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# Quantum Theory

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```									The Wave – Particle Duality

OR
Light Waves
Until about 1900, the classical wave theory of light described
most observed phenomenon.

Light waves:

Characterized by:

 Amplitude (A)
 Frequency (n)
 Wavelength (l)

Energy of wave a A2
YOUNG’S DOUBLE SLIT EXPERIMENT

An artists impression
Previous to the photoelectric effect, Young demonstrated the wave nature of
light (1801)

YOU CAN SEE ALL THIS USING A JAVA APPLET:
http://micro.magnet.fsu.edu/primer/java/interference/doubleslit
And then there was a problem…

In the early 20th century, several effects were observed which
could not be understood using the wave theory of light.

Two of the more influential observations were:

2) The photoelectric effect
3) The Compton scattering
Studies of the blackbody radiation spectrum
Predicting the existence of the photon
What is a blackbody?
• A black body is a theoretical object that absorbs all electromagnetic radiation that
falls on it. No electromagnetic radiation passes through it and none is reflected.
Because no light (visible electromagnetic radiation) is reflected or transmitted, the
object appears black when it is cold.
• If the black body is hot, these properties make it an ideal source of thermal radiation.
If a perfect black body at a certain temperature is surrounded by other objects in
thermal equilibrium at the same temperature, it will on average emit exactly as much
as it absorbs, at every wavelength. Since the absorption is easy to understand—every
ray that hits the body is absorbed—the emission is just as easy to understand.
• A black body at temperature T emits exactly the same wavelengths and intensities
which would be present in an environment at equilibrium at temperature T, and which
would be absorbed by the body. Since the radiation in such an environment has a
spectrum that depends only on temperature, the temperature of the object is directly
related to the wavelengths of the light that it emits.
• At room temperature, black bodies emit mostly infrared light, but as the temperature increases past a
few hundred degrees Celsius, black bodies start to emit visible wavelengths, from red, through
orange, yellow, and white before ending up at blue, beyond which the emission includes increasing
amounts of ultraviolet.
• The term "black body" was introduced by Gustav Kirchhoff in 1860. The light emitted by a black
In the laboratory, black-body radiation is approximated by the radiation from a small hole
entrance to a large cavity, a hohlraum. (this technique leads to the alternative term cavity
Any light entering the hole would have to reflect off the walls of the cavity multiple times before
it escaped, in which process it is nearly certain to be absorbed.

This occurs regardless of the wavelength of the radiation entering (as long as it is small
compared to the hole). The hole, then, is a close approximation of a theoretical black body
and, if the cavity is heated, the spectrum of the hole's radiation (i.e., the amount of light
emitted from the hole at each wavelength) will be continuous, and will not depend on the
material in the cavity, but only on the temperature of the cavity wall.
Studying the blackbody radiaton was a major challenge in theoretical physics during the
late nineteenth century, that ultimately led to Quantum Theory.
Starting from Stefan Boltzmann’s Law, we will move to the studies that led to the
quantisation of the electromagnetic field by Max Planck:
Stefan and Boltzman stated that the total
I  T 4      Stefan - Boltzman constant       intensity (power per unit area) over all wavelengths
  5.67039910-8W / m2  K 4                    for a blackbody radiator is proportional to
the fourth power of the absolute temperature.

For hot objects other than ideal radiators, the law is expressed in the form:

I  eT                 4

where e is the emissivity of the object (e = 1 for ideal radiator).
If the hot object is radiating energy to its cooler surroundings at temperature Tc, the net radiation
loss rate takes the form

I  e T 4 - To
4

Note:

The unusually rapid increase in radiation with
temperature is a consequence of the massless nature of
photons, the carriers of electromagnetic energy (the same
kind of behavior governs
phonons, the quanta of acoustic energy in solids and
liquids).
Although this relationship describes the total energy
emitted, it does not predict the spectral distribution.

Wien’s Displacement Law
Wien’s displacement law:

Wien's Law tells us that objects of different temperature emit spectra that peak at different
wavelengths. Hotter objects emit most of their radiation at shorter wavelengths; hence they will
appear to be bluer . Cooler objects emit most of their radiation at longer wavelengths; hence they
will appear to be redder.
Furthermore, at any wavelength, a hotter object radiates more (is more luminous) than a cooler
one.

