Particle Wave Duality What is a particle? What is a wave?
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Particle Wave Duality
What is a particle? What is a wave?
Problems with Classical Physics
•Nature of Light?
•Discrete Spectra?
•Blackbody Radiation?
•Photoelectric Effect?
•Compton Effect?
•Model of Atom?
Double Slit is VERY IMPORTANT because it is evidence
of waves. Only waves interfere like this.
Thomas Young 1804
sind m
REVIEW! Derive Fringe Equations
• For bright fringes
• For dark fringes
bright ( 0 1 2 ), ,λL
y m md
dark
1( 0 1 2 )
2, ,
λLy m m
d
Double Slit for Electrons
shows Wave Interference!
Key to Quantum Theory!
James Clerk Maxwell 1860s
Light is an electromagnetic wave.
The medium is the Ether.
8
0
13.0 10 /
o
c x m s
The Electromagnetic Spectrum
Michelson-Morely
Experiment
1887 The speed of light is independent of the motion and
is always c. The speed of the Ether wind is zero.
OR….
Lorentz Contraction
The apparatus shrinks by a factor :
2 21 / v c
Special Relativity 1905
2 2 2 2 2
0( ) ( ) ( ) E pc m c pc
2 E mc p mu
If m = 0 (photon)
/p E cphoton momentum:
2
0 E mc
2 2( )u u
pc muc m c mc Ec c
22 2
0( ) E pc E
p E
2 2 22 2 2 2 2 2 2 2 2 2 2
0 02 22
2
(1 ) ( )
1
mc u uE mc E mc E E E E E E
c cu
c
Why Continuous vs Discrete?
This is a continuous spectrum of colors: all colors are present.
This is a discrete spectrum of colors: only a few are present.
Kirkoff’s Rules
Kirkoff’s Rules for Spectra: 1859
Bunsen
German physicist who developed the spectroscope and the science of
emission spectroscopy with Bunsen.
Kirkoff
* Rule 1 : A hot and opaque solid, liquid or highly compressed gas emits a continuous spectrum.
* Rule 2 : A hot, transparent gas produces an emission spectrum with bright lines.
* Rule 3 : If a continuous spectrum passes through a gas at a lower temperature, the transparent
cooler gas generates dark absorption lines.
Compare absorption lines in a source with emission
lines found in the laboratory!
Kirchhoff deduced that elements were present in the atmosphere of the Sun
and were absorbing their characteristic wavelengths, producing the absorption
lines in the solar spectrum. He published in 1861 the first atlas of the solar
spectrum, obtained with a prism ; however, these wavelengths were not very
precise : the dispersion of the prism was not linear at all.
Anders Jonas Ångström 1869 Ångström measured the wavelengths on the
four visible lines of the hydrogen spectrum,
obtained with a diffraction grating, whose
dispersion is linear, and replaced
Kirchhoff's arbitrary scale by the
wavelengths, expressed in the metric
system, using a small unit (10-10 m) with
which his name was to be associated.
Line color Wavelength
red 6562.852 Å
blue-green 4861.33 Å
violet 4340.47 Å
violet 4101.74 Å
Balmer Series: 1885 Johann Balmer found an empirical equation that correctly
predicted the four visible emission lines of hydrogen
H 2 2
1 1 1
2R
λ n
RH is the Rydberg constant
RH = 1.097 373 2 x 107 m-1
n is an integer, n = 3, 4, 5,…
The spectral lines correspond to different
values of n
Johannes Robert Rydberg generalized
it in 1888 for all transitions:
Hα is red, λ = 656.3 nm
Hβ is green, λ = 486.1 nm
Hγ is blue, λ = 434.1 nm
Hδ is violet, λ = 410.2 nm
Why this shape? Why the drop?
When an object it heated it will
glow first in the infrared, then the
visible. Most solid materials break
down before they emit UV and
higher frequency EM waves.
Frequency ~ Temperature
Long
Short
All objects radiate energy continuously
in the form of electromagnetic waves
due to thermal vibrations of their
molecules.
