Review of Basic Physics Background

1
Review of Basic Physics Background
2
Basic physical quantities units

• Unit prefixes
• Basic quantities
• Units of measurement
• Planck units
• Physical constants

3
Unit Prefixes

• See http//www.bipm.fr/enus/3_SI/si-prefixes.html
  for the official international standard unit
  prefixes.
• When measuring physical things, these prefixes
  always stand for powers of 103 (1,000).
• But, when measuring digital things (bits bytes)
  they often stand for powers of 210 (1,024).
• See also alternate kibi, mebi, etc. system at
  http//physics.nist.gov/cuu/Units/binary.html
• Dont get confused!

4
Three fundamental quantities
5
Some derived quantities
6
Electrical Quantities

• Well skip magnetism related quantities this
  semester.

7
Information, Entropy, Temperature

• These are important physical quantities also
• But, are different from other physical quantities
• Based on statistical correlations
• But, well wait to explain them till we have a
  whole lecture on this topic later.
• Interestingly, there have been attempts to
  describe all physical quantities entities in
  terms of information (e.g., Frieden, Fredkin).

8
Unit definitions conversions

• See http//www.cise.ufl.edu/mpf/physlim/units.txt
  for definitions of the above-mentioned units,
  and more. (Source Emacs calc software.)
• Many mathematics applications have built-in
  support for physical units, unit prefixes, unit
  conversions, and physical constants.
• Emacs calc package (by Dave Gillespie)
• Mathematica
• Matlab - ?
• Maple - ?

9
Fundamental physical constants

• Speed of light c 299,792,458 m/s
• Plancks constant h 6.626075510?34 J s
• Reduced Plancks constant ? h / 2?
• h circle ? radian
• Newtons gravitational constant G 6.6725910?11
  N m2 / kg
• Others permittivity of free space, Boltzmanns
  constant, Stefan-Boltzmann constant to be
  introduced later as we go along.

10
Physics you should already know

• Basic Newtonian mechanics
• Newtons laws, motion, energy, etc.
• Basic electrostatics
• Ohms law, Kirchoffs laws, etc.
• Also helpful, but not prerequisite (well
  introduce them as we go along)
• Basic statistical mechanics thermodynamics
• Basic quantum mechanics
• Basic relativity theory

11
Generalized Classical Mechanics
12
Generalized Mechanics

• Classical mechanics can be expressed most
  generally and concisely in the Lagrangian and
  Hamiltonian formulations.
• Based on simple functions of the system state
• Lagrangian Kinetic minus potential energy.
• Hamiltonian Kinetic plus potential energy.
• The dynamical laws can be derived from either
  energy function.
• This framework generalizes to quantum mechanics,
  quantum field theories, etc.

13
Euler-Lagrange Equation
Note the over-dot!
or just

• Where
• L(qi, vi) is the systems Lagrangian function.
• qi Generalized position coordinate indexed i.
• vi Velocity of generalized coordinate i,
• (as
  appropriate)
• t Time coordinate
• In a given frame of reference.

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Euler-Lagrange example

• Let q qi be the ordinary x, y, z coordinates of
  a point particle with mass m.
• Let L ½mvi2 - V(q). (Kinetic minus potential.)
• Then, ?L/?qi - ?V/?qi Fi
• The force component in direction i.
• Meanwhile, ?L/?vi ?(½mvi2)/?vi mvi pi
• The momentum component in direction i.
• And,
• Mass times acceleration in direction i.
• So we get Fi mai (Newtons 2nd law)

15
Least-Action Principle
A.k.a.Hamiltonsprinciple

• The action of an energy means the integral of
  that energy over time.
• The trajectory specified by the Euler-Lagrange
  equation is one that locally extremizes the
  action of the Lagrangian
• Among trajectories s(t)between specified
  pointss(t0) and s(t1).
• Infinitesimal deviations from this trajectory
  leave the action unchanged to 1st order.