1
l peak 
T

o
We define the proportionality constant t be :
l peakT  2.898 103 m  K

Wien’s displacement Law:
The wavelength at which the intensity reaches its maximum (peak)
value decreases as the temperature is increased, in inverse proportion
to the temperature.
RAYLEIGH AND JEANS: AIMED TO PROVIDE A THEORETICAL
EXPRESSION TO FIT WIEN’S EXPERIMENTAL EVIDENCE PROVIDED BY
WIEN.FOR THAT THEY USED CLASSICAL CONCEPTS OF ELECTROMAGNETISM
AND STATISTICAL MECHANICS

Although the derivation of this formula is beyond the scope of this course, the form can be easily
explained. In classical statistical mechanics, every possible degree of freedom should acquire an average
energy of 1/2 kT. For example, a monatomic gas molecule has energy 3/2 kT because it can move in
three independent directions. The total intensity at a given frequency is the product of the number of
modes available and the average energy in each. As the frequency f becomes large, the predicted
intensity increases without limit, even for objects at modest temperature. This is called "the ultraviolet
catastrophe". On the other hand, at low frequencies, the formula gave accurate predictions of
experimental results, indicating that at least some aspects were basically correct.

Lets take a closer look
• Step 1: Calculate the amount of radiation (number of waves/modes) at each
wavelength.
• Step 2: The contribution of each wave to the total energy in the box.
• Step 3: The radiancy corresponding to that energy.
STEP 1: Cavity modes
Rayleigh and Jeans showed that the number of modes N per
unit volume, per unit wavelength was inverse proportional to the
4th power of wavelength.

dN 8
 4
dl l
STEP 2: Energy density (energy per unit volume) U
per unit wavelength

Modes per unit    Average energy
volume per unit
CLASSICAL     wavelength        per mode
TREATMENT                                        dU 8
 4 kT
dN 8                             dl l
 4               kT
dl l
STEP 3:
dI  Rl dl
Rayleigh-Jeans Law
8                Rl   radiancy (intensity per unit length interval)
Rl   4 kT , c  λv
c
l    4
Classical theory: Rayleigh Jeans Law
R   as v   or l  0
Experimental Evidence

R  0 as v   or l  0

Planck suggested a formula to fit
the experimental data.

CLASSICAL THEORY LED TO
THE ULTRAVIOLET CATASTROPHE
Quantum Treatment: proposed by Max Planck

CLASSICAL                 Modes per unit                        Average energy
TREATMENT                 volume per unit                       per mode
wavelength
dN 8                               kT
 4
dNl 8
d     l
 4
QUANTUM                    dl l
dN 8                                   hc            1
TREATMENT                      4                                  
dl l                                   l 
hc
e   lkT
1
He postulated that electromagnetic energy was emitted in discrete units or quanta, each with energy given by hf
where h is the Planck constant, 6.6256 x 10-34 joule-second. For photon energies with hf>kT, it would no longer
be possible to populate each mode with kT average energy since a fraction of hf is no longer allowed.

The consequence is an additional factor that reduces the spectral distribution, approaches
unity for small f, preserving the long wavelength Rayleigh-Jeans behavior but squelches the
ultra-violet divergence.
Plancks idea lead to the idea of quantisation of the electromagnetic radiation.
The basic quantity of energy hv was associated with the notion of the photon, the
quantum of light.
E= hv , h=Planck’s constant

Based on the above result he recalculated the radiancy to be:


8 c       
1 
Rl   4  hv hv  
l 4        
 e 1 
KT

which agreed with Wien’s experimental evidence!!!!!
An experiment demonstrating the quantum
nature of light
Experimental verification of Planck’s
ideas
What is a photon according to
Planck’s hypothesis ?(1912 - 1915)
A photon is a quantum of the
electromagnetic field with energy E=hv.
What is photoelectric emission?
The process of using light to eject electrons
from a metal surface.
Red light incident on a clean sodium plate in vacuum

SODIUM PLATE

Surface at 0 potential

Shining Red Light on
my metal plate I get
no current on my
ammeter. How can I
Ammeter registers current I       change that?
Photoelectric Effect (I)
“Classical” Method                  What if we try this ?
Increase energy by            Vary wavelength, fixed amplitude
increasing amplitude
electrons                            electrons
emitted ?                            emitted ?
No
No
No                                  Yes, with
low KE
No
Yes, with
No                                  high KE

No electrons were emitted until the frequency of the light exceeded
a critical frequency- threshold frequency, at which point
electrons were emitted from the surface! (Recall: small l  large n)
CHANGING THE WAVELENGTH
Metal         X-rays        UV         Blue          Red
Light         Light
Zinc           Yes         Yes         No            No

Sodium            Yes         Yes         Yes            No

Even though there was no current present when illuminating the
sodium plate with red light, that changed when the wavelength was
changed to blue or UV.
Thus the ability of light to kick out electrons, depends on the
frequency of the light, which is associated with their KE.
There is a minimum frequency required to eject electrons:
threshold frequency (a characteristic for different
metals)
Photoelectric Effect - Explanation
 Electrons are attracted to the (positively charged) nucleus by the
electrical force

 In metals, the outermost electrons are not tightly bound, and can
be easily “liberated” from the shackles of its atom.

