CHAPTER 2: Special Theory of Relativity

1
Review Modern Physics, Ph 311
It was found that there was no displacement of
the interference fringes, so that the result of
the experiment was negative and would, therefore,
show that there is still a difficulty in the
theory itself - Albert Michelson, 1907
1/3 to 2/3 of our modern economy !!!
2
Inertial Reference Frame

• A reference frame is called an inertial frame if
  Newton laws are valid in that frame.
• Such a frame is established when a body, not
  subjected to net external forces, is observed to
  move in rectilinear motion at constant velocity.

3
Newtonian Principle of Relativity

• If Newtons laws are valid in one reference
  frame, then they are also valid in another
  reference frame moving at a uniform velocity
  relative to the first system.
• This is referred to as the Newtonian principle of
  relativity or Galilean invariance/relativity.

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The Galilean Transformation

• For a point P
• In system K P (x, y, z, t)
• In system K P (x, y, z, t)

P
x
K
K
x-axis
x-axis
6
Conditions of the Galilean Transformation

• Parallel axes
• K has a constant relative velocity in the
  x-direction with respect to K
• Time (t) for all observers is a Fundamental
  invariant, i.e., the same for all inertial
  observers

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The Inverse Relations

• Step 1. Replace with .
• Step 2. Replace primed quantities with
• unprimed and unprimed with
  primed.

8
Results of Maxwell's electrodynamics

• Visible light covers only a small range of the
  total electromagnetic spectrum
• All electromagnetic waves travel in a vacuum with
  a speed c given by
• (where µ0 and e0 are the respective permeability
  and permittivity of free space)

9
Need for Ether

• The wave nature of light suggested that there
  existed a propagation medium called the
  luminiferous ether or just ether.
• Ether had to have such a low density that the
  planets could move through it without loss of
  energy
• It also had to have an enormous
  elasticity/toughness to support the high velocity
  of light waves
• According to classical physics ideas, the ether
  frame would be a preferred frame, the only one in
  which Maxwells equation would be valid as
  derived

10
An Absolute Reference System

• Ether was proposed as an absolute reference
  system in which the speed of light was this
  constant and in all frames moving with respect
  that that frame, there needed to be modifications
  of Maxwells laws.
• The Michelson-Morley experiment was an attempt to
  figure out Earths relatives movement through
  (with respect to) the ether so that Maxwells
  equations could be corrected for this effect.

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1. AC is parallel to the motion of the Earth
inducing an ether wind2. Light from source S
is split by mirror A and travels to mirrors C and
D in mutually perpendicular directions3. After
reflection the beams recombine at A slightly out
of phase due to the ether wind as viewed by
telescope E.
0
The Michelson Interferometer
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NEVER OBSERVED !!!!
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The Lorentz-FitzGerald Contraction

• Another hypothesis proposed independently by both
  H. A. Lorentz and G. F. FitzGerald suggested that
  the length l1, in the direction of the motion was
  contracted by a factor of
• thus making the path lengths equal to account
  for the zero phase shift.
• This, however, was an ad hoc assumption that
  could not be experimentally tested. It turned out
  to be less than half of the story

14
Length contracted for the moving muon, its own
life time just 2.2 micro seconds
Life time of the muon delayed for observer on
Earth so that it can travel the whole distance as
observed from Earth Great thing about special
relativity is that one can always take two
viewpoints, moving with the experiment, watching
the experiment move, the observations need to be
consistent in both cases
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Lorentz Transformation Equations
So there is four-dimensional space time !!!
16
Mary has a light clock. A suitable clock is just
any periodic process, the time it takes for one
cycle of the process is the period, its inverse
is the frequency. Tom watching Mary go by figures
that her time is delayed due to her moving in a
straight line with a constant high velocity.
17
Atomic Clock Measurement

• Figure 2.20 Two airplanes took off (at different
  times) from Washington, D.C., where the U.S.
  Naval Observatory is located. The airplanes
  traveled east and west around Earth as it
  rotated. Atomic clocks on the airplanes were
  compared with similar clocks kept at the
  observatory to show that the moving clocks in the
  airplanes ran slower.