A good absorber reflects little and appears Black
A good absorber is also a good emitter.
Blackbody Radiation • A black body is an ideal system that
absorbs all radiation incident on it
• The electromagnetic radiation emitted by a
black body is called blackbody radiation
c
Blackbody Approximation
• A good approximation of a
black body is a small hole
leading to the inside of a
hollow object
• The hole acts as a perfect
absorber
• The nature of the radiation
leaving the cavity through
the hole depends only on the
temperature of the cavity
Stefan’s Law: 1879
Rate of radiation of a Black Body
• P = σAeT 4 – P is the rate of energy transfer, in Watts
– σ = 5.6696 x 10-8 W/m2 . K4
– A is the surface area of the object
– e is a constant called the emissivity • e varies from 0 to 1
• The emissivity is also equal to the absorptivity
– T is the temperature in Kelvins
– With his law Stefan determined the temperature of the Sun’s surface and he calculated a value of 5430C. This was the first sensible value for the temperature of the Sun.
– Boltzmann was his student and derived Stefan’s Law from Thermodynamics in 1884 and extended it to grey bodies.
Jožef Stefan
(1835–1893)
Maxwell-Boltzmann Distribution: 1877
• The observed speed distribution of gas molecules in thermal equilibrium is shown at right
• NV is called the Maxwell-Boltzmann speed distribution function
• The distribution of speeds in N gas molecules is
• The probability of finding the molecule in a particular energy state varies exponentially as the negative of the energy divided by kBT
2
3 / 2
/ 22
B
42
Bmv k ToV
mN N v e
k T
nV (E ) = noe –E /kBT
Ludwig Boltzmann
1844 – 1906
• Temperature ~ Ave KE of each particle
• Particles have different speeds
• Gas Particles are in constant RANDOM motion
• Equipartition of Energy: Average KE of
each particle is: 3/2 kT
• Pressure is due to momentum transfer
Speed ‘Distribution’ at
CONSTANT Temperature
is given by the
Maxwell Speed Distribution
23/ 2 1/ 2 rmskT KE mv
k =1.38 x 10-23 J/K Boltzmann’s Constant
4P e T A
Radiant heat makes it impossible to stand close to a hot lava
flow. Calculate the rate of heat loss by radiation from 1.00
m2 of 1200C fresh lava into 30.0C surroundings, assuming
lava’s emissivity is 1.
The net heat transfer by radiation is: 4 4
2 1( )P e A T T
4 4
2 1( )P e A T T
8 4 2 4 41(5.67 10 / )1 ((303.15 ) (1473.15 ) )x J smK m K K
266P kW
Blackbody Experiment Results • The total power of the radiation emitted from the
surface increases with temperature
– Stefan’s law: P = AeT4
– P is the power and is the Stefan-Boltzmann constant:
= 5.670 x 10-8 W / m2 . K4 (0<e < 1, for a blackbody, e = 1)
• The peak of the wavelength distribution shifts to
shorter wavelengths as the temperature increases
– Wien’s displacement law
(T must be in kelvin):
Finding peak wavelengths
Finding peak wavelengths
The heating effect of a medium such as glass or the Earth’s
atmosphere that is transparent to short wavelengths but opaque
to longer wavelengths: Short get in, longer are trapped!
Intensity of Blackbody Radiation
• The intensity increases with increasing temperature
• The amount of radiation emitted increases with increasing temperature
– The area under the curve
• The peak wavelength decreases with increasing temperature
• Combining gives the Rayleigh-Jeans law:
I = P/A = T4
I , ~4
1λ T
λ
Problems with the Wein’s World
• At short wavelengths, there
was a major disagreement
between the Rayleigh-Jeans
law and experiment
• This mismatch became
known as the ultraviolet
catastrophe
– You would have infinite
energy as the wavelength
approaches zero
I , ~4
1λ T
λ
Max Planck: Father of Quantum
• Introduced the concept of “quantum of action” in 1900 to solve the black body mystery
• In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy
The possible frequencies and energy states of a wave on
a string are quantized.