 It just takes sufficient energy…
Classically, we increase the energy
of an EM wave by increasing the
intensity (e.g. brightness)

Energy a A2

But this doesn’t work ??
 An alternate view is that light is acting like a particle

 The light particle must have sufficient energy to “free” the
electron from the atom.

 Increasing the Amplitude is simply increasing the number
of light particles, but its NOT increasing the energy of each one!
 Increasing the Amplitude does nothing

 However, if the energy of these “light particle” is related to their
frequency, this would explain why higher frequency light can
knock the electrons out of their atoms, but low frequency light cannot…
 In this “quantum-mechanical” picture, the energy of the
light particle (photon) must overcome the binding energy of the
electron to the nucleus.

 If the energy of the photon exceeds the binding energy, the
electron is emitted with a KE = Ephoton – Ebinding.

 The energy of the photon is given by E=hn, where the
constant h = 6.6x10-34 [J s] is Planck’s constant.

“Light particle”
For example:
Electron 1 at the surface requires the least possible energy to be liberated, so it escapes
with the most possible kinetic energy.

Electron 2 deep in the material has lost too much kinetic energy by the time it reaches the
Surface and is attracted back.

Electron 3 escapes, but has less kinetic energy than electron 1

Electron 4 gained enough energy to escape, but was moving in the wrong direction and was
absorbed by the metal.
3
1
2
Photon

Metal
ion                               4
Electron
Einstein’s photoelectric equation
• Einstein suggested that when a photon causes photoemission
from a metal surface, some of the photon energy is used to
overcome the work function, while the remainder appears at
the kinetic energy of the ejected electron. He expressed that
in his famous photoelectric equation:
Kmax = hv – Φ

Einstein was awarded
the Nobel Prize for explaining
the
photoelectric effect in
1921
Work function (Ф)= Binding Energy
• Is the work necessary to remove an electrons from the
surface of the material.

• Different metals have different work functions, but the
work function of any metal is a characteristic of the metal
it self, and does not change for different frequencies of
light.
Stopping potential
• The photoelectric equation corresponds to a metal surface at
zero potential.

• Let the surface be held at a positive potential. If we gradually
increase the positive potential, we come to a point where no
electrons can escape any more.

prevention
 of photoem ission

      
K m ax  hv    eV             
         hv    eVs  0
m axim umenergyreducedto zero

h     h    h                            
 Vs  v     v  vo                 for vo 
e    e  e   e                            h

STOPPING POTENTIAL
Vs

vo                                       v

-Φ/e
Plotting the stopping potential against frequency
• Wave theory of light:

1. Any frequency of light would
eject an electron.

2. A higher light intensity, would
cause an increase in current.

3. The procedure of ejecting the
electrons would not be immediate.
• Particle theory of light:

1. Minimum frequency of light required to eject
electrons: threshold frequency.

2. If the frequency is sufficient to eject electrons,
increasing the intensity of light would increase the
current. If the frequency is not sufficient, then any
increase in the light intensity would have no effect
in producing a current. Changing the frequency,
would cause a change in the electron KE and
stopping potential.

3. The procedure of ejecting the electrons is
immediate.
Photons
 Quantum theory describes light as
a particle called a photon

 According to quantum theory, a
photon has an energy given by

E = hn = hc/l           h = 6.6x10-34 [J s] Planck’s constant,
after the scientist Max Planck.

 The energy of the light is proportional to the frequency (inversely
proportional to the wavelength) ! The higher the frequency (lower
wavelength) the higher the energy of the photon.

 10 photons have an energy equal to ten times a single photon.

 Quantum theory describes experiments to astonishing precision,
whereas the classical wave description cannot.
The Electromagnetic Spectrum
Shortest wavelengths
(Most energetic photons)

E = hn = hc/l
h = 6.6x10-34 [J*sec]
(Planck’s constant)

Longest wavelengths
(Least energetic photons)
Can a particle exist that has no rest mass, but
which nevertheless exhibits such particle-like
properties as energy and momentum?

According to classical theory   Special Relativity nevertheless
the answer is no                allows for it.
• Photons are particles that are never found at rest. That
means that their rest mass is zero.

• Photons move with the speed of light c.

• Using the relationship between momentum and energy
from Special Relativity, we find:

E  E  c p  mo c
2    2
o
2   2
     
2 2
c p c p
2   2   2   2

       E  pc
The Compton Effect
In 1924, A. H. Compton performed an experiment where X-rays
impinged on matter, and he measured the scattered radiation.

Incident X-ray
wavelength        M
A
l1            T                  Scattered X-ray
T
E
wavelength         l2 > l1
R                         l2
e

Application of Planck’s ideas               Electron comes flying out

Problem: According to the wave picture of light, the incident X-ray
should give up some of its energy to the electron, and emerge with a
lower energy (i.e., the amplitude is lower), but should have l2l1.
It was found that the scattered X-ray did not have the same wavelength ?
Quantum Picture to the Rescue
Electron                            Scattered X-ray
Incident X-ray
initially at                           E2 = hc / l2
rest (almost)
E1 = hc / l1

e
l2 > l1
e

Ee

Compton found that if you treat the photons as if they were particles
of zero mass, with energy E=hc/l and momentum p=h/l

 The collision behaves just as if it were 2 billiard balls colliding !