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No simultaneity if not also at the same position,
just a consequence of the Lorentz transformaitons
19
The Lorentz Velocity Transformations

• In addition to the previous relations, the
  Lorentz velocity transformations for ux, uy ,
  and uz can be obtained by switching primed and
  unprimed and changing v to v

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Einsteins Two Postulates

• With the belief that Maxwells equations (and
  with it all of the known physics of the time)
  must be
• valid in all inertial frames, Einstein proposes
  the
• following postulates
• The principle of relativity The laws of physics
  are the same in all inertial systems. There is no
  way to detect absolute motion, and no preferred
  inertial system exists.
• The constancy of the speed of light Observers in
  all inertial systems measure the same value for
  the speed of light in a vacuum.

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Relativistic Momentum

• Rather than abandon the conservation of linear
  momentum, let us look for a modification of the
  definition of linear momentum that preserves both
  it and Newtons second law.
• To do so requires reexamining mass to conclude
  that

Relativistic dynamics can be derived by assuming
that mass is increasing with velocity. The
Lorentz factor gets larger when velocities get
larger and so does mass apparently as we can see
from the relativistic momentum equation. Einstein
derived relativistic dynamics that way. His
derivations are sure correct, but the foundations
are somewhat shaking as there is no really good
definition for mass.
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Relativistic Energy

• Due to the new idea of relativistic mass, we must
  now redefine the concepts of work and energy.
• Therefore, we modify Newtons second law to
  include our new definition of linear momentum,
  and force becomes

23
Relativistic Kinetic Energy

• Equation (2.58) does not seem to resemble the
  classical result for kinetic energy, K ½mu2.
  However, if it is correct, we expect it to reduce
  to the classical result for low speeds. Lets see
  if it does. For speeds u ltlt c, we expand in a
  binomial series as follows
• where we have neglected all terms of power (u/c)4
  and greater, because u ltlt c. This gives the
  following approximation for the relativistic
  kinetic energy at low speeds
• which is the expected classical result. We show
  both the relativistic and classical kinetic
  energies in the next Figure. They diverge
  considerably above a velocity of 0.1c. Best to
  use relativistic dynamics as soon as the speed of
  something is larger than 1 of the speed of
  light.

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Relativistic and Classical Kinetic Energies
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Total Energy and Rest Energy

• We rewrite in the form
• The term mc2 is called the rest energy and is
  denoted by E0.
• This leaves the sum of the kinetic energy and
  rest energy to be interpreted as the total energy
  of the particle. The total energy is denoted by E
  and is given by

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Momentum and Energy

• We square this result, multiply by c2, and
  rearrange the result.
• We replace ß2 and find

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Momentum and Energy (continued)

• The first term on the right-hand side is just E2,
  and the second term is E02. The last equation
  becomes
• We rearrange this last equation to find the
  result we are seeking, a relation between energy
  and momentum.
• or
• is a useful result to relate the total energy of
  a particle with its momentum. The quantities (E2
  p2c2) and m are invariant quantities. Note that
  when a particles velocity is zero and it has no
  momentum, accelerator Equation correctly gives
  E0 as the particles total energy.

There can be mass less particles that still have
momentum. These can collide with massive
particles. For such a collision one needs to
invoke special relativity!
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Binding Energy

• The binding energy is the difference between the
  rest energy of the individual particles and the
  rest energy of the combined bound system.

A couple of eV for chemical reactions. A couple
of MeV for nuclear reactions.
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A Conducting Wire
0
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Principle of Equivalence

• The principle of equivalence is an experiment in
  non-inertial reference frames.
• Consider an astronaut sitting in a confined space
  on a rocket placed on Earth. The astronaut is
  strapped into a chair that is mounted on a
  weighing scale that indicates a mass M. The
  astronaut drops a safety manual that falls to the
  floor.

• Now contrast this situation with the rocket
  accelerating through space. The gravitational
  force of the Earth is now negligible. If the
  acceleration has exactly the same magnitude g on
  Earth, then the weighing scale indicates the same
  mass M that it did on Earth, and the safety
  manual still falls with the same acceleration as
  measured by the astronaut. The question is How
  can the astronaut tell whether the rocket is on
  earth or in space?
• Principle of equivalence There is no experiment
  that can be done in a small confined space that
  can detect the difference between a uniform
  gravitational field and an equivalent uniform
  acceleration.

