Wave Particle Duality

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Wave - Particle Duality

• Subhalakshmi Lamba

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Till the nineteenth century

• Established theories in Physics
• Analytical Mechanics
• Thermodynamics
• Maxwellian Electrodynamics

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Why a new theory ?
By the end of the nineteenth century and the
early years of the twentieth century a number of
experiments had been carried out which could not
be explained by the classical theories. So A
new way of thinking was required !
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Experimental problems confronting classical
physics

• Discovery of Electrons
• Alpha Scattering
• Atomic Spectroscopy
• Blackbody Radiation
• Photoelectric Effect
• Compton Effect
• Electron Diffraction

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Cathode Rays
Cathode rays are deflected off their paths by
magnetic and electric fields.
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Discovery of electrons

• He proposed that
• Cathode rays are actually streams of tiny
• negatively charged particles (much smaller than
  atoms).
• Their charge to mass
• ratio 1.7 1011 C/kg.

1906
J. J. Thomson (1856 1940)
Electrons are a fundamental constituent of
matter.
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Alpha Scattering Experiment

• Established that
• Almost all the mass
• of an atom was concentrated in a
  positively charged
• nucleus.
• Most of the atom was empty space.

Ernest Rutherford
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Nuclear Model of the Atom
An atom is made up of
Nucleus

• A minute positively charged nucleus.
• An equal and opposite
• negative charge is distributed around the nucleus
  in the form of electrons.

Electrons
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Can the electrons be stationary?

• Can the atom contain
• stationary positive
• and negative charges ?

NO !
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Can the electrons be stationary?
is much greater than
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There is a problem..
The electron would fall into the nucleus.
An electric charge cannot be in equilibrium, at
rest under the action of electric forces alone.
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Next ?

• Consider that the electron revolves around the
  nucleus and the
• attractive electrostatic force provides the
  necessary centripetal force.

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There is a problem
An accelerating electron, however, would
radiate energy.
Eventually it would fall into the nucleus.
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Atomic Spectroscopy

• Balmer series of Hydrogen
• Line spectrum is obtained when light from a gas
  through which an electric discharge is passed is
  dispersed by a prism or a grating
• spectrometer.

There are four lines in the Balmer series of
Hydrogen.
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What is surprising is that

• Instead of a continuous band of colors only a few
  colors appear.
• The wavelengths of the lines are characteristic
  of the element that is emitting the light.
• Each element has its own particular line spectra.

Johannes Rydberg (1854-1919)
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Blackbody Radiation

• At very low and very high temperatures the
  emissive power is very small.
• At intermediate temperatures there is a maximum.
• The height of the maximum increases with
  temperature.
• The maximum shifts to smaller wavelengths.

Distribution of energy in the spectrum of a
blackbody radiation at different temperatures.
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Plancks theory
1918
Emission and absorption of radiation caused by
oscillators present in the walls of the black
body. The walls of the blackbody contain
oscillators of all frequencies.
Max Planck (1858 1947)
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Plancks theory
1918

• He sought
• To find the average energy of a harmonic
  oscillator at a given temperature.
• To modify the statistical distribution of energy
  between the oscillators.

Max Planck (1858 1947)
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Plancks Theory

• Oscillators can radiate energy only in discrete
  amounts like 0, ?0 , 2?0, 3?0.. n ?0.

?0 h n is a QUANTUM of energy. h is a
universal constant. (Plancks constant)
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Plancks Theory

• Drastic departure from classical ideas.
• Average energy of the oscillator

h6.62618 X 10-34 Js
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How much is a quantum of energy ?
What is the magnitude of energy E associated
with a quantum ? E hc/? (?c/ ? ) For a
quantum of visible light of wavelength 5000 Å
the energy is E 4 10-19 J
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The birth of QUANTUM PHYSICS
The year was 1900

• Energy of an oscillator can vary only in discrete
  jumps.
• Emission discontinuous transition
• between states n h n
• In general any physical system capable of
  emitting electromagnetic radiation has a discrete
  set of allowed energy values
• or energy levels.

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Photoelectric effect
Light with a frequency gt threshold frequency
Electrons emitted
Metal surface

• Electron current varies with the intensity of
  the light.
• The emission of electrons is immediate (3 10 -9
  secs).
• The maximum kinetic energy of the emitted
  electrons is a linear function of n and is
  independent of the intensity .

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More about the threshold frequency

• For any given metallic surface,
• if the frequency of the incident light
• is less than the threshold frequency,
• then, no matter
• how long the light is incident or
• how great its intensity,
• electrons are not emitted.

