The wanning of the classical world view

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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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• 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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Intensity versus wavelength for different
temperatures
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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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Ultra violet Catastrophe
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• The failure of the Rayleigh-Jeans formula at the
  high energy end of the spectrum is spectacular
• The energy density of radiation should increase
  with frequency all the way to infinity.
• A heated object should give off more ultraviolet
  light than visible and more X-rays than either.
• Toasting marshmallows should kill you!

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Intensity versus wavelength for different
temperatures
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• He further showed that

frequency
constant
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Discovery of the electron
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y
E
x
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• In the Thomson experiment the electric field was
  generated by defecting plates of length l hence
  the deflection at the end of the plates is
• yq0El2/2mv20
• If the electric force is equal and opposite to
  the magnetic force then
• v0E/B

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• Established the existence of a new particle
• The electron

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Millikans oil drop experiment

• Basically Robert Millikans design was just a
  uniform electric field, which is a pair of
  parallel plates that lie horizontal with large
  potential difference. Then the oil drops are
  dropped in to the plates and the drops are
  suspended between the plates. By changing the
  voltage you can make the oil drops rise and fall.
  A ring of insulating material is used to hold the
  plates together. The plates have four holes cut
  into it and three have a bright light shining
  through them, and the other has a microscope
  placed through it.
• The oil is a type that is usually used in vacuum
  apparatus. This is because this type of oil has
  an extremely low vapor pressure. Ordinary oil
  would evaporate away under the heat of the light
  source, so the mass of the oil drop would not
  remain constant over the course of the
  experiment.Using Stokes Law he was able to
  establish that the drops always had A charge Ne
  where
• e1.599 X10-19C/-0.01 X 10-19C
• present day value1.602 X10-19C

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• Rutherford discovered the nucleus
• hyperphysics.phy-astr.gsu.edu/hbase/rutsca.html

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• Natural to think of atoms as little planetary
  systems

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• White light

Continuous Spectrum
White light
Emission Spectrum
Hot GAS
Absorption Spectrum
Cold Gas
White light
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Bohrs argument
Assume nucleus infinitely heavy Assume electron
is moving in circular orbit of radius,r,with
speed v then
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Bohr assumed that the orbital angular momentum
was quantized i.e. Consequently we obtain
fixed values of
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For Hydrogen Z1
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De Broglie Hypothesis
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Recall

• Standing waves are non-traveling vibrations of
  certain wavelength and frequency which occur on a
  medium of certain size. The size of the medium
  controls the wavelengths of the standing waves.
  Also, the way that the medium is held at its
  ends, either fixed or open, controls the
  wavelengths of the standing waves.

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Double Slit Waves
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Two Slits Particles(bullets)
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Two Slits Electrons
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Some more mathematics

• Separation of variables

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Laplacian
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• Frequently in Physics we need to solve partial
  differential equations,
• e.g.

The Helmholtz Equation
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• Physical problems consist of a formal
  differential equation together with boundary
  conditions,

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• From the general theory of partial differential
  equations one can establish rules for when such
  an equation has a non-trivial solution(F 0)
• and if such a solution is unique.

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• You may assume unless otherwise stated that any
  p.d.e. you have to deal with has a solution and
  it is unique
• Thus it doesnt matter how you find your solution
  it just has to satisfy the diff eqn and the
  boundary conditions.

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Depends only on x
Depends only on y and z, independent of x
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Depends only on x
Depends only on y and z, independent of x
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Function only of z
Function only of y
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• We have turned our partial differential equation
  into sets of ordinary differential equations

Chosen to enforce boundary conditions
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Different coordinates

• Cartesian coordinates are good for boxes
• But we may be interested in problems with
  spherical symmetry or cyclindrical symmetry or
  other more complex symmetries
• The Laplacian can be transformed

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Cylindrical symmetry
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Spherical Symmetry
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Schrödinger Equation(Chapter 2 Landshoff)

• Schrödinger began from the assumption of matter
  waves, and by analogy with geometrical optics he
  introduced a variational principle which lead him
  to postulate that the matter waves satisfied the
  equation

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Time dependent wave function We will assume that
potential V is independent of time and purely
radial
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Look for a stationary solution, i.e look for a
solution by separation of variables
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Easy to see solution is of the form
E is a constant that comes from the separation of
variables which we identify with the energy And
satisfies the eigenvalue equation
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Example
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Particle in a box
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