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Gravitational Time Dilation

• Since the frequency of the clock decreases near
  the Earth, a clock in a gravitational field runs
  more slowly (it takes longer for a hand to move
  on a clock so in aggregate the clock gets
  slower) according to the gravitational time
  dilation. This is because 4D space-time is bend
  non-Euclidian, so there are no Euclidian
  straight lines to follow but Geodesics in a space
  whit Riemanns coordinates
• A very accurate experiment was done by comparing
  the frequency of an atomic clock flown on a Scout
  D rocket to an altitude of 10,000 km with the
  frequency of a similar clock on the ground. The
  measurement agreed with Einsteins general
  relativity theory to within 0.02.

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Tests of General Relativity

• Bending of Light
• During a solar eclipse of the sun by the moon,
  most of the suns light is blocked on Earth,
  which afforded the opportunity to view starlight
  passing close to the sun in 1919. The starlight
  was bent as it passed near the sun which caused
  the star to appear displaced.
• Einsteins general theory predicted a deflection
  of 1.75 seconds of arc, and the two measurements
  found 1.98 0.16 and 1.61 0.40 seconds.
• Since the eclipse of 1919, many experiments,
  using both starlight and radio waves from
  quasars, have confirmed Einsteins predictions
  about the bending of light with increasingly good
  accuracy.

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Light Retardation

• As light passes by a massive object, the path
  taken by the light is longer because of the
  spacetime curvature.
• The longer path causes a time delay for a light
  pulse traveling close to the sun.
• This effect was measured by sending a radar wave
  to Venus, where it was reflected back to Earth.
  The position of Venus had to be in the superior
  conjunction position on the other side of the
  sun from the Earth. The signal passed near the
  sun and experienced a time delay of about 200
  microseconds. This was in excellent agreement
  with the general theory.

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Spacetime Curvature of Space

• Light bending for the Earth observer seems to
  violate the premise that the velocity of light is
  constant from special relativity. Light traveling
  at a constant velocity implies that it travels in
  a straight line.
• Einstein recognized that we need to expand our
  definition of a straight line.
• The shortest distance between two points on a
  flat surface appears different than the same
  distance between points on a sphere. The path on
  the sphere appears curved. We shall expand our
  definition of a straight line to include any
  minimized distance between two points.
• Thus if the spacetime near the Earth is not flat,
  then the straight line path of light near the
  Earth will appear curved.

35
Perihelion Shift of Mercury

• The orbits of the planets are ellipses, and the
  point closest to the sun in a planetary orbit is
  called the perihelion. It has been known for
  hundreds of years that Mercurys orbit precesses
  about the sun. Accounting for the perturbations
  of the other planets left 43 seconds of arc per
  century that was previously unexplained by
  classical physics.
• The curvature of spacetime explained by general
  relativity accounted for the 43 seconds of arc
  shift in the orbit of Mercury.

36
Gravitational Wave Experiments

• Taylor and Hulse discovered a binary system of
  two neutron stars that lose energy due to
  gravitational waves that agrees with the
  predictions of general relativity.
• LIGO is a large Michelson interferometer device
  that uses four test masses on two arms of the
  interferometer. The device will detect changes in
  length of the arms due to a passing wave.

• NASA and the European Space Agency (ESA) are
  jointly developing a space-based probe called the
  Laser Interferometer Space Antenna (LISA) which
  will measure fluctuations in its triangular
  shape.

No success so far, perhaps general relativity
(and special relativity with it) are not really
true, just very very good approximations to
something else?
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BUT, thank you very much indeed Albert !!!
everybody loves this !!!!
38
Dual nature of light (electromagnetic radiation)
both/neither wave and/nor particle
http//usatoday30.usatoday.com/tech/science/geneti
cs/2008-05-08-platypus-genetic-map_N.htm Australi
a's unique duck-billed platypus is part bird,
part reptile and part mammal according to its
gene map. The platypus is classed as a mammal
because it has fur and feeds its young with milk.
It flaps a beaver-like tail. But it also has bird
and reptile features a duck-like bill and
webbed feet, and lives mostly underwater. Males
have venom-filled spurs on their heels.
39
Light according to Maxwell
Fig. 3-2, p. 67
40
Wiens Displacement Law

• The intensity (?, T) is the total power
  radiated per unit area per unit wavelength at a
  given temperature.
• Wiens displacement law The maximum of the
  distribution shifts to smaller wavelengths as the
  temperature is increased.

When u(x) is plotted over x, there is only one
peak! One universal curve for all wavelengths and
T
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Two fitting parameters and no physical theory
behind them !!
42
3.5 Blackbody Radiation

• When matter is heated, it emits radiation.
• A blackbody is a cavity in a material that only
  emits thermal radiation. Incoming radiation is
  absorbed in the cavity.