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Classical Roadblock

• In metals the outermost electrons in the atoms
  are not tightly bound to the nucleus and can be
  removed.
• Just sufficient energy is required.

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Light is, after all, an electromagnetic wave
Increase energy by increasing the amplitude
Why doesnt this work?
Increasing wavelength ?
Decreasing frequency ?
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Theory of Photoelectric effect.

• Any given source could
• absorb or emit
• radiant energy only in units
• or quanta all
• exactly equal to h ?.
• Light itself consisted of
• quanta of energy h ?
• which move through space
• with the velocity of light.
• This quantum of electromagnetic radiation is
  called a photon.

1921
Albert Einstein 1879 - 1955
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So Photoelectric Effect is
A collision between a photon and a bound
electron in which a photon is completely
absorbed and the energy of the electron increases
by h ?. The photoelectric equation
Workfunction of the metal W h?0 So
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Photoelectric Effect

• Electrons can be emitted from the metal only
  when ? gt ?0.
• The energy of the electrons varies linearly
• with the frequency (? - ?0).
• The energy of the electrons is independent of
  the intensity of the radiation.
• The number of electrons ejected is proportional
  to the intensity of the radiation.

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Structure of the atom
Rutherfords model of the atom was intrinsically
unstable. Bohr applied the quantum ideas of
Planck and Einstein to Rutherfords nuclear atom.
His model for the atom is a hybrid of
classical and quantum ideas
1922
Niels Bohr (1885-1962)
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Postulates of Bohrs Atomic Model

• Orbit of the electron around the nucleus.
• The electrostatic attraction between the nucleus
  and the electron, similar to the gravitational
  attraction in its spatial properties could lead
  to stable circular or elliptical orbits for the
  electron.
• Electron in an atom moves in a circular orbit
  about the nucleus with the centripetal force
  being supplied by the Coulomb attraction between
  the nucleus and the electron

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Postulates of Bohrs Atomic Model

• Allowed Orbits.
• The allowed orbits are separated from the
  forbidden ones by a quantum condition, which is
  imposed on the angular momentum and not on the
  energy.
• Only those orbits are allowed for which the
  angular momentum of the electron L is an
  integral multiple of ? (h/2p).

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Postulates of Bohrs Atomic Model

• Stationary States.
• The problem of the stability for the circular
  orbit of the electron was solved by postulating
  that in an allowed orbit, the electron must have
  a constant energy.
• An electron in an allowed orbit does not emit
  any radiation. These constant energy states are
  called stationary states.

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Postulates of Bohrs Atomic Model

• IV. Emission/Absorption of Energy.
• The mechanism of emission /absorption of energy
  from/by an electron in an atom was by transitions
  between these constant energy states.
• Energy is emitted (or absorbed) from an atom
  only when the electron jumps from one allowed
  orbit to another.
• Einsteins frequency relation
• h? Ei - Ef

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A logical progression

• The frequency of the emitted radiation is
• The energies vary as 1/n2
• Radii of the orbits vary as n2
• Angular momentum varies as nh

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Radius Energy for Bohrs Orbits

• With these postulates we can write

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Radius Energy for Bohrs Orbits
Radius of the nth orbit
Energy of the nth stationary state
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Explanation of Atomic Spectra

• The frequency of the emitted radiation when the
  electron jumps from a state n to am state m can
  be found from the Einstein frequency relation by
  substituting for the energies in the two states.
  So,

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Successes of the Bohr Model

• Bohrs theory could explain
• The spectra of one electron atoms.
• (hydrogen and single ionized Helium)
• And
• Gave a physical interpretation for the spectral
  lines in terms of the stationary states of the
  atom.

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Extension of Bohrs Theory

• The spectra of the neutral hydrogen atom and
  the singly ionized helium atom also have
  fine lines.
• This could not be explained within Bohrs
  theory which has only a single quantum number
  n.

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Extension of Bohrs Theory

• It was explained by Sommerfield by
• 1. Postulating elliptic as well as
  circular orbits thus adding a new quantum
  number.
• 2. Accounting for the relativistic
  variation of electronic mass .

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Serious Discrepancies

• Were found between theory and experiment when
• Bohrs theory was applied to
• two-electron atoms and
• In trying to account for the splitting of
  spectral lines in a magnetic field. Another
  quantum number was required.
• The atomic model itself was held to be at fault
  and QUANTUM MECHANICS developed.

