Download Chapter 2 Particle properties of waves

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.
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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.
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Constructive interference
Destructive interference
same phase, greater amplitude
different phase, partial or
completely cancellation of waves
Interference
wave characteristic
Young’s diffraction experiments:
diffraction
wave characteristic
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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.
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Considering the radiation inside a cavity of absolute temperature
T whose walls are perfect reflectors to be a series standing EM
waves.
L  n* / 2
Density of standing waves in cavity
G( )d  8 2d / c3
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
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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
u ( )d

 G( )d  (8kT / c3 ) 2d
increase
 +d
Rayleigh-Jeans formula
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
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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.
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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
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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.
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(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.
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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
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 What is light
Wave model: light intensity
Particle model: light intensity
N

E2

N(#of photons/sec.area)
 E2
.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.
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Wave & quantum theory complement each other.
photoelectric effect : Ephotons
yes
x-ray
faster e΄
more x-ray
# of e΄ increase
Intensity of x-ray increase
.for given accelerating V
V
.most of e΄
Ee΄
λ min
λ min
heat
A few e΄ lose E in single collisions
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x-ray
 .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=(1.24  10-6)/V(m)
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hνmax = Ve = hc/λmin
λmin = hc/Ve = (1.24  10-6)/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
Bragg’s condition (2dsinθ=λ)
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constructively interference
 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
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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
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&
E = KE + moc2
E=
mo c 4  P 2c 2
2
(KE + moc2)2 = mo2c4 + 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
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 Relativistic formulas
Total energy
E=
mo c 2
mo= rest mass
v2
1 2
c
mo v
Relativistic momentum P =
When
1 v
c2
mo = 0 (massless particle) & v<c
How about v=c , mo= 0
E2 =
2
2
mo c 4
2
1 v 2
c
,
E –P c
2
2 2
P2=
2
mo c 4
= 2
1 v 2
c
E=P=0
E=0/0 , P=0/0 (any values)
2
mo v 2
v2
1 2
c
P2c2 =
2
mo v 2 2
c
v2
1 2
c
(1-v2/c2) = mo2c4
E2 = mo2c4 +P2c2
For all particles
E=
m0 c 4  p 2c 2
2
=
Eo  p 2c 2
2
Restriction of massless particles : E = Pc (mo=0)
Total energy
2
2
mc =moc +KE =
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mo c 2
1 v
2
c2
 Pair production
A photon give an e΄ all of its energy
part of its energy
a photon materialize into an e΄ & position
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photoelectric
compton
(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 energy  1.02Mev
 pair production cannot occur in empty space
concervation of energy hν = 2mc2
momentum concervation
P = mv
hν/c = 2Pcosθ
hν = 2mc2(v/c) cosθ
v/c <1 & cosθ  1
hν < 2mc2
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hν = 2Pc(cosθ)
linear attenuation coefficient
 dI
 udx
I
I = Ioexp(-ux)
absolute thickness
x=
ln(
Io
I
)
u
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