2
vf n
l
Strings are Quantized
Planck’s Two Assumptions
• The energy of an oscillator can have only certain discrete values En= nhƒ
– This says the energy is quantized
– Each discrete energy value corresponds to a different quantum state
• The oscillators emit or absorb energy when making a transition from one quantum state to another
– The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation
Energy-Level Diagram
• An energy-level diagram
shows the quantized energy
levels and allowed
transitions
• Energy is on the vertical axis
• Horizontal lines represent
the allowed energy levels
• The double-headed arrows
indicate allowed transitions
More About Planck’s Model
• The average energy of a wave is the average
energy difference between levels of the oscillator,
weighted according to the probability of the wave
being emitted
• This weighting is described by the Boltzmann
distribution law and gives the probability of a state
being occupied as being proportional to
BE k Te
Planck’s Wavelength
Distribution Function
• Planck generated a theoretical expression
for the wavelength distribution
– h = 6.626 x 10-34 J.s
– h is a fundamental constant of nature
2
5
2
1I ,
Bhc λk T
πhcλ T
λ e
Planck’s
Model,
Graphs
Intensity of Blackbody Radiation P40.61
The total power per unit area radiated by a black body at a temperature T is the area under the I(λ, T)-versus-λ curve, as shown in Figure 40.3. (a) Show that this power per unit area is
where I(λ, T) is given by Planck’s radiation law and σ is a constant independent of T.
This result is Stefan’s law.
To carry out the integration, you should make
the change of variable x = hc/λkT and use the
fact that
4
0λ λ, TdTI
0
43
151
xe
dxx
2
5
2
1I ,
Bhc λk T
πhcλ T
λ e
34
, n= 0,1,2,3,...
6.626 10
E nhf
h x Js
Atomic Energy is quantized.
It comes in chunks of Planck’s constant, h.
Max Planck NEVER liked the idea
of quantized energy states.
In classical physics particles have
continuous energy states….to say they
have discrete energy states would mean
that you can only drive at 10mph and
20mph but not at 15mph or at any speed
in between 10 and 20 mph! In classical
physics only special bound states have
discrete or quantum energy states….
Why doesn’t Planck like Quantum?????
The Photoelectric Effect • In 1886 Hertz noticed, in the
course of his investigations, that a
negatively charged electroscope
could be discharged by shining
ultraviolet light on it.
• In 1899, Thomson showed that the
emitted charges were electrons.
• The emission of electrons from a
substance due to light striking its
surface came to be called the
photoelectric effect.
• The emitted electrons are often
called photoelectrons to indicate
their origin, but they are identical
in every respect to all other
electrons.
The Problem with Waves:
Increasing the intensity of a low frequency
light beam doesn’t eject electrons. This
didn’t agree with wave picture of light
which predicts that the energy of waves
add so that if you increase the intensity of
low frequency light (bright red light)
eventually electrons would be ejected –
but they don’t! There is a cut off
frequency, below which no electrons will
be ejected no matter how bright the beam!
Also there is no time delay in the ejection
of electrons as the waves build up!
The PROBLEM with the
Photoelectric Effect
The Problem with Waves:
Increasing the intensity of a low frequency
light beam doesn’t eject electrons. This
didn’t agree with wave picture of light
which predicts that the energy of waves
add so that if you increase the intensity of
low frequency light (bright red light)
eventually electrons would be ejected –
but they don’t! There is a cut off
frequency, below which no electrons will
be ejected no matter how bright the beam!
Also there is no time delay in the ejection
of electrons as the waves build up!
The Photoelectric Effect Proof that Light is a Particle
Characteristics of the
Photoelectric Effect 1. The current I is directly proportional to the light intensity.
2. Photoelectrons are emitted only if the light frequency f
exceeds a threshold frequency f0.
3. The value of the threshold frequency f0 depends on the
type of metal from which the cathode is made.
4. If the potential difference ΔV is positive, the current does
not change as ΔV is increased. If ΔV is made negative, the
current decreases until, at ΔV = −Vstop the current reaches
zero. The value of Vstop is called the stopping potential.