16
Hamiltons Equations
Implicitsummationover i.

• The Hamiltonian is defined as H vipi - L.
• Equals Ek Ep if L Ek - Ep and vipi 2Ek
  mvi2.
• We can then describe the dynamics of (qi, pi)
  states using the 1st-order Hamiltons equations
• These are equivalent to but often easier to solve
  than the 2nd-order Euler-Lagrange equation.
• Note that any Hamiltonian dynamics is
  bideterministic
• Meaning, deterministic in both the forwards and
  reverse time directions.

17
Field Theories

• Space of indexes i is continuous, thus
  uncountable. A topological space T, e.g., R3.
• Often use f(x) notation in place of qi.
• In local field theories, the Lagrangian L(f) is
  the integral of a Lagrange density function L(x)
  over the entire space T.
• This L(x) depends only locally on f, e.g.,
• L(x) L(f(x), (?f/?xi)(x), (x))
• All successful physical theories can be
  explicitly written down as local field theories!
• There is no instantaneous action at a distance.

18
Special Relativity and the Speed-of-Light Limit
19
The Speed-of-Light Limit

• No form of information (including quantum
  information) can propagate through space at a
  velocity (relative to its local surroundings)
  that is greater than the speed of light, c,
  3108 m/s.
• Some consequences
• No closed system can propagate faster than c.
• Although you can define open systems that do by
  definition
• No given piece of matter, energy, or momentum can
  propagate faster than c.
• All of the fundamental forces (including gravity)
  propagate at (at most) c.
• The probability mass that is associated with a
  quantum particle flows in an entirely local
  fashion, no faster than c.

20
Early History of the Limit

• The principle of locality was anticipated by
  Newton
• He wished to get rid of the action at a
  distance aspects of his law of gravitation.
• The finiteness of the speed of light was first
  observed by Roemer in 1676.
• The first decent speed estimate was obtained by
  Fizeau in 1849.
• Weber Kohlrausch derived a velocity of c from
  empirical electromagnetic constants in 1856.
• Kirchoff pointed out the match with the speed of
  light in 1857.
• Maxwell showed that his EM theory implied the
  existence of waves that always propagate at c in
  1873.
• Hertz later confirmed experimentally that EM
  waves indeed existed
• Michaelson Morley (1887) observed that the SoL
  was independent of the observers state of
  motion!
• Maxwells equations apparently valid in all
  inertial reference frames!
• Fitzgerald (1889), Lorentz (1892,1899), Larmor
  (1898), Poincaré (1898,1904), Einstein (1905)
  explored the implications of this...

21
Relativity Non-intuitive but True

• How can the speed of something be a fundamental
  constant? Seemed broken...
• If Im moving at velocity v towards you, and I
  shoot a laser at you, what speed does the light
  go, relative to me, and to you? Answer both c!
  (Not vc.)
• Newtons laws were the same in all frames of
  reference moving at a constant velocity.
• Principle of Relativity (PoR) All laws of
  physics are invariant under changes in velocity
• Einsteins insight The PoR is consistent w.
  Maxwells theory! Change def. of spacetime.

22
Some Consequences of Relativity

• Measured lengths and time intervals in a system
  vary depending on the systems velocity relative
  to observers.
• Lengths are shortened in direction of motion.
• Clocks run slower.
• Sounds paradoxical, but isnt!
• Mass is amplified.
• Energy and mass are the same quantity measured in
  different units Emc2.
• Nothing (incl. energy, matter, information, etc.)
  can go faster than light! (SoL limit.)

23
Three Ways to Understand c limit

• Energy of motion contributes to mass of object.
• Mass approaches ? as velocity?c.
• Infinite energy needed to reach c.
• Lengths, times in a faster-than-light moving
  object would become imaginary numbers!
• What would that mean?
• Faster than light in one reference frame ?
  Backwards in time in another reference frame
• Sending info. backwards in time violates
  causality, leads to logical contradictions!

