Minkowski Space

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Minkowski Space

• Consider a 4 dimensional vector space

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Example
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Example
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Time Dilation Again
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Four velocity
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The relativistic Addition of velocities
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Relativistic addition of velocities
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The momentum Energy 4 vector

• As we have seen the classical momentum is not
  relativistically invariant

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Doppler again
y
x
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Relativistic Center of Mass System
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• The energy available for inelastic processes is

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End of Special Relativistic Section
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The wanning of the classical world view
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Classical Physics

• The physical universe is deterministic, given
  enough information one can predict exactly the
  evolution of the system
• Light consists of electromagnetic waves while
  ordinary matter consists of discrete particles
• Physical quantities like postion momentum,
  angular momentum and energy are continuous
  variables
• Newtonian Mechanics and Electromagnetism depend
  typically on second order differential equations

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Thermal Radiation

• We see objects by scattering electromagnetic
  radiation from them
• When we heat an object it can also emit radiation

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Observations

• As the Temperature of a body is increased the
  intensity of the thermal radiation rises
• The higher the temperature the shorter the wave
  length of the most intense emitted radiation
• A body becomes red hot and then white hot

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• Stefan showed that the total power emitted per
  unit area,R, called the total emissive power or
  total emittance is given by the empirical
  formula

Constant independent of surface
Emissivity,characterisic of surface, 1?e
Temperature on absolute scale
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• If a body is in thermal equilibrium with its
  surroundings, it must absorb and admit the same
  amount of radiant energy(otherwise temperature
  would rise)
• A blackbody is a perfect absorber so if it is
  emitting thermal radiation we must have
• e1

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• Early attempts to study these observations
  quantatively ran into difficulties because it was
  found that the thermal radiation emitted from a
  given body at a given temperature depended on
  the material from it was made, the roughness of
  the surface etc.

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Cavity Radiator

• To avoid these problems the idea of a cavity
  radiator was introduced.
• Idea form a cavity in a material with its walls
  held at a constant temperature
• A small hole is created which allows radiation to
  escape
• The radiation emerging from this hole does not
  depend on the nature of the cavity or the
  material just on the temperature

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• "Blackbody radiation" refers to an object or
  system which absorbs all radiation incident upon
  it and re-radiates energy which is characteristic
  of this radiating system only, not dependent upon
  the type of radiation which is incident upon it.
  The radiated energy can be considered to be
  produced by standing wave or resonant modes of
  the cavity which is radiating.

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Cavity blackbody radiation

• The radiation emitted from a cavity through a
  small hole ("cavity radiation") is very close to
  the theoretical blackbody curve for the same
  temperature. In the cavity, the radiation is in
  equilibrium with the material - most of the
  radiation stays inside the cavity, being
  continually emitted and re-absorbed by the walls.
  Radiation emitted from the outer surface of a
  material will not necessarily be fully
  thermalized - some frequencies corresponding to
  certain transitions of the material, will be
  emitted preferentially. So, the blackbody curve
  is not material-specific, but the actual emission
  from an object will be. Cavity radiation will
  depend less on the material, and the smaller the
  hole, the closer it will correspond to the
  theoretical blackbody curve.

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Intensity versus wavelength for different
temperatures
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Rayleigh-Jeans distribution

• The radiation detected outside the hole will be a
  sample of the radiation inside the box, so some
  analysis is required to understand whats
  happening inside the box.
• The box is filled with electromagnetic standing
  waves. If the walls are metal, the radiation
  bounces around inside the box with the electric
  field stopping at each wall, creating a node at
  each wall.
• The number of standing waves with wavelengths
  between ?? and ?d? is N(??) d? (8? V / ?4) d?
  where V is the volume of the box.
• This can be proven by regular analysis of
  standing waves and expanding it to three
  dimensions.
• Each individual wave contributes an energy kT to
  the radiation in the box. From classical
  thermodynamics, we know that the radiation in the
  box is in thermal equilibrium with the walls at
  temperature T. Radiation is absorbed and quickly
  reemitted by the walls, which creates
  oscillations in the frequency of the radiation.
  The mean thermal kinetic energy of an oscillating
  atom is 0.5kT. Since these are simple harmonic
  oscillators, the mean kinetic energy is equal to
  the mean potential energy, so the total energy is
  kT.
• The radiance is related to the energy density
  (energy per unit volume) u(?) in the relationship

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Some ideas from Thermodynamics

• Consider a collection of electromagnetic waves
  inside a blackbody cavity of temperature T.
• The energy density is just the average energy of
  the waves multiplied by their number density

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• k is the Boltzmann constant

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From all this we get
known as the Rayleigh-Jeans formula)
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Ultra violet Catastrophe
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Ultra violet Catastrophe

• Plank rederived the formula and avoided the
  catastrophe by assuming that the oscillators
  could only take energies which were integer
  multiples of some energy

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• He further showed that

frequency
constant