5. The value of Vstop is the same for both weak light and
intense light. A more intense light causes a larger current,
but in both cases the current ceases when ΔV = −Vstop.
The Photoelectric Effect
Photoelectric Effect Problem 1
• Dependence of photoelectron kinetic energy on light intensity
– Classical Prediction • Electrons should absorb energy continually from the
electromagnetic waves
• As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy
– Experimental Result • The maximum kinetic energy is independent of light intensity
• The maximum kinetic energy is proportional to the stopping potential (DVs)
Photoelectric Effect Problem 2
• Time interval between incidence of light and
ejection of photoelectrons
– Classical Prediction
• At low light intensities, a measurable time interval should pass
between the instant the light is turned on and the time an
electron is ejected from the metal
• This time interval is required for the electron to absorb the
incident radiation before it acquires enough energy to escape
from the metal
– Experimental Result
• Electrons are emitted almost instantaneously, even at very low
light intensities
Photoelectric Effect Problem 3
• Dependence of ejection of electrons on light
frequency
– Classical Prediction
• Electrons should be ejected at any frequency as long as the light
intensity is high enough
– Experimental Result
• No electrons are emitted if the incident light falls below some
cutoff frequency, ƒc
• The cutoff frequency is characteristic of the material being
illuminated
• No electrons are ejected below the cutoff frequency regardless
of intensity
Photoelectric Effect Problem 4
• Dependence of photoelectron kinetic energy on
light frequency
– Classical Prediction
• There should be no relationship between the frequency of the
light and the electric kinetic energy
• The kinetic energy should be related to the intensity of the light
– Experimental Result
• The maximum kinetic energy of the photoelectrons increases
with increasing light frequency
Einstein’s Postulates: Light Quanta
Einstein framed three postulates about light quanta and
their interaction with matter:
1. Light of frequency f consists of discrete quanta, each
of energy E = hf, where h is Planck’s constant
h = 6.63 × 10−34 J s. Each photon travels at the
speed of light c = 3.00 × 108 m/s.
2. Light quanta are emitted or absorbed on an all-or-
nothing basis. A substance can emit 1 or 2 or 3
quanta, but not 1.5. Similarly, an electron in a metal
can absorb only an integer number of quanta.
3. A light quantum, when absorbed by a metal, delivers
its entire energy to one electron.
Light is quantized in chunks of Planck’s constant. Electrons will not be ejected in the Photoelectric Effect unless every
photon has the right energy. One photon is completely absorbed by
each electron ejected from the metal. As you increase the intensity of
the beam, more electrons are ejected, but their energy stays the same.
Photons
E hf
EX 39.2 The energy of a light quantum
Einstein’s Explanation of the
Photoelectric Effect An electron that has just absorbed a quantum of light
energy has Eelec = hf. (The electron’s thermal energy at
room temperature is so much less than that we can neglect
it.) This electron can escape from the metal, becoming a
photoelectron, if
In other words, there is a threshold frequency
for the ejection of photoelectrons because each light
quantum delivers all of its energy to one electron.
Einstein’s Explanation of the
Photoelectric Effect • A more intense light delivers a larger number of light
quanta to the surface. These quanta eject a larger number
of photoelectrons and cause a larger current.
• There is a distribution of kinetic energies, because
different photoelectrons require different amounts of
energy to escape, but the maximum kinetic energy is
The stopping potential Vstop is directly proportional to Kmax.
Einstein’s theory predicts that the stopping potential is
related to the light frequency by
max 0hf KE E
Photon
Energy
Max KE of
ejected electron
Work to eject
(Work Function)
or Binding Energy
Cutoff Frequency • The lines show the linear
relationship between K and ƒ The slope of each line is h
• The x-intercept is the cutoff frequency. This is the frequency below which no photoelectrons are emitted
• The cutoff frequency is related to the work function through ƒc = φ / h
• The cutoff frequency corresponds to a cutoff wavelength
ƒc
c
c hcλ
φ
Work Function
The Photoelectric Effect What is the maximum velocity of electrons ejected from
a material by 80nm photons, if they are bound to the
material by 4.73eV? Ignore relatavistic effects.