24
The c limit in quantum physics

• Sometimes you see statements about nonlocal
  effects in quantum systems. Watch out!
• Even Einstein made this mistake.
• Described a quantum thought experiment that
  seemed to require spooky action at a distance.
• Later it was shown that this experiment did not
  actually violate the speed-of-light limit for
  information.
• These nonlocal effects are only illusions,
  emergent phenomena predicted by an entirely local
  underlying theory respecting SoL limit..
• Widely-separated systems can maintain quantum
  correlations, but that isnt true non-locality.

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The Lorentz Transformation
Actually it was written down earlier e.g., one
form by Voigt in 1887

• Lorentz, Poincaré All the laws of physics
  remain unchanged relative to the reference frame
  (x',t') of an object moving with constant
  velocity v ?x/?t in another reference frame
  (x,t) under the following conditions

Where
Note our ? here is the reciprocal of the
quantity denoted ? by other authors.
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Consequences of Lorentz Transform

• Length contraction (Fitzgerald, 1889, Lorentz
  1892)
• An object having length ? in its rest frame
  appears, when measured in a relatively moving
  frame, to have the (shorter) length ??. (For
  lengths parallel to direction of motion.)
• Time dilation (Poincaré, 1898)
• If time interval t is measured between two
  co-located events in a given frame, a larger time
  t t/? will be measured between those events in
  a relatively moving frame.
• Mass expansion (Einsteins fix for Newtons
  Fma)
• If an object has mass m0 in its rest frame, then
  it is seen to have the larger mass m m0/? in a
  relatively moving frame.

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Lorentz Transform Visualization
x'0
Original x,t(rest) frame
Line colors
Isochrones(space-like)
t'0
Isospatials(time-like)
New x',t'(moving) frame
Light-like
In this example v ?x/?t 3/5? ?t'/?t
4/5vT v/? ?x/?t' 3/4
The tourists velocity.
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An Alternative View Mixed Frames
t'
t
t'
StandardFrame 1
MixedFrame 1
t
x
x
In this example v ?x/?t 3/5 vT ?x/?t'
3/4 ? ?t' /?t 4/5 Note that (?t)2 (?x)2
(?t')2by the PythagoreanTheorem!
(Light pathsshown ingreen here.)
x
t'
t
x'
StandardFrame 2
MixedFrame 2
x'
x'
Note the obvious complete symmetryin the
relation between the two mixed frames.
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Relativistic Kinetic Energy

• Total relativistic energy E of any object is E
  mc2.
• For an object at rest with mass m0, Erest
  m0c2.
• For a moving object, m m0/?
• Where m0 is the objects mass in its rest frame.
• Energy of the moving object is thus Emoving
  m0c2/?.
• Kinetic energy Ekin Emoving - Erest m0c2/?
  - m0c2 m0c2(1 - ?)
• Substituting ? (1-ß2)1/2 and Taylor-expanding
  gives

Pre-relativistic kinetic energy ½ m0v2
Higher-orderrelativistic corrections
30
Spacetime Intervals

• Note that the lengths and times between two
  events are not invariant under Lorentz
  transformations.
• However, the following quantity is an invariant
  The spacetime interval s, where
• s2 (ct)2 - xi2
• The value of s is also the proper time t
• The elapsed time in rest frame of object
  traveling on a straight line between the two
  events. (Same as what we were calling t'
  earlier.)
• The sign of s2 has a particular significance
• s2 gt 0 - Events are timelike separated (s is
  real) May be causally connected.
• s2 0 - Events are lightlike separated (s
  is 0) Only 0-rest-mass signals may connect
  them.
• s2 lt 0 - Events are spacelike separated (s
  is imaginary) Not causally connected at all.

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Relativistic Momentum

• The relativistic momentum p mv
• Same as classical momentum, except that m m0/?.
• Relativistic energy-momentum-rest-mass
  relation E2 (pc)2 (m0c2)2If we use units
  where c 1, this simplifies to just E2 p2
  m02
• Note that if we solve for m02, we get
• m02 E2 - p2
• This is another relativistic invariant!
• Later we will show how it relates to the
  spacetime interval s2 t2 - x2, and to a
  computational interpretation of physics.