• Blackbody radiation is theoretically interesting
  because the radiation properties of the
  blackbody are independent of the particular
  material. Physicists can study the properties of
  intensity versus wavelength at fixed temperatures.

43
Rayleigh-Jeans Formula

• Lord Rayleigh used the classical theories of
  electromagnetism and thermodynamics to show that
  the blackbody spectral distribution should be
• It approaches the data at longer wavelengths, but
  it deviates badly at short wavelengths. This
  problem for small wavelengths became known as
  the ultraviolet catastrophe and was one of the
  outstanding exceptions that classical physics
  could not explain.

k Boltzmanns constant 8.614 10-5 eV/K
44
Plancks Radiation Law

• Planck assumed that the radiation in the cavity
  was emitted (and absorbed) by some sort of
  resonators that were contained in the walls.
  These resonators were modeled as harmonic
  oscillators. He effectively invented new physics
  in the process. His result cannot be explained
  with classical Boltzmann-Maxwell statistics.
• Planck made two modifications to the classical
  theory
• The oscillators (of electromagnetic origin) can
  only have certain discrete energies determined by
  En nhf, where n is an integer, f is the
  frequency, and h is called Plancks constant. h
  6.6261 10-34 Js.
• The oscillators can absorb or emit energy in
  discrete multiples of the fundamental quantum of
  energy given by

Plancks radiation law, only one fundamental
constant h left that can explain Wiens and
Stephans constants , significant progress
45
Photoelectric effect
46
Experimental Results
Only if the energy threshold to get electrons out
of the metal (work function) is exceeded.
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Einsteins Theory

• Einstein suggested that the electromagnetic
  radiation field is quantized into particles
  called photons. Each photon has the energy
  quantum
• where f is the frequency of the light and h is
  Plancks constant. Also he came up with the wave
  particle duality of light, at long wavelengts it
  looks more like a wave at short wavelenght, high
  frequency, high enerly it looks more like a
  particle
• The photon travels at the speed of light in a
  vacuum, and its wavelength is given by

48
Einsteins Theory

• Conservation of energy yields
• where is the work function of the metal.
• Explicitly the energy is
• The retarding potentials measured in the
  photoelectric effect are the opposing potentials
  needed to stop the most energetic electrons.

49
X-Ray Production

• An energetic electron passing through matter will
  radiate photons and lose kinetic energy which is
  called bremsstrahlung, from the German word for
  braking radiation. Since linear momentum must
  be conserved, the nucleus absorbs very little
  energy, and it is ignored. The final energy of
  the electron is determined from the conservation
  of energy to be
• An electron that loses a large amount of energy
  will produce an X-ray photon. Current passing
  through a filament produces copious numbers of
  electrons by thermionic emission. These electrons
  are focused by the cathode structure into a beam
  and are accelerated by potential differences of
  thousands of volts until they impinge on a metal
  anode surface, producing x rays by bremsstrahlung
  as they stop in the anode material.

50
Inverse Photoelectric Effect.

• Conservation of energy requires that the electron
  kinetic energy equal the maximum photon energy
  where we neglect the work function because it is
  normally so small compared to the potential
  energy of the electron. This yields the
  Duane-Hunt limit which was first found
  experimentally. The photon wavelength depends
  only on the accelerating voltage and is the same
  for all targets.

Lets have 10 50 keV, very short wavelengths,
very energetic photons
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Braggs law
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No way !!!
Just a relativistic collision between a mass less
particle and a massive particle.
53
Compton Effect

• When a photon enters matter, it is likely to
  interact with one of the atomic electrons. The
  photon is scattered from only one electron,
  rather than from all the electrons in the
  material, and the laws of conservation of energy
  and momentum apply as in any elastic collision
  between two particles. The momentum of a particle
  moving at the speed of light is
• The electron energy can be written as
• This yields the change in wavelength of the
  scattered photon which is known as the Compton
  effect

54
X-Ray Scattering, modern crystallography

• Max von Laue suggested that if x rays were a form
  of electromagnetic radiation with wavelengths on
  the 0.1 nm scale, interference effects should be
  observed for a crystal, which can be thought of
  as a 3D diffraction grating.
• Friedrich and Knipping did the experiments and
  modern crystallography was born !!! Almost all of
  of our knowledge of atomic structures comes from
  such (and electron and neutron) diffraction
  experiments

