1

Chapter 2 Particle properties of waves

Electronics: particles charge , mass Wave?

Electromagnetic wave: wave diffraction, interference,

Polarization particle?

Wave-particle Duality

2.1Emwaves

Changing magnetic field current (or voltage)

Maxwell proposed: changing electric field magnetic field

Hertz created EM waves and determined the wavelength and

speed of the wave, and showed that they both have E and B

component, and that they could be reflected, refracted, and

diffracted. Wave characteristic.

2

Principle of Superposition:

When two or more waves of the same nature travel past a point

at the same time, the instantaneous amplitude is the sum of the

instantaneous amplitude of the individual waves.

3

Constructive interference same phase, greater amplitude

Destructive interference different phase, partial or

completely cancellation of waves

Interference wave characteristic

Young’s diffraction experiments:

diffraction wave characteristic

4

2.2Blackbody radiation

Is light only consistent of waves? Amiss: understand the origin

of the radiation emitted by bodies of matter.

Blackbody: a body that absorbs all radiation incident upon it,

regardless of frequency.

A blackbody radiates more when it is hot than it is cold, and the

spectrum of a hot blackbody has its peak at a higher frequency

than that of a cooler one.

5

Considering the radiation inside a cavity of absolute temperature

T whose walls are perfect reflectors to be a series standing EM

waves.

2/*nL

Density of standing waves in cavity

32 /8)( cddG

The higher ν, the shorter the wavelength, and the greater the

number of possible standing waves.

The average energy per degree of freedom of an entity that is a

member of a system of such entities in thermal equilibrium at T

is 1/2kT. K is Boltzmann’s constant=1.381*10-23J/k

6

An idea gas molecular has three degree of freedom: kinetic

energy in three independent directions 3/2kT

One dimensional harmonic oscillator has two degree of freedom:

kinetic energy and potential energy.

Each standing wave in a cavity originates in an oscillating

electric charge in the cavity wall. Two degree of freedom.

Classic average energy per standing wave kT

Total energy per unit volume in the cavity in and +d

u dckTdGd 23)/8()()( Rayleigh-Jeans formula

increase energy density increase with 2.

In the limit of infinitely high frequencies, u( )d goes to

infinity. In reality, the energy density(and the radiation rate)falls

to 0 as goes to infinity. Ultraviolet catastrophe

7

Plank Radiation Formula

u( )d =(8πh/c3)( 3d )/(eh /kT-1)

h is planck’s constant=6.626*10-34Js

h >>kT eh /kT u( ) 0

No ultraviolet catastrophe.

In general, ex=1+x+x2/2+………

When h << kT, 1/(eh /kT-1)~1/((1+(h /kT)-1)~kT/h

u( )d ~(8πh/c3)( 3d )/( kT/h )~(8πkT/c3) 2d

which is Rayleigh-Jeans formula.

8

How to justify the Plank radiation formula

The oscillators in the cavity walls could not have a continuous

Distribution of possible energy ε but must have only specific

energies εn=nh n=0,1,2……

An ocillator emits radiation of frequency when it drops from

one energy state to the next lower one, and it jumpsto the next

higher state when it absorbs radiation of . Each discrete bundle

of energy h is called a quantum.

With oscillator energies limited to nh , the average energy per

oscillator in the cavity walls turn out to be not kT as for a

continuous distribution of oscillator energies, but

ε=h /(eh /kT-1) average energy per standing wave

9

2.3 Photoelectric effect

Some of photoelectrons that emerge from the metal surface have

enough energy to reach the cathode despite its negative polarity

Current

When V is increased to a certain value V0, no more

photoelectrons arrive. V0 correspond to the max photoelectron

kinetic energy.

Three experimental finding:

(1) No delay between the arrival of the light at the metal surface

and the emission of photoelectrons.

(2) A bright light yields more photoelectrons than a dim one,

but highest electron energy remain the same.

10

(3) The higher the frequency of the light, the more energy the

photoelectrons have. At the frequencies smaller than 0,

which is a characteristic of the specific metal, no more

electrons are emitted.

Quantum theory of light

Einstein proposed Photons. The energy in light is not spread out,

but is concentrated in small packets. Each photon of light of

frequency has the energy h .

Einstein proposed that energy was not only given to em waves

in separate quanta but was also carried by the waves in separate

quanta.

11

Explanation of experiments:

(1) Since em wave energy is concentrated in photons and not

spread out, there should be no delay in the emission.

(2) All photons of frequency have the same energy h .

Changing the intensity of light only change the number of

photoelectrons but not their energy.

(3) The higher , the greater photon energy and so the more

energy the photoelectrons have.

