Quantum PhysicsQuantum Physics

The Quantization of Light

§19-1 Thermal Radiation and Plank’s theory o

f Radiation 热辐射 普朗克的量子假设

§19-2 The Photoelectric Effect and Einstein’s Quantum Theory 光电效应 爱因斯坦的光子理论

§19-3 The Compton Effect 康普顿效应

I. Thermal radiation

§19-1 Thermal Radiation and Plank’s theory of Radiation

-- At any temperature, a body emits radiation of all wavelength, but the distribution in wavelength, the spectral distribution, depends on temperature.

Some concepts about thermal radiationSome concepts about thermal radiation

1. The spectral radiancy1. The spectral radiancy ( ( 光谱辐射出射度光谱辐射出射度 )) ee

(( , ,TT)) Let Let dede((,T,T):): the energy emitted per unit time in the energy emitted per unit time in radiation of wavelength in the interval radiation of wavelength in the interval ++dd from an unit area of the surface at absolute temfrom an unit area of the surface at absolute temperature perature TT . .

d

TdeTe

),(),(

ee (( , ,TT)) specifies the spectral distribution of an specifies the spectral distribution of an

body radiation at body radiation at TT..

Then,Then,

2. Radiancy2. Radiancy ( ( 辐射出射度辐射出射度 )) EE((TT))

The total energy emitted per unit time per unit The total energy emitted per unit time per unit

area from a body at temperature area from a body at temperature TT..

dTeTE ),()(0

The integral of spectral radiancy The integral of spectral radiancy ee (( , ,TT)) over all over all

..

All bodies emit radiation to their surrounding All bodies emit radiation to their surrounding and radiation may fall on a body.and radiation may fall on a body.

3. Black body3. Black body(( 黑体黑体 ))

When radiation falls on a body,When radiation falls on a body,

absorbed

reflected

Black bodyBlack body: can absorb : can absorb all radiation falling on it.all radiation falling on it. no any reflection.no any reflection.

----ideal model.----ideal model.

外壳外壳

热电偶热电偶

保温层保温层 加热线圈加热线圈

腔体腔体

腔芯腔芯

热屏蔽套管热屏蔽套管

II. The experiment results of black body radiationII. The experiment results of black body radiation

ee (( , ,TT)) varies continuovaries continuo

usly with usly with . Each . Each ee

(( , ,TT)) curve has a peak.curve has a peak.

ee (( , ,TT)) curve increases rcurve increases r

apidly with the increasinapidly with the increasing of g of TT..

mm decreases linearly decreases linearly

withwith TT increasing increasing

),( Te

1T

2T3T

4T

m1

Quantitatively,Quantitatively,

(1) (1) Stefan’s lawStefan’s law4)( TTE

=5.67=5.671010-8-8 W/m W/m22·K·K44

--Stefan constant--Stefan constant

(2) (2) Wien’s displacement lawWien’s displacement law

bTm

bb=2.898=2.8981010-3-3 m mKK

),( Te

T

m

--Wien constant.--Wien constant.

[[ExampleExample] Stefan’s law can be used to determine ] Stefan’s law can be used to determine the radius of a sun in astronomy. It is known thathe radius of a sun in astronomy. It is known that the radiation power of a sun arriving to unit art the radiation power of a sun arriving to unit area of the earth is ea of the earth is 1.21.21010-8-8 W/m W/m22. The distance b. The distance between the sun and the earth is etween the sun and the earth is 4.34.310101717 m . m . TheThe temperature of the sun’s surface istemperature of the sun’s surface is 5200 5200KK. The s. The sun can be regarded as a black body.Find its radiun can be regarded as a black body.Find its radius.us.

SolutionSolutionLet Let RR: sun’s radius, : sun’s radius, RR : distance betwee: distance betwee

n sun and earthn sun and earth R

Sun earth'R

R

'R

Neglecting absorption, we haveNeglecting absorption, we have

''44 242 ERTR 2

1

4

2 ''

T

ERR

2

1

48

8217

52001067.5

102.1)103.4(

m1026.7 9

The total radiation power of the sun is The total radiation power of the sun is

)(4 2 TERW 424 TR

4)( TTE

III. Classical physics encountered difficulty for III. Classical physics encountered difficulty for explaining the radiation of black body.explaining the radiation of black body.

How can we deduce the quantitative expression How can we deduce the quantitative expression ofof ee (( ,T ,T )) theoretically and make it agree with theoretically and make it agree with

the experiment?the experiment?

cc11,,cc22: : constants constants

determined by determined by experiment.experiment.

