Tài liệu Chapter XX Quantum theory of light pdf

But then it came as a great shock when some simple phenomena wereobserved which could not be explained by classical physics → a new theory, quantum theory, was developed at the beginnin

GENERAL PHYSICS III

Optics

&

Quantum Physics

Chapter XX Quantum theory of light

§ 1 Blackbody radiation Planck’s theory of radiation

§ 2 Photoelectric effect Einstein’s theory of light

§ 3 Compton scattering

At the end of the 19-th century, physics was at its most confidence

situation Classical phyics, as formulated in Newton’s law of mechanicsand Maxwell’s theory of electromagnetism, have proved very successful

in solving every problem

→ At that time there seemed to be no question for which physics couldnot provide an answer !!!

But then it came as a great shock when some simple phenomena wereobserved which could not be explained by classical physics

→ a new theory, quantum theory, was developed at the beginning of the20-th century

We begin our study of quantum physics by two following phenomena:

• Blackbody radiation

• Photoelectric effect

We will see what were the failures of classical physics and how a newtheory had been developed

§1 Blackbody radiation Planck’s theory of radiation:

• Heat bodies emit electromagnetic radiation in the infra-red region of thespectrum (see the next slide) In this region the radiation is not visible

• As the temperature of a body is increased to any value, the body begins

to glow red and then white, emitting visible electromagnetic radiation

(an example is the variation of the radiation of a filament of electric lampwhen the electric current varies)

• Observation of the spectrum emitted by a solid shows that the radiationextends over a continous range of frequency Such a spectrum is called

• The relation between intensity and temperature is given by the followingformula:

I = T4 ,

where I → the average power of radiation per unit surface area,

→ a fundamental physics constant called the Stephan-Boltzmann

constant = 5,67 x 10-8 W m-2 K-4

→ a dimensionless number (0 <  < 1) called the emissivity, which

depends on the nature of the radiating surface

It is found that for any surface the absorption is the exact

reverse of the emission process It means that the ability of

emission is proportional to that of absorption:

→ 0, the surface has low emissivity and low ability of absorption

 → 1, the surface has high emissivity and high ability of absorption

The idealized case, when = 1, such a body is called absolute blackbody

A blackbody absorbs completely all the incident radiation and gives noreflecting radiation (that is why the body is black !)

• Therefore we have for a blackbody: I =  T4

This is the Stephan-Boltzmann law for a blackbody

• Remark: For a blackbody the total intensitydepends only on absolute temperature

• A cavity with a small aperture is an example

of a blackbody Electromagnetic radiationentering the cavity is eventually absorbedafter successive reflections → the cavity is

a perfect absorber of e-m radiation

1.1.2 Wien displacement law:

Note that the intensity I in the Stephan-Boltzmann is the total intensity,that is, the radiation intensity for all wavelengths

Denote by I() dthe intensity corresponding to wavelengths in the

interval and + dwe can write

I I

I() is the distribution function

over all wavelengths It is called

the spectral emittance

• The form of I() depends on

temperature By experimental

measures one has IT() shown

in the picture

• Each curve has a peak at =m

It is observed that as the

temperature T increases, the

peak grown larger, and shifts to

shorter wavelengths

• By experiment Wien showed

that m is inversely proportional

to T, and

m T = 2.90 x 10-3 m.K This is Wien displacement law

1.2 Rayleigh–Jeans formula Ultraviolet catastrophe:

• Rayleigh and Jeans attemped to explain the observed blackbody spectrum

on the base of the concept that radiation is a e-m wave They derived thefollowing formula for the intensity distribution:

4

2 )

• Comparing the Rayleigh-Jeans

spectrum with experiment one

can see that:

The R-J formula agrees well

with experiment at large 

But there is a serious

disagreement at small 

The unrealistic behavior of

the Rayleigh-Jeans distribution

at short wavelengths is known in

physics as “ultraviolet catastrophe”

A more impressive indication of the complete failure of the Jeans spectrum is the result on the total intensity:

1 lim 3

2 2

) (

d I

I

It means that I →  , an unceptable result !!!

The blackbody radiation spectrum could not be explained

by classical physics

1.2 Planck’s theory of radiation:

The discrepancy between experiment and theory was resolved in 1900

by Planck, by introducing a postulate which was revolutionary with

respect to certain concepts of classical physics

Planck’s postulate: “Electromagnetic radiation consists of simple harmonicoscillations which can possess only energies

= nh (n = 0, 1, 2, 3, ….)where n is the frequancy of the oscillation, and h is an universal constant”

Energy level diagrams for classical sipmple harmonic oscillations and for that obeying Planck’s postulates.

