T), it is a function of for a constant T. What is Wien’s displacement law?

bTpeak

For the radiation power and the fixed temperature, two formulas were obtained using classical theories, which one is correct for longer wavelength and which one is valid for shorter wavelength?

Rayleigh-Jeans’ line

Wein’s lineTCTM 41)(

TC

eCTM

352)(

1

12)( 52

Tkhc

ehcTM

•Planck’s Magic formula

In 1900, after studying the above two formulas carefully, Planck proposed (提出 ) an empirical formula.

How to get Stefan-Boltzmann’s Law using above formula?

How to obtain Wein’s displacement law by Planck’s formula?

How could you get Rayleigh and Jeans’ formula, Wein’s formula by Planck’s formula?

1

12)( 52

Tkhc

ehcTM

• Planck-Einstein Energy Quantization Law and photoelectric effect.

hc

hE

2

2

1mvAh

22 c

h

c

Emp

h

c

h

c

Ep

9.3 Compton effect

• A phenomenon called Compton scattering (observed in 1924) provides additional direct confirmation of the quantum nature of electromagnetic radiation

• 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).

• Scattered radiation has smaller frequency and longer wavelength than the incident radiation.

• The wavelength of scattered radiation depends on the scattered angle φ .

• Classical explanation

Based on the classical principles, the scattering mechanism is induced by 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 way that the frequency can be shifted by this mechanism.

• Quantum explanation

The quantum theory, by contrast, provides a beautifully simple explanation. We imagine the scattering process as a collision of two particles, the incident photon and an electron at rest as shown in Fig. 8.4. 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.

• 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:

• Using relativistic theory;

• Energy conservation;

• Momentum conservation (using cosine principle at the same time);

• The key points of derivation procedure:

(1) Relativistic theory:

220

22

2

)()(

cmcpE

mcEhc

hE

When m0 = 0, the momentum is equal to

h

c

h

c

Ep

(2) Energy conservation

mch

cmh

mchcmh

00

2200

For electron, we have 220

22 )()( cmcpE 2

022 )()()( cmmcmV

(1)

(2)

(3) The momentum conservation

h/

h/0

mV

cos20

22

0

2

hhhhmV

Vmnh

nh

0

0

So the (mV)2 can be deleted from (2) and (3), so we have

(3)

cos20

222

0

20

2 hhhcmmc

(4)

Squaring (1) and minus (4), we have

cos2200

2

00

hhhcm

The above equation can be rewritten as

2sin2

2sin2

cos1

2c

2

0

00

cm

h

cm

h

where m0 is the electron mass. Fig. 8.4 Schematic diagram of Compton scattering.

··P

0, p0

p

φ

nmmc

hc 00243.0 called Compton

wavelength.

9.4 The duality of light

The concept that waves carrying energy may have a corpuscular (particle) aspect and that particles may have a wave aspect; which of the two models is more appropriate will depend on the properties the model is seeking to explain. For example, waves of electromagnetic radiation need to be visualized as particles, called photons to explain the photoelectric effect. Now you may confuse the two properties of light and ask what the light actually is?

The fact is that the 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.

P

OS0

=d sin = x/L

d

This means that light interacts with itself. The light shows photon property when it interact with other materials.

We can say that light is wave when it is involved in its propagation only like interference and diffraction.

9.5 Line spectra and Energy quantization in atoms

The quantum hypothesis, used in the preceding section for the analysis of the photoelectric effect, also plays an important role in the understanding of atomic spectra.

9.5.1 Line spectra of Hydrogen atoms

• Hydrogen always gives a set of line spectra in the same position.

(nm)

H H H H H

656.3 486.1

434.1

410.2

364.6

Fig. 9.5 the Balmer series of atomic hydrogen.

• It is impossible to explain such a line spectrum phenomenon without using quantum theory.

• For many years, unsuccessful attempts were made to correlate the observed frequencies with those of a fundamental and its overtones (谐波 ) (denoting other lines here). Finally, in 1885, Balmer found a simple formula which gave the frequencies of a group lines emitted by atomic hydrogen. Since the spectrum of this element is relatively simple, and fairly typical of a number of others, we shall consider it in more detail.

Under the proper conditions of excitation, atomic hydrogen may be made to emit the sequence of lines illustrated in Fig. 9.5. This sequence is called series.

(nm)

H H H

H H

656.3 486.1

434.1

410.2

364.6

Fig. 9.5 the Balmer series of atomic hydrogen.

There is evidently a certain order in this spectrum, the lines becoming crowded more and more closely together as the limit of the series is approached. The line of

longest wavelength or lowest frequency, in the red, is known as H, the next, in the blue-green, as H, the third as H, and so on.

Balmer found that the wavelength of these lines were given accurately by the simple formula

,1

2

1122

nR

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,

1710097.1 mR

(9.5.1)

(9.5.2)

Substituting R and n = 3 into the above formula, one obtains the wavelength of the H-line:

nm3.656

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.

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

Lymann series: ,4,3,2,1

1

1122

n

nR

Paschen series:

,6,5,4,1

3

1122

n

nR

,7,6,5,1

4

1122

n

nR

Brackett series:

,8,7,6,1

5

1122

n

nR

Pfund series:

(9.5.3)

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.

,3,2,1,111

22

kkkn

nkR

All the spectra of atomic hydrogen can be described by this simple formula. As no one can explain this formula, it was ever called Balmer formula puzzle.

(9.5.4)

9.5.2 Bohr’s atomic theory

Bohr’s theory was not by any means the first attempt to understand the internal structure of atoms. Starting in 1906, Rutherford and his co-workers had performed experiments on the scattering of alpha Particles by thin metallic. These experiments showed that each atom contains a massive nucleus whose size is much smaller than overall size of the atom.