Wien’s semi-experiment formulaWien’s semi-experiment formula ：：

T

c

ec

Te

2

51),(

--It agrees with experiment only in short --It agrees with experiment only in short wavelength range.wavelength range.

),( Te

Wien’s line

4

2),(

ckT

Te

Rayleigh-Jeans formulaRayleigh-Jeans formula

It agrees with It agrees with experiment only experiment only in longer in longer wavelength rangewavelength range

----“ultraviolet ultraviolet catastrophecatastrophe”

),( Te

Wein’sWein’s

Classical physics cannot explain the Classical physics cannot explain the radiation of black body.radiation of black body.

R-J line

IV. Plank’s hypothesis and formulaIV. Plank’s hypothesis and formula

1. “Quantum of energy 1. “Quantum of energy (( 能量子能量子 ) ) ” hypothesis” hypothesis There are many oscillators in the black body. There are many oscillators in the black body.

The energy of an oscillator of a given frequenThe energy of an oscillator of a given frequency cy cannot take arbitrary values, but can onlcannot take arbitrary values, but can only take on the discrete values y take on the discrete values nhnh. where . where nn is a is a positive integer or zero. positive integer or zero.

= = nhnh is a finite amount, or quantum, of energy. is a finite amount, or quantum, of energy.

00 = = hh is the minimum energy of an oscillator. is the minimum energy of an oscillator.

---- ---- quantumquantum ofof energyenergy

n n ---- quantum number(---- quantum number( 量子数量子数 ))

sJ1063.6 34 h ----Plank’s constant----Plank’s constant

Applying his hypothesis, Plank obtainedApplying his hypothesis, Plank obtained

1

12),(

/5

2

Tkhce

hcTe

----Plank’s black body radiation formula----Plank’s black body radiation formula

Plank obtained great agreement with the Plank obtained great agreement with the experiment over the entire range of experiment over the entire range of wavelength. wavelength.

Furthermore, Furthermore,

Plank got the Nobel Prize in Plank got the Nobel Prize in physics in 1918.physics in 1918.

dformulasPlank )'(0

---Stenfa’s law---Stenfa’s law

4T

)(TE

)'( formulasPlankd

d

0

Wien’s displacement lawWien’s displacement law

bTm can be obtained.can be obtained.

[[ExampleExample] A spring-particle oscillator system wi] A spring-particle oscillator system with th kk=15N/m, =15N/m, m=m=1kg 1kg andand AA==0.01m0.01m ，， Calculate Calculate the total energy of the system the total energy of the system EE=?=? the quantthe quantum number of the system um number of the system nn== ？？ ifif n n changes frchanges from om nn to to nn+1+1 or or nn-1-1 , , E/EE/E=?=?

SolutionSolution Total energy isTotal energy is

2

2

1kAE 201.015

2

1 J105.7 4

According to Plank’s hypothesis,According to Plank’s hypothesis,

nhE

If nn changes an unit, energy changes changes an unit, energy changes

hE

nEE 1 29106.5 No any instrument can distinguish the changing.No any instrument can distinguish the changing.

----Quantum effect disappear for macr----Quantum effect disappear for macro-system(o-system( 宏观系统宏观系统 ).).

m

k

2

1And And

hEn 30108.1

1s617.0

Electrons are ejected from metal surface Electrons are ejected from metal surface when it is radiated by high frequency when it is radiated by high frequency electromagnetic waveselectromagnetic waves

§19-2 The photoelectric effect§19-2 The photoelectric effect

photoelectron

Cathode( 阴极 )

Quartzwindow

I. The results of the experiments I. The results of the experiments

GV

KA

V

I1sI2sI

0V

光强较强光强较强

光强较弱光强较弱

饱和饱和电流电流

截止截止电压电压

G—sensitive ammeter

Is—saturated current

V0—stopping potential

The saturated current is proportional with the incident light intensity.

--the number of photoelectrons ejected from --the number of photoelectrons ejected from cathode in an unit time is proportional with cathode in an unit time is proportional with the incident light intensity.

The photoelectric current The photoelectric current =0 =0 when an invewhen an inverse stopping potentialrse stopping potential -V-V00 is suppliedis supplied

--photoelectrons have the maximum initial --photoelectrons have the maximum initial kinetic energy.kinetic energy.