νo corresponds to the min energy Φ for the electron to escape

from the metal surface. This energy is called work function.

Φ=hνo

Photoelectric effect hν=kEmax+Φ

h = kEmax+hνo

kEmax=h(ν-νo).

Photo energy

E=(6.626*10-34Js/1.602*10-19J/eV )ν=(4.136*10-15)νeVs

ν=c/λ E=1.24*10-6eVm/λ

12

What is light

Wave model: light intensity 2E

Particle model: light intensity N(#of photons/sec.area)

N 2E

.N is large interference pattern

N is small a series of random flashes

.if keep track of flashes for long time

same as large N

intensity of wave at a given place on the specific space

the probability of finding photons.

13

Wave & quantum theory complement each other.

photoelectric effect : Ephotons Ee΄

yes x-ray

faster e΄ more x-ray

# of e΄ increase Intensity of x-ray increase

.for given accelerating V λ min

V λ min

.most of e΄ heat

A few e΄ lose E in single collisions x-ray

14

.x-ray are em waves

EM theory predicts that an accelerated electric charge will

radiate em waves, and a rapidly moving e΄ suddenly brought to

rest is certainly accelerated Bremsstranlung (“braking

radiation”)

x-ray at specific λ nonclassical

* different targets give different characteristic x-ray

* for the same V, λmin is the same for different materials

λmin=(1.2410-6)/V(m)

15

hνmax = Ve = hc/λmin λmin = hc/Ve = (1.2410-6)/V

Scattering by an atom (wave model)

atom in E polarized distorted charge distribution

electric dipole

em wave with ν on atom polarization charge with ν

oscillating electric dipole radiate em wave

x-ray falls on a crystal will be scattered in all directions because

of regular arrangement of atoms constructively interference

Bragg’s condition (2dsinθ=λ)

16

Compton effect

Loss in photon energy=gain in e΄ energy

hν -hν΄ =kE

for massless particle E= Pc (P=momentum)

photon momentum P =E/c = hν/c

17

hν/c = (hν΄/c)cosΦ+ Pcosθ (parallel)……..(1)

0 = (hν΄/c)sinΦ – Psinθ () ……………….(2)

(1) &(2)x c

Pc(cosθ) = hν -hν΄cosΦ

Pc(sinθ) = (hν΄)sinΦ

P2c2 = (hν)2 – 2(hν)(hν΄)cosΦ +(hν΄)2

18

& E = KE + moc2 E = 2242

cPcmo

(KE + moc2)2 = mo

2c4 + P2c2

P2c2 = KE2 + 2moc2KE

Because KE = hν - hν΄

P2c2 = (hν - hν΄)2 + 2moc2KE

2moc2(hν - hν΄) = 2(hν)(hν΄) (1 –cosΦ)……(3)

(3)/2h2c2 moc/h(ν/c - ν΄/c) =(ν/c)( ν΄/c)(1 –cosΦ)

moc/h(1/λ – 1/λ΄) =(1 –cosΦ)/( λλ΄)

λ΄ - λ = (h/moc)(1 –cosΦ) = λc(1 –cosΦ)

λ=Compton wavelength

19

Relativistic formulas

Total energy E=

2

2

2

1c

v

cmo

mo= rest mass

Relativistic momentum P = 2

2

1c

v

vmo

When mo = 0 (massless particle) & v<c E=P=0

How about v=c , mo= 0 E=0/0 , P=0/0 (any values)

E2 =2

2

42

1c

v

cmo

, P2=

2

2

22

1c

v

vmo

P2c2 =

2

2

22

1c

v

vmo

c2

E2 – P2c2 =

2

2

42

1c

v

cmo

(1-v2/c2) = mo

2c4

E2 = mo2c4 +P2c2

For all particles E= 2242

0 cpcm = 222cpEo

Restriction of massless particles : E = Pc (mo=0)

Total energy mc2=moc2 +KE =

2

2

2

1c

v

cmo

20

Pair production

A photon give an e΄ all of its energy photoelectric

part of its energy compton

a photon materialize into an e΄ & position

21

(momentum is conserved with the help of the nucleus which

carries away enough photon momentum)

rest energy moc2 of electron or position is 0.51Mev pair

production requires a photon energy1.02Mev

pair production cannot occur in empty space

concervation of energy hν = 2mc2

momentum concervation hν/c = 2Pcosθ hν = 2Pc(cosθ)

P = mv hν = 2mc2(v/c) cosθ

v/c <1 & cosθ 1 hν < 2mc2

22

linear attenuation coefficient

udxI

dI

I = Ioexp(-ux)

absolute thickness x =u

IIo )ln(