• The atomic model of Rutherford

The nucleus is surrounded by a swarm ( 一大群 ) of electrons. To account for the fact, Rutherford postulated that the electrons revolve (旋转 ) about the nucleus in orbits, more or less as the planets in the solar system revolve around the sun, but with electrical attraction providing the necessary centripetal force ( 向心力 ). This assumption, however, has an unfortunate consequence.

(1) Accelerated electron will emit electromagnetic waves; Its energy will be used up sometimes later. Dead atoms

A body moving in a circle is continuously accelerated toward the center of the circle and, according to classical electromagnetic theory, an accelerated electron radiates energy. The total energy of the electrons would therefore gradually decrease, their orbits would become smaller and smaller, and eventually they would spiral (盘旋 ) into the nucleus and come to rest. Go to next

(2) The emitted frequency should be that of revolution and they should emit continuous frequency.

Furthermore, according to classical theory, the frequency of the electromagnetic waves emitted by a revolving electron is equal to the frequency of revolution. Their angular velocities would change continuously and they would emit a continuous spectrum (a mixture of frequencies), in contradiction to the line spectrum actually observed.

In order to solve the above contradictions, Bohr made his hypotheses:

• Static (or stable-orbit) postulate: Faced with the dilemma, Bohr concluded that , in spite of the success of electromagnetic theory in explaining large scale phenomenon, it could not be applied to the processes on an atomic scale. He therefore postulated that an electron in an atom can revolve in certain stable orbits, each having a definite associated energy, without emitting radiation.

• The angular momentum mvr of the electron on the stable orbits is supposed to be equal to the integer multiple of h/2π. This condition may be stated as

,3,2,12

nh

nmvr

where n is quantum number, this is the hypothesis of stable state and it is called the quantization condition of orbital angular momentum.

(9.5.5)

• Transition hypothesis,

Bohr postulated that the radiation happens only at the transition of electron from one stable state to another stable state. The radiation frequency or the energy of the photon is equal to the difference of the energies corresponding to the two stable states.

h

EEEEh kn

kn

(9.5.6)

• Corresponding principle

The new theory should come to the old theory under the limited conditions.

• Important conclusions:

Another equation can be obtained by the electrostatic force of attraction between two charges and Newton’s law:

r

vm

r

e 2

2

2

04

1

(9.5.7)

Solving the simultaneous equation of (9.5.5) and (9.5.7), we have

),3,2,1(2

,0

22

2

20 n

hn

evn

me

hr nn

So the total energy of the electron on the nth orbit is

220

4

0

22

842

1

nh

me

r

emv

EEE

nn

pkn

It is easy to see that all the energy in atoms should be discrete. When the electron transits from the nth orbit to kth orbit, the frequency and wavelength can be calculated as

(9.5.8)

2222320

4

22320

4

1111

8

1

11

8

nkR

nkch

me

c

knnkh

me

h

EE kn

(9.5.9)

where ch

meR

320

4

8

is Rydberg constant.

It is found that the value of R is matched with experimental data very well. Till then, the 30-years puzzle of line spectra of atoms had been solved by Bohr since equation (9.5.9) is exactly the general Balmer formula.

When Bohr’s theory met problems in explaining a little bit more complex atoms (He) or molecules (H2), Bohr realized that his theory is full of contradictions as he used both quantum and classical theories. The problem was solved completely after De Broglie proposed that electron also should have the wave-particle duality. Since then, the proper theory describing the motion of the micro-particles, quantum mechanics, has been gradually established by many scientists.

9.6 De Broglie Wave

In the previous sections we traced the development of the quantum character of electromagnetic waves. Now we will turn to the consequences of the discovery that particles of classical physics also possess a wave nature. The first person to propose this idea was the French scientist Louis De Broglie.

9.6.1 De Broglie wave

De Broglie’s result came from the study of relativity. He noted that the formula for the photon momentum can also be written in terms of wavelength

h

c

hc

c

hcmp p

2(9.6.1)

If the relationship is true for massive particles as well as for photons, the view of matter and light would be much more unified.

De Broglie’s point was the assumption that momentum-wavelength relation is true for both photons and massive particles.

So De Broglie wave equations are

hEmv

h

p

h (9.6.2)

Where p is the momentum of particles, λ is the wavelength of particles. At first sight, to claim that a particle such as an electron has a wavelength seems somewhat absurd. The classical concept of an electron is a point particle of definite mass and charge, but De Broglie argued that the wavelength of the wave associated with an electron might be so small that it had not been previously noticed. If we wish to prove that an electron has a wave nature, we must perform an experiment in which electrons behave as waves.

9.6.2 Electron diffraction

In order to show the wave nature of electrons, we must demonstrate interference and diffraction for beams of electrons. At this point, recall that interference and diffraction of light become noticeable when light travels through slits whose width and separation are comparable with the wavelength of the light. So let us first look at an example to determine the magnitude of the expected wavelength for some representative objects.

For example, de Broglie wavelength for an electron whose kinetic energy is 600 eV is 0.0501nm. The de Broglie wavelength for a golf ball of mass 45g traveling at 40m/s is: 3.68 ×10-34 m. Such a short wave is hardly observed.

We must now consider whether we could observe diffraction of electrons whose wavelength is a small fraction of a nanometer. For a grating to show observable diffraction, the slit separation should be comparable to the wavelength, but we cannot rule a series of lines that are only a small fraction of a nanometer apart, as such a length is less than the separation of the atoms in solid materials.

When electrons pass through a thin gold or other metal foils ( 箔 ), we can get diffraction patterns. So it indicates the wave nature of electrons. Look at the pictures on page 265 and 231 in your Chinese.

Introduce: Electron single and double slits experiments; electronic microscope, nuclear reactor etc.