02

2

1eVmvm

VV00 depends linearly on the frequency depends linearly on the frequency of the iof the i

ncident light and is independent of its intensitncident light and is independent of its intensity.y.

i.e.0V 02

2

1eVmvm

0

0V

0cutoff frequency

Below0 0 , no photoelectric effect occurs.

Electron emission takes place Electron emission takes place immediatelyimmediately as as the light is incident on the surfacethe light is incident on the surface with no detewith no detectable time delay.ctable time delay.

II. The classical wave theory of light cannot II. The classical wave theory of light cannot explain the results.explain the results.

According to the classical theory,According to the classical theory,

the initial kinetic energy would be decided the initial kinetic energy would be decided by the intensity of light by the intensity of light instead of the instead of the frequency of light.frequency of light.

photoelectric effect would take place for any photoelectric effect would take place for any frequency of light as long as its intensity is frequency of light as long as its intensity is large enough instead of large enough instead of existing a cutoff existing a cutoff frequency frequency 00..

the photoelectron would not escape from the the photoelectron would not escape from the

metal metal immediatelyimmediately if the intensity of light is if the intensity of light is

very small.very small.

III. Einstein’s quantum theory of the PE-effectIII. Einstein’s quantum theory of the PE-effect

The electromagnetic field itself is quantized The electromagnetic field itself is quantized and that light consists of corpuscles, called and that light consists of corpuscles, called light quantalight quanta or or photonsphotons. Each photon travels . Each photon travels with the speed of light with the speed of light c c and carries a and carries a quantum of energy of magnitude quantum of energy of magnitude EE==hh

The energy flow density ( or intensity ) of The energy flow density ( or intensity ) of

light is light is S S ==N N hh

Einstein offered his “photon postulate” for Einstein offered his “photon postulate” for explanation the results of PE-effect in 1905.explanation the results of PE-effect in 1905.

Wmvh m 2

2

1the energy of photon

the work function of electron

--Einstein’s photoelectric effect equation--Einstein’s photoelectric effect equation

If If vvmm==0,0, Whh 0

h

W 0

When a photon falls on a metallic surface, When a photon falls on a metallic surface,

then,then,

IV.Einstein’s explanation for photoelectric effectIV.Einstein’s explanation for photoelectric effect

larger intensity of lightlarger intensity of light

i.e. larger number of photonsi.e. larger number of photons

larger number of photoelectronslarger number of photoelectrons

i.e. larger photoelectric current.i.e. larger photoelectric current.

asas Wmvh m 2

2

1

2

2

1mmvi.e.i.e.

there is a cutoff frequencythere is a cutoff frequency ，hW0no photoelectric effect fono photoelectric effect for r WW//h h ，，

AA photon is absorbed by photon is absorbed by an electron immediately an electron immediately if if 0 0 and the electron and the electron

will eject immediately.will eject immediately.

ConclusionConclusion ：： light is light is the flow of particles.the flow of particles.

Einstein got Noble prize on physics in 1921Einstein got Noble prize on physics in 1921

[Example] A beam of ultraviolet light with [Example] A beam of ultraviolet light with =250=25000ÅÅ, intensity , intensity SS==22W/mW/m2 2 irradiates on a potassium firradiates on a potassium foil. The work function of potassium is oil. The work function of potassium is WW==2.21eV2.21eV. . Find Find the maximum kinetic energy of the photoethe maximum kinetic energy of the photoelectrons, lectrons, the maximum number of the photoelethe maximum number of the photoelectrons per unit time per area from the surface of ctrons per unit time per area from the surface of the potassium foil.the potassium foil.

SolutionSolution Using Einstein’s equation,Using Einstein’s equation,

Wc

hmv m

2

2

121.2

106.1105.2

1031063.6197

834

21.297.4 eV76.2

Because one photon can knock out one electron Because one photon can knock out one electron only, only,

the maximum number of the photoelectrons is the maximum number of the photoelectrons is

hcS

N

2118 ms1052.2 1910957

2

.

The energyThe energy of a photon isof a photon is

eV97.4c

h J106.197.4 19J1095.7 19

I. The Compton effectI. The Compton effect

§19-3 The Compton Effect§19-3 The Compton Effect

When a beam of When a beam of xx-rays with sharply defined -rays with sharply defined wavelength wavelength 00 falls on some target (such as falls on some target (such as

graphite) , it will be scattered and the scattered graphite) , it will be scattered and the scattered radiation have two components of radiation have two components of wavelength:the origin wavelength wavelength:the origin wavelength 00 and the and the

larger wavelength larger wavelength ..