55
Wave particle duality, wavical
Taoism Taijitu (literally "diagram of the
supreme ultimate"
No problem, Bohrs complementarily
56
Dual nature of quantum mechanical
objectsboth/neither particle and/nor wave
http//usatoday30.usatoday.com/tech/science/geneti
cs/2008-05-08-platypus-genetic-map_N.htm Australi
a's unique duck-billed platypus is part bird,
part reptile and part mammal according to its
gene map. The platypus is classed as a mammal
because it has fur and feeds its young with milk.
It flaps a beaver-like tail. But it also has bird
and reptile features a duck-like bill and
webbed feet, and lives mostly underwater. Males
have venom-filled spurs on their heels.
57
Thomsons Atomic Model

• J. J. Thomsons plum-pudding model of the atom
  had the positive charges spread uniformly
  throughout a sphere the size of the atom, with
  electrons embedded in the uniform background.
• In J. J. Thomsons view, when the atom was
  heated, the electrons could vibrate about their
  equilibrium positions, thus producing
  electromagnetic radiation.

Not quite, electrons repulse each other as much
as possible but what is the dough?
58
More experiments, looking at large angles where
one would not expect any scattering to show up,
BUT
59
There is no dough, just lots and lots of empty
space and a tiny tiny heavy nucleus where all of
the positive charges reside.
60
Planetary model of the atom would not work on the
basis of classical physics, would not explain why
atoms are forever, when a molecule breaks up, the
atoms are just as before
Fig. 4-21, p. 131
61
Angular momentum must be quantized in nature in
units of h-bar, from that follows quantization of
energy levels .
62
Fig. 4-23, p. 133
63
Fig. 4-24, p. 134
64
Can all be explained from Bohrs model as he puts
physical meaning to the Rydberg equation.
Fig. 4-20, p. 129
65
Bohrs second paper in 1913. There should be
shells, idea basically correct, helps explaining
basic chemistry
66
The Correspondence Principle
Classical electrodynamics
Bohrs atomic model

Determine the properties of radiation

• Need a principle to relate the new modern results
  with classical ones. Mathematically h -gt 0

In the limits where classical and quantum
theories should agree, the quantum theory must
reduce the classical result.
Bohrs correspondence principle
Bohrs third paper in 1913
67
4.6 Characteristic X-Ray Spectra and Atomic
Number

• Shells have letter names
• K shell for n 1
• L shell for n 2
• The atom is most stable in its ground state.
• When it occurs in a heavy atom, the radiation
  emitted is an x ray.
• It has the energy E (x ray) Eu - El.

An electron from higher shells will fill the
inner-shell vacancy at lower energy.
68
Atomic Number Z, X-ray spectroscopy on the basis
of the characteristic X-rays

• L shell to K shell Ka x ray
• M shell to K shell Kß x ray
• Atomic number Z number of protons in the
  nucleus.
• Moseley found a relationship between the
  frequencies of the characteristic x ray and Z.
• This holds for the Ka x ray.

Explanation on the basis of Bohrs model for H
and shielding for all other atoms !!!!
69
Moseleys Results support Bohrs ideas for all
tested atoms

• The x ray is produced from n 2 to n 1
  transition.
• 
  
• In general, the K series of x ray wavelengths are
• Moseleys research clarified the importance of
  the electron shells for all the elements, not
  just for hydrogen.

70
Frank-Hertz experiment

• Accelerating voltage is below 5 V.
• electrons did not lose energy as they are
  scattered elastically at the much heavier Hg
  atoms.
• Accelerating voltage is above 5 V.
• sudden drop in the current because there is now
  inelastic scattering instead.

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There are also matter waves, not only classical
and electromagnetic waves !!!Wave particle
duality for matter leads us into quantum
mechanics, condensed matter physics .
1/3 to 2/3 of our modern economy !!!
73
Instantaneous (linear) momentum is quantized as
well in a bound system
momentum h / wavelength for particles with
mass as well, not only photons
32 ao
Fig. 5-2, p. 153
74
5.3 Electron Scattering

• Davisson and Germer experimentally observed that
  electrons were diffracted much like x rays in
  nickel crystals.

• George P. Thomson (18921975), son of J. J.
  Thomson, reported seeing the effects of electron
  diffraction in transmission experiments. The
  first target was celluloid, and soon after that
  gold, aluminum, and platinum were used. The
  randomly oriented polycrystalline sample of SnO2
  produces rings as shown in the figure at right.

