The fall of Classical Physics

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The fall of Classical Physics
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Classical physics Fundamental Models

• Particle Model (particles, bodies)
• Motion in 3 dimension for each time t, position
  and speed are known (they are well-defined
  numbers, regardless we know them). Mass is known.
• Systems and rigid objects
• Extension of particle model
• Wave Model (light, sound, )
• Generalization of the particle model energy is
  transported, which can be spread (de-localized)
• Interference

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Classical physics at the end of XIX Century

• Scientists are convinced that the particle and
  wave model can describe the evolution of the
  Universe, when folded with
• Newtons laws (dynamics)
• Description of forces
• Maxwells equations
• Law of gravity.
• We live in a 3-d world, and motion happens in an
  absolute time. Time and space (distances)
  intervals are absolute.
• The Universe is homogeneous and isotropical time
  is homogeneous.
• Relativity
• The physics entities can be described either in
  the particle or in the wave model.
• Natura non facit saltus (the variables involved
  in the description are continuous).

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Something is wrongRelativity, continuity,
wave/particle (I)

• Maxwell equations are not relativistically
  covariant!
• Moreover, a series of experiments seems to
  indicate that the speed of light is constant
  (Michelson-Morley, )

A speed!
5
Something is wrong Relativity, continuity,
wave/particle (IIa)

• In the beginning of the XX century, it was known
  that atoms were made of a heavy nucleus, with
  positive charge, and by light negative electrons
• Electrostatics like gravity planetary model
• All orbits allowed
• But electrons, being accelerated, should radiate
  and eventually fall into the nucleus

6
Something is wrong Relativity, continuity,
wave/particle (IIb)

• If atoms emit energy in the form of photons due
  to level transitions, and if color is a measure
  of energy, they should emit at all wavelengths
  but they dont

7
Something is wrong Relativity, continuity,
wave/particle (III)

• Radiation has a particle-like behaviour,
  sometimes
• Particles display a wave-like behaviour,
  sometimes
• gt In summary, something wrong involving the
  foundations
• Relativity
• Continuity
• Wave/Particle duality

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Need for a new physics

• A reformulation of physics was needed
• This is fascinating!!! Involved philosophy,
  logics, contacts with civilizations far away from
  us
• A charming story in the evolution of mankind
• But just a moment I leaved up to now with
  classical physics, and nothing bad happened to
  me!
• Because classical physics fails at very small
  scales, comparable with the atoms dimensions,
  10-10 m, or at speeds comparable with the speed
  of light, c 3 108 m/s
• Under usual conditions, classical physics makes
  a good job.
• Warning What follows is logically correct,
  although sometimes historically inappropriate.

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ILight behaves like a particle, sometimes
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1) Photoelectric Effect

• The photoelectric effect occurs when light
  incident on certain metallic surfaces causes
  electrons to be emitted from those surfaces
• The emitted electrons are called photoelectrons
• When the system is kept in the dark, the ammeter
  reads zero
• When plate E is illuminated, a current is
  detected by the ammeter

• The current arises from photoelectrons emitted
  from the negative plate (E) and collected at the
  positive plate (C)

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Photoelectric Effect, Interpretation

• Electrons are trapped in the metal, by a
  potential V gt Ve
• Light might give to the electrons enough energy
  Eg to escape
• Electrons ejected possess a kinetic energy
• K Eg - eV
• Kmax Eg f
• f eVe is called the work function
• The work function represents the minimum energy
  with which an electron is bound in the metal
• Typically, f 4 eV

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• At large values of DV, the current reaches a
  maximum value
• All the electrons emitted at E are collected at C
• The maximum current increases as the intensity of
  the incident light increases

• When DV is negative, the current drops
• When DV is equal to or more negative than DVs,
  the current is zero

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Photoelectric Effect Feature 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 current goes to zero at the same negative
  voltage for all intensity curves

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Photoelectric Effect Feature 2

• Time interval between incidence of light and
  ejection of photoelectrons
• Classical Prediction
• For very weak light, 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
• Less than 10-9 s

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Photoelectric Effect Feature 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

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Photoelectric Effect Feature 4

• Dependence of photoelectron kinetic energy on
  light frequency
• Classical Prediction
• There should be no relationship between the
  frequency of the light and the electron maximum
  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

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Cutoff Frequency

• The lines show the linear relationship between K
  and ƒ
• The slope of each line is independent of the
  metal
• h 6.6 10-34 Js
• The absolute value of the y-intercept is the work
  function
• The x-intercept is the cutoff frequency
• This is the frequency below which no
  photoelectrons are emitted

Kmax hƒ f
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Photoelectric Effect Featuresand Photon Model
explanation