探测器探测器

石墨石墨

光阑光阑 入入射射光光

散射光散射光

x x 射射线管线管

DeviceDeviceResultsResults

045

090

0135

0

0

Results of experimentResults of experiment

==--00 increase with the incre increase with the incre

ment of scattering angle ment of scattering angle . . ha has nothing to do with s nothing to do with 00 and scatte and scatte

r (target).r (target).

the intensity of the intensity of 00 decreases and decreases and

the intensity of the intensity of increase with t increase with the increment of he increment of ..

045

090

0135

0

0

14Si

16S

19K

20Ca

24Cr

26Fe

28Ni

29Cu

the intensity of the intensity of is is larger for the target larger for the target with lighter atoms twith lighter atoms than the target with han the target with heavier atoms.heavier atoms.

Classical theory:Classical theory:

xx-ray is electromagnetic wave.-ray is electromagnetic wave.

It acts on the free electrons of the target.It acts on the free electrons of the target.

There should not be the component of There should not be the component of in scattered rays.in scattered rays.

II.The explanations for Compton effect.II.The explanations for Compton effect.

Forcing the electrons oscillate at same Forcing the electrons oscillate at same frequency.frequency.

radiate electromagnetic wave radiate electromagnetic wave with same same frequency. with same same frequency.

Photon collides with electron in elastic.Photon collides with electron in elastic.

Photon theoryPhoton theory ：：

Photons collide with the Photons collide with the outer bound electronsouter bound electrons in the atoms of the target, in the atoms of the target,

outer bound electronouter bound electron

looked as free electronlooked as free electron

Ze

h0

Ze

h

mc2

Part of energy Part of energy

is transferred is transferred

to electronto electron

0

Photons collide with the Photons collide with the inner bound electronsinner bound electrons in the atoms of the target, in the atoms of the target,

inner bound electroninner bound electron

looked as free electronlooked as free electron

Ze

h0

equivalent to collide equivalent to collide with the whole with the whole atoms of target.atoms of target.

As As mmatomatom>> >> mmphotonphoton, ,

photon does not lphoton does not l

ose energy. So we hose energy. So we h

ave ave ==0 0 or or ==00

Ze

h0

Compton effect is evident for smaller Z than bigger Z.

III. The Compton equationIII. The Compton equation

1.1.Photons collide with the Photons collide with the outer bound electronsouter bound electrons

Before collision

photonphoton：：

energy

00 hE

momentum

0P

electronelectron：：

20cm 0

After collision

photonphoton：：

hE P

electronelectron：：

eE eP

x

y

00 PEe

20cm

x

y

e

E,P

Ee ,P

e

Conservation of momentumConservation of momentum

Conservation of energyConservation of energy

eEEcmE 200

ePPP

0

00 PE

x

y

e

Considering relativistic energy and momentumConsidering relativistic energy and momentum

420

222 cmcPE ee electronelectron：：

For photonFor photon ：： cPE 00

and can be re-written2

00 cmEEEe

cos2 022

02 PPPPPe

substitutingPcE

We getWe get ：： cos1)11

(0

0 PP

cm

andand ：：

h

P )cos1(0

0 cm

h

00 PE

x

y

e

oror2

sin2 2 c

cm

hc

0

m1043.2 12

----Compton wavelength of the electron----Compton wavelength of the electron

Here Here

----depend on ----depend on onlyonly

The magnitude of c is closer to the magnitude of the wavelength 0 of x-ray (0.1~100Å)

=2.4310-2 Å

So is evident .

2.2.Photon collides with the Photon collides with the innerinner bound electron bound electron

Mc

hc

The mass of atom >>m0

0 c

10

So Compton effect is not evident .

[Example] [Example] xx-ray with -ray with 00=1.00=1.001010-10-10m m is is

scattered by electron. Find scattered by electron. Find The Compton shift The Compton shift at the scattering angle at the scattering angle =90=9000 and and The energy The energy that each electron gets from x-ray.that each electron gets from x-ray.

SolutionSolution ： ：

2sin2 2 c

0212 45sin1043.22

m1043.2 12

The energy that each electron gets from x-rayThe energy that each electron gets from x-ray

)()( 0 hh )(0

cch

)11

(00

hc

J1072.4 17So the energy that each electron getsSo the energy that each electron gets

)( hEk J1072.4 17

== the energy that each photon lose the energy that each photon lose