MaxK BE BEhc
hf
1 2
18
2 6
31
2 1.729 10 J1 2KEK 1.95 10 m s
2 9.11 10 kgmv v
m
34 8 19
9
6.63 10 J s 3.00 10 m s 1.60 10 J4.73 eV
80.0 10 m 1 eV
181.7295 10 J
(SR: ) 2( 1)K mc
Maximum photoelectron speed
Compton Effect, Classical
Predictions • According to the
classical theory, EM
waves incident on
electrons should:
– have radiation pressure
that should cause the
electrons to accelerate
– set the electrons
oscillating
Compton Effect, Observations
• Compton’s
experiments showed
that, at any given
angle, only one
frequency of radiation
is observed
Compton Effect, Explianed
• The results could be explained by treating the photons as point-like particles having energy hƒ
• Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved
• This scattering phenomena is known as the Compton effect
Compton Shift Equation
• The graphs show the scattered
x-ray for various angles
• The shifted peak, λ’ is caused
by the scattering of free
electrons
– This is called the Compton shift
equation
1' coso
e
hλ λ θ
m c
Arthur Holly Compton
• 1892 - 1962
• Director of the lab at
the University of
Chicago
• Discovered the
Compton Effect, 1923
• Shared the Nobel
Prize in 1927
hp
The phenomenon in which an X-ray photon is scattered from an
electron, the scattered photon having a smaller frequency than the
incident photon is called The Compton Effect.
hcE hf pc
2 E mc p mv
E pcDivide:
incident scattered electronp p p
The photon transfers momentum, acts like a particle.
The Compton wavelength of a particle is
equivalent to the wavelength of a photon whose
energy is the same as the rest mass of the particle.
It gives the limits of measuring the position of a
particle using traditional QM and not QED.
The compton wavelength of the elctron is:
1' coso
e
hλ λ θ
m c
. 122 43 10e
hx m
m c
Compton Wavelength
The incident X-ray photon has an energy of 3.98 keV and is scattered by an angle of 140.0 degrees.
a) What is the wavelength of incident X-ray?
b) What is the wavelength of the scattered X-ray?
c) What is the energy of the scattered X-ray?
d) What is the kinetic energy of the recoil electron?
e) What is the de Broglie wavelength of the recoil electron?
82.998 10 /c x m s319.109 10em x kg
346.626 10h x Js
191.602 10 /x J eV
,
Joseph John Thomson
“Plum Pudding” Model 1904
• Received Nobel Prize in
1906
• Usually considered the
discoverer of the electron
• Worked with the
deflection of cathode rays
in an electric field
• His model of the atom
– A volume of positive
charge
– Electrons embedded
throughout the volume
1911: Rutherford’s
Planetary Model of the
Atom
(Couldn’t explain the stability or spectra of atoms.)
•A beam of positively charged alpha
particles hit and are scattered from a
thin foil target.
•Large deflections could not be
explained by Thomson’s pudding
model.
1911: Rutherford’s Planetary
Model of the Atom
(Couldn’t explain the stability or spectra of atoms.)
•A beam of positively charged alpha
particles hit and are scattered from a
thin foil target.
•Large deflections could not be
explained by Thomson’s model.
Classical Physics at the Limit
WHY IS MATTER (ATOMS)
STABLE?
Electrons exist in quantized orbitals with energies given by
multiples of Planck’s constant. Light is emitted or absorbed when
an electron makes a transition between energy levels. The energy of
the photon is equal to the difference in the energy levels:
i fE E E hf
34
, n= 0,1,2,3,...
6.626 10
E nhf
h x Js
Light Absorption & Emission i fE E E hf
1. Electrons in an atom can occupy only certain discrete quantized
states or orbits.
2. Electrons are in stationary states: they don’t accelerate and they
don’t radiate.
3. Electrons radiate only when making a transition from one
orbital to another, either emitting or absorbing a photon.