The constant h is called the Planck constant It’s value was determined

by fitting theoretical consequences with experiment data (see below)

1.2.1 Planck’s postulate and radiation law:

With this postulate Planck derived the following spectrum distributionfunction

2 )

1.2.2 The agreement of the Planck law with experiment:

The Wien displacement law could be derived from the Planck law:

This coincides with the Wien displacement law,

if the value of h satisties the equation

K

m k

hc

.10

9

2965

.4

4 5

2 )

h c

k d

/ 10

67

5 15

2

K m

W h

c

k     



0 )

hc

m

1.965.4





is solved numerically,

(**)

fits the Planck law curve

with experiment leads to

the same result for the value

of the Planck’s constant:

h = 6.626x10-34 J.s

Therefore, the experimental

laws for blackbody radiation

could have a satisfactory

explanation with the Planck’s

theory The root idea is that

electromagnetic energy

emitted by bodies can have

only discrete values n.h

→ one say that energy is

quantized The discrete nature

of energy is the foundation of quantum theory

One more remark: The large wavelength limit of the Planck’s spectrum

formula is the Rayleigh’s formula (recall ex ≈1 + x with small x) This meansthat at very long wavelengths (very small quanta energies), quantum effectsbecome unimportant

§2.Photoelectric effect Einstein’s theory of light:

Binding potential

2.1 Photoelectric effect:

 If you shine light on the metal surface, electrons absorb energy fromthe incident light, and have enough energy to escape Eshtablish on themetal surface a electric field → the escaped electrons create an currentthat is called photoelectric current

How will the photoelectric current depend on the intensity I and

the frequency of the incident light ?

 Consider a metal surface Electrons in

a metal are “bound” by the binding

potential energy Introduce the

quantity , called “work function”

is the minimum amount of energy an

induvidual electron has to gain to escape:

• E < → the electron is confined inside metal

• E > → the electron can escape from metal

(E: the energy

of the electron )

 Experiment 1: Measure the maximum

energy of ejected electrons

 The electric field between the

“collector” and the metal willrepel ejected electrons

 Increase negative voltage untilflow of ejected electrons

 Experiment 2: Measure the maximum energy vs. 

The results:

 Stopping voltage Vstop (and the maximum kinetic energy of electrons)

decreases with decreasing (linear dependence)

 Below a certain frequency o, no electrons are emitted, even for intenselight!

Incident Light (variable frequency )

0 5 10 15

f00

Summary of results:

 Energy of electrons emitted depends on frequency, not intensity

 Electrons are not ejected for frequencies below 0

 Electrons are emitted as soon as any light with   

0 5 10 15

f0

h/e 1

This results are hard to understand on the basis of classical physics !

(According to classical physics: if the light intensity increases, electronsgain more energy → have more chance to escape Don’t understand why

is there a limit frequency )

2.2 Einstein’s quantum theory of light:

To overcome the difficulty of classical physics, Albert Einstein introducedthe quantum theory of light, and developed the correct analysis of the

2.2.2 Analysis of the photoeffect by the quantum theory:

A photon arriving at the surface is absorbed by an electron (one by one).After that the electron gets all the photon’s energy (h)

For an electron we can write the following equation:

2

v2max

m

h    

The energy of the electron

after absoption of photon

The energy part for escaping from the metal surface

The remaining part, the kinetic energy of the motion outside metal

Kinetic energy of the electron outside metal is telled by the value Vstop

By this analysis it is understandable that

• Energy of electrons emitted depends on frequency, not intensity

• The limit frequency 0 for the photoeffect is determined by theequation h0 =  Under this light frequency the electron has notenough energy to escape

discovered a shift in wavelength:  ’ ≠

The wavelength shift of scattered X-rays is called the Compton effect

• It is found that the wavelength difference   ’ - varies withthe scattering angle according to the equation

(A.H Compton 1923)

See the graph of the

The Compton effect could not be interpreted by classical physics !!But on the basis of the photon model of light one has a satisfactoryexplanation of this effect

3.2 Compton effect as the result of photon-electron collision:

On the basis of the photon model of light, weconsider the Compton scattering as the

following collision process

+ e → ’ + e’

and e - photon and electron before collision;

’ and e’ - photon and electron after collision

• Regarding a photon as a particle with energy Eand momentum p, we have from the theory ofrelativity

• The conservation of the total momentum vectorgives

e

p p

p    '  

Photon momentum before scattering Photon momentum after scattering

Electron momentum after scattering

Photon has the rest mass m = 0, so we have



c

h c

E

cos ' 2 '2

2 2

pp p

h

e

This equation agrees well with experiment

Therefore, together with the phenomena of blackbody radiation andthe photoeffect, the Compton effect gave a more evidence for the

“particle” nature of light

 At the end of 19-th century classical physics encountered a serious

challenge: the experimental laws for the blackbody radiation could not beunderstood by the concepts of classical physics

 In 1900 Planck introduced his hypothesis about the discrete nature of

radiation energy He stated that energy can be emitted in the form ofenergy quanta, each quantum is h

Planck’s theory gave the consequences that agree well with the

experimental laws for the blackbody radiation

Planck’s hypothesis is recognised as the beginning of a new theory, thequantum theory

• In 1905 Einstein developed an analogue theory for light: Light consists

of photons, the energy of each photon is h

Einstein’s theory gave a satisfactory interpretation of the experimentalresults of the photoeffect, which could not explained by classical physics

 In 1923 Compton observed the wavelength shift of scattered X-rays.This effect was interpreted excelently by the photon model of light.The formula for the wavelength shift was derived by using the

equations which express the conservation of the total energy and thetotal momentum of the collision of two particles – photon and electron

• The mentioned phenomena showed the “particle” nature of

electromagnetic waves (thermal radiation, visible light, X-rays,…)

Electromagnetic waves can be considered as flows of photons

• In the limit of large wavelengths (high frequencies, small photon

energies) quantum effects become unimportant The “particle” behavior

of electromagnetic waves is remarkable in short-wavelength regions ofthe continuum of electromagnetic waves