• The experimental results contradict all four
  classical predictions
• Einstein interpretation All electromagnetic
  radiation can be considered a stream of quanta,
  called photons
• A photon of incident light gives all its energy
  hƒ to a single electron in the metal

• h is called the Planck constant, and plays a
  fundamental role in Quantum Physics

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Photon Model Explanation

• Dependence of photoelectron kinetic energy on
  light intensity
• Kmax is independent of light intensity
• K depends on the light frequency and the work
  function
• The intensity will change the number of
  photoelectrons being emitted, but not the energy
  of an individual electron
• Time interval between incidence of light and
  ejection of the photoelectron
• Each photon can have enough energy to eject an
  electron immediately
• Dependence of ejection of electrons on light
  frequency
• There is a failure to observe photoelectric
  effect below a certain cutoff frequency, which
  indicates the photon must have more energy than
  the work function in order to eject an electron
• Without enough energy, an electron cannot be
  ejected, regardless of the light intensity

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Photon Model Explanation of the Photoelectric
Effect, final

• Dependence of photoelectron kinetic energy on
  light frequency
• Since Kmax hƒ f, as the frequency increases,
  the maximum kinetic energy will increase
• Once the energy of the work function is exceeded
• There is a linear relationship between the
  kinetic energy and the frequency

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Cutoff Frequency and Wavelength

• The cutoff frequency is related to the work
  function through ƒc f / h
• The cutoff frequency corresponds to a cutoff
  wavelength
• Wavelengths greater than lc incident on a
  material having a work function f do not result
  in the emission of photoelectrons

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2) The Compton Effect

• Compton dealt with Einsteins idea of photon
  momentum
• Einstein a photon with energy E carries a
  momentum of E/c hƒ / c
• According to the classical theory,
  electromagnetic waves of frequency ƒo incident on
  electrons should scatter, keeping the same
  frequency they scatter the electron as well

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• Comptons experiment showed that, at any given
  angle, only one frequency of radiation is
  observed
• The graphs show the scattered x-ray for various
  angles
• Again, treating the photon as a particle of
  energy hf explains the phenomenon. The shifted
  peak, lgt l0, is caused by the scattering of free
  electrons
• This is called the Compton shift equation (wait
  the relativity week)

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Compton Effect, Explanation

• The results could be explained, again, by
  treating the photons as point-like particles
  having
• energy hƒ
• momentum hƒ / c
• Assume the energy and momentum of the isolated
  system of the colliding photon-electron are
  conserved
• Adopted a particle model for a well-known wave
• The unshifted wavelength, lo, is caused by x-rays
  scattered from the electrons that are tightly
  bound to the target atoms
• The shifted peak, l', is caused by x-rays
  scattered from free electrons in the target

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3) Blackbody radiation

• Every object at T gt 0 radiates electromagnetically
  , and absorbes radiation as well
• Stefan-Boltzmann law
• Blackbody the
• perfect absorber/emitter

Black body

• Classical interpretation atoms in the object
  vibrate since ltEgt kT, the hotter the object,
  the more energetic the vibration, the higher the
  frequency
• The nature of the radiation leaving the cavity
  through the hole depends only on the temperature
  of the cavity walls

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Experimental findings classical calculation

• Wiens law the emission peaks at
• Example for Sun T 6000K
• But the classical calculation (Rayleigh-Jeans)
  gives a completely different result
• Ultraviolet catastrophe

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Experimental findings classical calculation

• Classical calculation (Raileigh-Jeans) the
  blackbody is a set of oscillators which can
  absorb any frequency, and in level transition
  emit/absorb quanta of energy
• No maximum a ultraviolet catastrophe should
  absorb all energy

Experiment
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Plancks hypothesis

• Only the oscillation modes for which
• E hf
• are allowed

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Interpretation

• Elementary oscillators can have only quantized
  energies, which satisfy Enhf (h is an universal
  constant, n is an integer quantum- number)
• Transitions are accompanied by the emission of
  quanta of energy (photons)

• The classical calculation is accurate for large
  wavelengths, and is the limit for h -gt 0

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Which lamp emits e.m. radiation ?

• 1) A
• 2) B
• 3) A B
• 4) None

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4) Particle-like behavior of lightnow smoking
guns

• The reaction

has been recorded millions of times
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Bremsstrahlung

• "Bremsstrahlung" means in German "braking
  radiation it is the radiation emitted when
  electrons are decelerated or "braked" when they
  are fired at a metal target. Accelerated charges
  give off electromagnetic radiation, and when the
  energy of the bombarding electrons is high
  enough, that radiation is in the x-ray region of
  the electromagnetic spectrum. It is characterized
  by a continuous distribution of radiation which
  becomes more intense and shifts toward higher
  frequencies when the energy of the bombarding
  electrons is increased.