Photon behaves like a particle with energy & momentum as given above!
BEFORE THE COLLISION              AFTER THE COLLISION

Scattered photon
INCIDENT PHOTON
P’=hf’/c
ELECTRON AT REST
hf                                        

Pe=mu
P=hf/c
Scattered electron

Kinetic energy of the scattered electron: Ke=mc2-moc2
Conservation of linear momentum and
energy

hf hf 
X : pbefore  pafter           cos  pe cos
c   c

hf 
Y : pbefore  pafter   0      sin   pe sin 
c

Ebefore  Eafter  hf  hf   K e
Squaring the following equations and adding them together, we eliminate


pe c cos  hf  hf  cos 
 pe c 2  hf   2hf hf  cos  hf 
2           2                            2

pe c sin   hf  sin     

Then weequate the following two expressions for the total energy of the
particle.

E  KE  mo c 2     


  KE  mo c
2
2
 mo c 4  pe c 2  pe c 2  KE 2  2mo c 2 KE
2        2        2

E  mo c 4  pe c 2 
2        2

But :

KE  hf  hf 

 pe c 2  KE 2  2mc 2 KE  hf   2hf hf   hf   2mc 2 hf  hf 
2                             2                   2

on
But from the conservati of momentum we have :

pe c 2  hf   2hf hf  cos  hf 
2          2                            2

 2mc 2 hf  hf   2hf hf 1  cos 
The above relationship is simpler when expressedin terms of the wavelength
l.
Dividing by 2h 2 c 2 we have :

2mo c 2 hf  hf   2hf hf 1  cos 

mo c  f f   f f 
                  1  cos 
h c c  c c

f 1     f 1
Since   and    we have :
c l     c l

mo c  1 1  1 - cos
             l -l 
h
1  cos 
h  l l      ll            mo c
l  - l  lC 1  cos  , lC 
h
mo c

The above equation gives the change in wavelength
expected for a photon that is scattered through the angle 
by a particle of rest mass m.

This change is independent of the wavelength λ of the
incident photon.

The quantity λC is called the Compton wavelength of the
scattering particle.
Compton effect-Summary

Compton was able to explain all he was seeing
by using the photon theory of light. As incident photons
collided with the electrons, they transferred some of their
energy to them. Compton applied the conservation of
momentum and energy to the experiment, and found the
results agreed. Photons conformed with the laws of
conservation of momentum and energy. This provided
support that electromagnetic radiation is quantised and has
momentum and energy associated with it.
Summary of Photons
 Photons can be treated as “packets of
light” which behave as a particle.

 To describe interactions of light with matter, one generally has to
appeal to the particle (quantum) description of light.

 A single photon has an energy given by
E = hc/l,
where
h = Planck’s constant = 6.6x10-34 [J s]   and,
c = speed of light    = 3x108 [m/s]
l = wavelength of the light (in [m])

 Photons also carry momentum. The momentum is related to the
energy by:                   p = E / c = h/l
So is light a
wave or a
particle ?

On macroscopic scales, we can treat a large number of photons
as a wave.

When dealing with subatomic phenomenon, we are often dealing
with a single photon, or a few. In this case, you cannot use
the wave description of light. It doesn’t work !
Matter Waves: The De Broglie hypothesis
• If there is a particle of momentum p, its motion
is associated with a wave of wavelength:
h  h
l              DE BROGLIE WAVELENGTH
p mu
MATTER WAVES
The De Broglie wavelength of a moving cow would be
trillions times smaller than atomic dimensions and
far too small to be detected.
The De Broglie wavelength of a C60 carbon molecule
(buckyball) travelling at a few meters per second is
approximately the size of the molecule itself (about 1nm).
The De Broglie wavelength of an electron travelling at
a few meters per second has a de Broglie wavelength
equivalent to the width of a human hair (a fraction of a
millimetre. That is large enough for the quantum wave to
actually show up in experiments.
Complementarity
•   Light exhibits both wave and particle bahaviour, and using different experimental
set ups and we can either detect one or the other.
e.g. the photoelectric effect demonstrates the particle nature of light (photons),
while Young’s double slit experiment demonstrates the wave nature of light.
Both the particle and wave picture of light are valid and under no circumstances
cancel each other. In the contrary they are complementary.

This is the principle of complementarity in quantum mechanics and we will see it
exhibiting itself in more examples as we move across the subject

The principle of complementarity also holds for particles. Both the matter
wave description of a moving atom, or it’s description as a localised particle are
equally valid.

```
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