75
TEM
One operation mode is transmission diffraction,
there is also electron energy loss spectroscopy
and X-ray spectroscopy
76
SEM
Short wavelength and nearly parallel fine
electron beam results in large depth of focus,
SEM images appear almost three-dimensional
77
One full cycle for envelop wave 2 pi
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Wave Packet Envelope

• The superposition of two waves yields a wave
  number and angular frequency of the wave packet
  envelope.
• The range of wave numbers and angular frequencies
  that produce the wave packet have the following
  relations
• A Gaussian wave packet has similar relations
• The localization of the wave packet over a small
  region to describe a particle requires a large
  range of wave numbers. Conversely, a small range
  of wave numbers cannot produce a wave packet
  localized within a small distance.

80
Modern physics backed up by experiments
Mathematical uncertainties
Heisenberg's uncertainties
81
Since the uncertainty principle is really a
statement about accuracy rather than precision,
there is a kind of systematic rest error that
cannot be corrected for In classical physics this
is simply ignored as things are large in
comparison to electrons, atoms, molecules,
nano-crystals
82
Probability, Wave Functions, and the Copenhagen
Interpretation

• The square of the wave function determines the
  likelihood (or probability) of finding a particle
  at a particular position in space at a given
  time.
• The total probability of finding the electron is
  1. Forcing this condition on the wave function is
  called normalization.

If wavefunction is normalized !!
dy for no particular reason, its just 1D dx
83
The Copenhagen Interpretation

• Copenhagens interpretation of the wave function
  (quantum mechanics in its final and current form)
  consisted of 3 (to 4) principles
• The complementarity principle of Bohr
• The uncertainty principle of Heisenberg
• The statistical interpretation of Born, based on
  probabilities determined by the wave function
• Bohrs correspondence principle (for quantum
  mechanics being reasonable
• Together these concepts form a logical
  interpretation of the physical meaning of quantum
  theory. According to the Copenhagen
  interpretation, physics needs to make predictions
  on the outcomes of future experiments
  (measurement) on the basis of the theoretical
  analysis of previous experiments (measurements)
• Physics is not about the truth, questions that
  cannot be answered by experiments (measurements)
  are meaningless to the modern physicist.
  Philosophers, priests, gurus, can be asked
  these questions and often answer them. Problem
  they tend to disagree

84
Probability of finding the Particle in a certain
region of space

• The probability of observing the particle between
  x and x dx in each state is
• Since there is dx, we need to integrate over the
  region we are interested in
• All other observable quantities will be obtained
  by integrations as well.
• Note that E0 0 is not a possible energy level,
  there is no quantum number n 0, so E1 is ground
  state also called zero point energy if in a
  quantum oscillator
• The concept of energy levels, as first discussed
  in the Bohr model, has surfaced in a natural way
  by using matter waves.

We analyze the same model in the next chapter
with operators on wave functions and expectation
value integrals (that tell us all there is to
know)
85
Particle in an infinitely deep Box, no potential
energy to be considered

• A particle of mass m is trapped in a
  one-dimensional box of width L.
• The particle is treated as a standing wave. It
  persist to exist just like a standing wave.
• The box puts boundary conditions on the wave. The
  wave function must be zero at the walls of the
  box and on the outside.
• In order for the probability to vanish at the
  walls, we must have an integral number of half
  wavelengths in the box.
• The energy of the particle is .
• The possible wavelengths are quantized which
  yields the energy
• The possible energies of the particle are
  quantized.

There is a ground state energy, zero point
energy, particles that are confined can never
stand still, always move, no way to utilize this
energy for mankind
86
L
a widths of slits, a lt d ? ltlt L
? Path difference (rad)
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91
time dependent Helmholtz
due to the uncertainty principle, we can only
make statistical inferences
92
Given the wave particle duality, we need a new
way of thinking. The whole physical situation is
described by a wave function (which is complex
for a traveling matter wave). That wave function
accounts for the physical boundary conditions,
which encode the nature of the physical
problem. To check if the wave function we came up
with makes physical sense, we put it to the
Schrödinger equation test. (Its a test if our
wave function obeys the conservation of total
energy (while ignoring rest energy and with that
special relativity if we need to include that,
i.e. v gt 0.01 c, we need to make the Dirac
equation test) If our wave function is OK, we can
calculate anything we are allowed to know about
the quantum mechanical system from it.