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Let us examine the following

• Our understanding of the physical world
• is that it is made up of two basic distinct
  entities.
• Waves
• Like sound waves,ripples on the surface of
  water, electromagnetic waves.
• Material objects
• Like a particle,a ball, a car, the planets.

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Are they very different ?

• Material Objects
• Can be located at a definite position at a given
  time.
• Can be at rest or moving or accelerating under
  an external force.
• When they collide, they either scatter or
  shatter. They definitely cannot pass through each
  other.

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It would seem so.

• Waves
• They are spread out in space and time.
• They are defined by their velocity, wavelength,
  frequency or amplitude.
• They can pass through one another. In the process
  the waves are either enhanced or reduced.

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Wave Nature of Material Objects

• Light, believed to be an electromagnetic wave
  shows both
• Wave like behavior interference
  diffraction and
• Particle like behavior photoelectric effect.
• Should not material particles then show wave
  like behaviour ?

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De Broglie Hypothesis
1929

• Particles of matter should exhibit both particle
  and wave nature.
• A material particle of energy E and a momentum p
  may exhibit the characteristics of a wave of
  wavelength
• ? h/p

Louis de Broglie (1892-1987)
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Louis de Broglie, Nobel Prize Speech
Determination of the stable motion of electrons
in the atom introduces integers, and up to this
point the only phenomena involving integers in
physics were those of interference and of normal
modes of vibration. This fact suggested to me the
idea that electrons too could not be considered
simply as particles, but that frequency (wave
properties) must be assigned to them also.
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Unification of Two Concepts

• This required the unification of two concepts
• Wavelength which has a clear cut meaning only
  for waves.
• Momentum which has a natural interpretation
  only for a moving particle.

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De Broglie Hypothesis
To complete the analogy we write
Particle Photon Momentum p h/? Energy
E h ? hc/?pc
Light
Wave matter wave Wavelength ?h/p Momentum
p h/?
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Is Light a Particle or a Wave ?
On a macroscopic scale (a large number of
photons) light can still be thought of as a
wave. In the interaction of light with matter on
the subatomic scale we must look at the
particle description of light.
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Some Typical De Broglie wavelengths
A ball weighing 100 g, and moving with a speed
of 25 m/s 2.65 x 10-34 meters.
An electron accelerated through a potential
difference of 50 V 1.73 x 10-10 meters.
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Compton Effect
Monochromatic X-rays were scattered by a
graphite block and the wavelength of the
scattered radiation was measured.
1929
Arthur Compton
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Compton Effect

• incident X-rays scattered X-rays
• electron
• At each scattering angle peaks
• were observed at two wavelengths
• One at the incident wavelength
• One at a longer wavelength
• ( ? dependent)

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Compton Effect
Elastic Collision between a Photon and an
Electron
Applying the principles of conservation of
energy and momentum he could derive the
expression for the wavelength shift.
Waves could behave like particles!
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Diffraction of Electrons from a Crystal
Accelerated electrons impinging on a Ni crystal
create a diffraction pattern.
1937
Davisson and Germer
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Diffraction of Electrons from a Crystal
Electrons are associated with a wave of
wavelength ? 1 A . If the planes of the Ni
crystal are considered to be a diffraction
grating then we can look upon the process as the
Diffraction of Electron Waves.
Particles could behave like Waves!
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Low Energy Electron Diffraction
LEED has developed as the principle technique
for examining surface structures.
Uses a beam of electrons (typically in the range
20 - 200 eV) incident normally on a crystal
sample. The diffraction pattern provides accurate
information about the atomic positions and the
unit cell.
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Low Energy Electron Diffraction
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Revisiting Bohrs Atomic Model
If electrons behave as waves the concept of
Bohrs orbit must change. Such a wave could
exist is if a whole number of its wavelengths fit
exactly around the circle.           
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Revisiting Bohrs Atomic Model
So orbits can have only certain sizes,
depending on the wavelength of the electron
--which is decided by their momentum.
Circumference of the circle (2 ? r) is an
integral multiple of the wavelength (? h/p )
of the electron. So 2 ? r n h/p n h / m v
m v r n h / 2 ? which is the condition
for quantization of angular momentum.
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Summary

• The idea of a quantum of energy for a harmonic
  oscillator is needed to explain blackbody
  radiation
• The concept of a quantum of light- the photon was
  able to explain photoelectric effect.
• Bohrs theory of atomic structure gave a
  physical interpretation for atomic spectra.
• Matter was proposed to have wave-like properties.