Bohr’s Assumptions
Postulate: The angular momentum of an electron is
always quantized and cannot be zero:
2
( 1,2,3,....)
hL n
n
: = ( 1,2,3,....)2
hFrom L n mvr n
Bohr’s Derivation of the Energy for Hydrogen:
E K U
F is centripetal:
Conservation of E:
Sub back into E:
From Angular
Momentum:
(1)
Sub r back into (1):
(2)
Sub into (2):
Why is it negative?
2 2 2
0 0
2 2
4 2
e e ekq q q
vnh hn hn
Bohr Line Spectra of Hydrogen
2
2 2
1 1 1( )
f i
RZn n
7 11.097 10R x m
Balmer: Visible
Lyman: UV
Paschen: IR
Bohr’s Theory derived the spectra equations that Balmer,
Lyman and Paschen had previously found experimentally!
2
0 13.6E Z eV
1. Bohr model does not explain why electrons don’t radiate in orbit.
2. Bohr model does not explain splitting of spectral lines.
3. Bohr model does not explain multi-electron atoms.
4. Bohr model does not explain ionization energies of elements.
1. Bohr model does not explain angular momentum postulate:
The angular momentum of an electron is
always quantized and cannot be zero*:
2
( 1,2,3,....)
hL n
n
*If L=0 then the electron travels linearly and does not ‘orbit’….
But if it is orbiting then it should radiate and the atom would be unstable…eek gads!
WHAT A MESS!
E hf hp
c c
If photons can be particles, then
why can’t electrons be waves?
e
h
p
Electrons are
STANDING
WAVES in
atomic orbitals.
deBroglie Wavelength:
2 nr n
2
( 1,2,3,....)
e n
hL m vr n
n
2e n
hm vr n
1924: de Broglie Waves
Explains Bohr’s postulate of angular
momentum quantization:
h
p 2 nr n
2 n
e
h hr n n
p m v
Electrons are STANDING WAVES in atomic orbitals.
h
p
1924: de Broglie Waves
2 nr n
De Broglie Wavelength
346.626 10h x J s
h
mv
Lynda’s De Broglie Wavelength
346.626 10
(75 )2 /
x J s
kg m s
364.4 10x m
Too small to notice or to interact with anything!
Particle-Wave: Light
A gamma ray photon has a momentum of 8.00x10-21 kg m/s.
What is its wavelength? What is its energy in Mev?
h
p
663 10
800 10829 10
34
21
14.
..
J s
kg m s m
21 88.00 10 kg m s 3.00 10 m sE pc
12
13
1 MeV2.40 10 J 15.0 MeV
1.60 10 J
Electron De Broglie Wavelength
for electron v = .1c
34
31 7
6.626 10
(9.1 10 )(3 10 / )
x J s
x kg x m s
/h mv
112.4 10x m
Limits of Vision
112.4 10e x m
Electron
Waves
Electron Microscope
Electron microscope picture of a fly.
The resolving power of an optical lens depends on the wavelength of
the light used. An electron-microscope exploits the wave-like
properties of particles to reveal details that would be impossible to see
with visible light.
Electron Microscope
Stem Cells
The fossilized shell of
a microscopic ocean
animal is magnified
392 times its actual
size.
Salmonella Bacteria
Double Slit for Electrons
shows Wave Interference
Double Slit for Electrons
A modified oscilloscope is used to
perform an electron interference
experiment. Electrons are incident on
a pair of narrow slits 0.060 0 μm
apart. The bright bands in the
interference pattern are separated by
0.400 mm on a screen 20.0 cm from
the slits. Determine the potential
difference through which the
electrons were accelerated to give
this pattern.
Particle-Wave Duality
Interference pattern builds one
electron at a time.
Electrons act like
waves going through
the slits but arrive at
the detector like a
particle.
Trying to see what slit an
electron goes through destroys
the interference pattern.
Feynman version of the
Uncertainty Principle
It is impossible to design an apparatus
to determine which hole the electron
passes through, that will not at the
same time disturb the electrons enough
to destroy the interference pattern.
-Richard Feynman
Next Time….
Electron waves are probability waves
in the ocean of uncertainty. - Richard Feynman:
Related Documents