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Summary

• The wave model cannot explain the behavior of
  light in certain conditions
• Photoelectric effect
• Compton effect
• Blackbody radiation
• Gamma conversion/Bremsstrahlung
• Light behaves like a particle, and has to be
  considered in some conditions as made by single
  particles (photons) each with energy
• h 6.6 10-34 Js is called the Plancks constant

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IIParticles behave like waves, sometimes
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Summary of last lecture

• The wave model cannot explain the behavior of
  light in certain conditions
• Photoelectric effect
• Compton effect
• Blackbody radiation
• Gamma conversion
• Light behaves like a particle, and has to be
  considered in some conditions as made by single
  particles (photons) each with energy
• h 6.6 10-34 Js is called the Plancks constant

36
Should, symmetrically, particles display
radiation-like properties?

• The key is a diffraction experiment do particles
  show interference?
• A small cloud of Ne atoms was cooled down to T0.
  It was then released and fell with zero initial
  velocity onto a plate pierced with two parallel
  slits of width 2 mm, separated by a distance of
  d6 mm. The plate was located H3.5 cm below the
  center of the laser trap. The atoms were detected
  when they reached a screen located D85 cm below
  the plane of the two slits. This screen
  registered the impacts of the atoms each dot
  represents a single impact. The distance between
  two maxima, y, is 1mm.
• The diffraction pattern is consistent with the
  diffraction of waves with

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Diffraction of electrons

• Davisson Germer 1925
• Electrons display diffraction patterns !!!

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de Broglies wavelength

• What is the wavelength associated to a particle?
• de Broglies wavelength
• Explains quantitatively the diffraction by
  Davisson and Germer
• Note the symmetry
• What is the wavelength of an electron moving at
  107 m/s ?
• (smaller than an atomic length note the
  dependence on m)

39
Atomic spectra

• Why atoms emit according to a discrete energy
  spectrum?

Balmer

• Something must be there...

40
Electrons in atoms a semiclassical model

• Similar to waves on a cord, lets imagine that
  the only possible stable waves are stationary
• 2 ??r n ??
  n1,2,3,

gt Angular momentum is quantized (Bohr
postulated it)
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Hydrogen (Z1)

• NB
• In SI, ke (1/4pe0) 9 x 109 SI units
• Total energy lt 0 (bound state)
• ltEkgt -ltEp/2gt (true in general for bound states,
  virial theorem)

Only special values are possible for the radius !
42
Energy levels

• The radius can only assume values
• The smallest radius (Bohrs radius) is
• Radius and energy are related

• And thus energy is quantized

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Transitions

• An electron, passing from an orbit of energy Ei
  to an orbit with Ef lt Ei, emits energy a photon
  such that f (Ei-Ef)/h

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Level transitions and energy quanta

• We obtain Balmers relation!

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Limitations

• Semiclassical models wave-particle duality can
  explain phenomena, but the thing is still
  insatisfactory,
• When do particles behave as particles, when do
  they behave as waves?
• Why is the atom stable, contrary to Maxwells
  equations?
• We need to rewrite the fundamental models,
  rebuilding the foundations of physics

46
Wavefunction

• Change the basic model!
• We can describe the position of a particle
  through a wavefunction y(r,t). This can account
  for the concepts of wave and particle (extension
  and simplification).
• Can we simply use the DAlembert waves, real
  waves? No

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Wavefunction - II

• We want a new kind of waves which can account
  for particles, old waves, and obey to Fma.
• And they should reproduce the characteristics of
  real particles a particle can display
  interference corresponding to a size of 10-7 m,
  but have a radius smaller than 10-10 m
• Waves of what, then? No more of energy,
• but of probability
• The square of the wavefunction is the intensity,
  and it gives the probability to find the particle
  in a given time in a given place.
• Waves such that Fma? Well see that they cannot
  be a function in R, but that C is the minimum
  space needed for the model.

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SUMMARY

• Close to the beginning of the XX century, people
  thought that physics was understood. Two models
  (waves, particles). But
• Quantization at atomic level became
  experimentally evident
• Particle-like behavior of radiation radiation
  can be considered in some conditions as a set of
  particles (photons) each with energy
• Wave-like property of particles particles behave
  in certain condistions as waves with wavenumber
• Role of Plancks constant, h 6.6 10-34 Js
• Concepts of wave and particle need to be unified
  wavefunction y (r,t).

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Lequazione di Schroedinger
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Proprieta della funzione donda
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Lequazione di S.
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Laboratorio virtualeOrigini della Meccanica
Quantistica

• Radiazione termica del corpo nero
• Diffrazione degli elettroni