WAVES AND QUANTUM PHYSICS:

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• WAVES AND QUANTUM PHYSICS
• Why Waves? Explain by a one-page physics summary
  from 1600-1900
• Galileo notes physics in a steadily moving boat,
  a spacetime transform to a moving reference
  frame, V x x Vt
• Newton gives law of motion of mass m under force
  F F m d²x/dt²
• Time has entered, quantitatively, but not in the
  force laws Fgrav -GmM/r², Felectric
  qe/(4??r²) of Newton and Coulomb.
• Time really enters when Faraday discovers
  electricity induced by time-varying magnetism and
  Maxwell codifies it, generated two crises
  relating to electromag Waves.

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• Canal boat

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• Good old Newton spotted both problems already, in
  his Opticks,
• his force law implies action at a distance
  which he thought strange
• he thought (al)chemistry required something
  different, and worked on it.
• Maxwells equations have built-in waves carrying
  fields to a distance
• in vacuum, i.e. no free charges.
• Proof Consider loops on two surfaces related to
  two planes separated in space and time, E(y,z)
• ?x

y
y
?x
Ey(x2)
Bz(x2)
Ey(x1)
Bz(x1)
x
x
z
z
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• We sum the integrand around the loop in the
  xy-plane, no contribution from the top and
  bottom, so
• The magnetic flux through the box is
• and the first Maxwell (Faraday) law
• then gives
• which shows that we can have time-varying fields
  in the vacuum.
• In the xz-plane, same procedure gives
• We can eliminate either
• by differentiating with respect to x or t and we
  will do the former

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• or
• which we call a wave equation. Why?
• Suppose solution is f(x, t). The left side has
  got to produce a c on each differentiation, right
  side no factor
• can do if f(x, t) f(x - ct) or f(x ct)
• and this gives a shape that just slides along x
  as a function of time, what we call a wave.
• As we see, f(x) can be any shape. We see that c
  is the speed of the wave.Maxwell said, c1/?(??)
  vlight 3.1010m/s Hey,
  this is light!!

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• Tremendous success radio waves, precision c
  precision v light, polarization, etc etc
  etc.Problem
• We chose x,y,z in our derivation quite
  arbitrarily.
• Suppose we choose x, moving at speed V with
  respect to x, then Galileo and common
  understanding says that the velocity of the light
  way changes by V, plus or minus depending on
  direction (like headwind/tailwind). But
  Maxwells equations say vlight c
• unless they only work in one frame, or something
  else crazy. Try measuring light in different
  frames, e.g. using motion of earth. Guess what!
  v c!!! Always. Big Puzzlefact versus
  definition.

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• Start with vc fact, follow logic
• Train moving with speed V struck by lightning at
  both ends at time t.
• Flashes from strike go at vc and meet at red
  dot, splashing to mark this event on rails, and
  in car. See picture in car, drawn v c/2

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• In the moving car, we see that the path from the
  source of the left flash to the meeting point is
  three times longer than the right hand flash to
  meeting distance. No problem if this were a
  moving ball, we would conclude that they had
  different speeds, BUT we know the light flashes
  move always with v c.
• The only possible conclusion is that they started
  at different times.
• Einstein concluded that the time variable can
  vary in different observer frames. Simultaneity
  is relative. In primed system, t ? t in
  general. The assumption that while x depends on
  the system, t does not, is found to be an
  assumption, and a wrong assumption.

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• Pythagoras taught how to survey using x, y, to
  find a space interval
• s2 x2 y2 Einstein-Minkowski taught how
  to survey in space-time
• s2 x2 y2 -c2t2 an invariant
• s2 gt 0 is space-like
• s2 lt 0 is time-like
• s2 0 is on the light-cone
• The intervals s are independent of the system of
  the observer, (inertial, non-accelerating
  systems, as in Galileo) and only these are
  physically meaningful, just as in surveying only
  the distance intervals are meaningful, not x,y
  etc, since these depend on arbitrary coordinate
  systems. Same for other observables, like
  m2c4E2 - p2c2

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• Sometimes, though, it is convenient to transform
  from one coordinate system to another, even if
  not invariant, so we need transforms.
• Consider a light pulse starting at xx0tt
  where x is moving with velocity V, then xct,
  xct, and the Galilean transforms must be
  modified, by a factor we name ?
• substitute x and x
• and by eliminating t from these eqs

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• And by eliminating x and x, we get t and t and
  writing them all out
• the Lorentz transformation.
• We see the non-invariance of simultaneity
  explicitly, and that time intervals will differ
  by a factor ?? also that space intervals are
  affected
• time dilation for the moving clock
• length contraction for moving rule
• Of course these transforms give the right
  answers, but they are prone to the generation of
  so-called paradoxes if the reasoning is not
  closely followed, so use invariant quantities if
  you can.

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• The twin paradox

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• A paradox usually called The Javelin Thrower in
  the Barn
• a one meter object enters a half meter barn at a
  velocity of 1/?2 cwhen the nose hits the far
  door, the back door closes we have trapped a one
  meter object in a half meter barn????????
• The turtles view of what happens

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• A paradox usually called The Javelin Thrower in
  the Barn
• a one meter object enters a half meter barn at a
  velocity of 1/?2 cwhen the nose hits the far
  door, the back door closes we have trapped a one
  meter object in a half meter barn????????