Thursday, October 18, 2018

Measurements in Quantum Mechanics.



-  2130  - Measurements in Quantum Mechanics.  Both extremes of space, to the very largest, to the very smallest, depend on understanding the nature of the smallest fundamental particles.  This review gets down to those small particles, which become waves.
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----------------------------------  2130  - Measurements in Quantum Mechanics.
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-  My last Review 2120 was about fundamental measurements in the macro world.   It is a prerequisite.  In it we learned that both extremes of space, to the very largest, to the very smallest, depend on understanding the nature of the smallest fundamental particles.  This Review 2130 gets down to those small particles, which become waves.
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-  96% of the Universe is made of Dark Matter ( 26%) and Dark Energy (70%).  Only 4% of the Universe is made of the stuff we understand, Ordinary Matter.  It is this 4% of the Universe that we can see with electromagnetic radiation.  We “see “ the 26% Dark Matter only because of its effect on gravity.
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-   We see Dark Energy only because of its effect on accelerating the expansion of the Universe.  We call them both “Dark” because today they are beyond our understanding to explain what they are.  However, the search is leading us to the smallest fundamental particles that make up the Universe.
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-  When particles become very small they behave as waves.  We believe the entire Universe is made of these waves but we just cannot see them until we look at dimensions smaller than the atom.
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-  An atom is 10^-10 meters.  When you get down to these dimensions everything becomes lumpy.  We no longer have a continuous Universe.  Light is radiation that is lumpy.  We call each lump a photon.  We have learned that all forms of energy is lumpy, not just light.  And, even space and time are lumpy.  The waves are lumps that come in distinct packets called quantum.  It is the world of Quantum Mechanics and it is very weird.
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-  The packets all have a constant unit of Action, called Planck’s Constant.  Action is Energy expended over time.  Planck’s constant uses the symbol “h” and

-----------------------------------  h  =  6.625 * 10^-34     kilograms * meters^2 / second. 
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-----------------------------------  h   =   E * t
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-----------------------------------  h   =  10^-34
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-   I will remove the numbers and the units and just use the magnitudes in this review.  The constant wave packet of Action is 10^-34.
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-  The wave packet for Energy is 10^-34 times the frequency of the wave.   E  =  h * f.   This is the energy of a single photon.  The formula says that the higher the frequency the greater the energy.  This should make sense to you because X-rays and Gamma rays  have a  lot greater energy than radio waves.  But, also blue light has more energy than red light.  Red light fells warm but blue, ultra-blue light will burn the skin.
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-----------------------------------  E  =  h * f.
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Since “h” is a unit of Action which is Energy times Time, and,  Frequency is cycles per second, then Action times Frequency = Energy.  But, the Action is a constant quantity,  just like the speed of light is a constant quantity.
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-----------------------------------  E = 10^-34 * f
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------------------------------------  Frequency  times  Wavelength = Velocity.
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-    In the case of electromagnetic radiation, Frequency * Wavelength = the velocity of light.  So, Frequency * Wavelength = 3*10^8 meters per second.  Following the convention that we agreed to Frequency* Wavelength= 10^8.  So, a wave packet of Energy is also equal to 10^-34 * 10^8 / Wavelength  =  10^-26 / Wavelength.
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-----------------------------------  E = 10^-26 / w
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------------------------------------  f  * w  =  c
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-----------------------------------  f * w  =  10^8
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Frequency * Wavelength = speed of light.  Frequency is the number of cycles per second and wavelength is the distance per cycle so the product of the two equals velocity.
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-----------------------------------  E = h*f
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-----------------------------------  E = h * c / w
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-----------------------------------  E = m*c^2    ,    This is Einstein’s formula for the conversion of mass to Energy.
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-----------------------------------  m*c^2 = h * c / w
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-----------------------------------  Wavelength  =  h / m*c
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- ---------------------------------   Wavelength is equal to 10^-34 / mass * 10^8
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-----------------------------------  Wavelength = 10^-26 / mass
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-  Where in this case “c” is the velocity of light = 3 * 10^8  meters per second.  At lower velocities we substitute “v” for “c”.
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-----------------------------------  Wavelength = h / m*v
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-  “m*v” is mass times velocity which is momentum, or inertia of a moving object.  “m*c” can be considered the momentum of a light beam.
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------------------------------------  Wavelength = 10^-34 / momentum. 
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-  Everything is a wave.  Your body is a wave.  A baseball is a wave.  Let’s try this formula out for a 0.1 kilogram baseball thrown at a slow pitch of 10 meters per second.  The Wavelength for  the baseball is 10^-34 / 10^-1*10^1  =  10^-34 meters.   The uncertainty of the baseball’s wave  at 10^-34 meters which would hardly cause you to miss it in a called STRIKE.
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-  Baseballs are too big to see the effect of the duality of particles becoming waves.  Let’s do the calculation for an electron orbiting a nucleus of an atom.  The mass of the electron is 10^-30 kilograms.  The velocity of the electron is 99% the speed of light, or 10^6 meters per second.  Wavelength = 10^-34 / momentum of the electron.  Wavelength = 10^-34 / mass times velocity of the electron.  Wavelength = 10^-34 / 10^-30 * 10^6.
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-  Wavelength = 10^-10 meters.  The uncertainty of the electron’s position inside an atom is about the diameter of the atom.  This gets us into another property of small wave particles in our Quantum World
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-  The next property we learn about the Quantum World is that it is not only lumpy it is not deterministic.  It is governed by uncertainties and probabilities.  
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-  Using the electron as our illustration, although the same applies to any fundamental particle, the electron’s position in space and its motion in space cannot be known exactly.  There is always a trade off being position and velocity. 
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-  The uncertainty of position times the uncertainty of velocity must always be less than h/4*pi, which is 1.1*10^-34.  So, there is always a trade-off.  The better you measure the position the more uncertain the velocity will be in your measurement.  And, vise versa.
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-   If you know the velocity with some high probability than the knowledge of the position is a low probability.  The same uncertainty trade-off exists between the measurements of energy and time.  It is as though these paired wave packets were complementary.  The product of the uncertainties must always be greater than 10^-34.  Let’s go back to E = h*f     and:
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------------------------------------    E = h*c/w
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------------------------------------    Energy = 10^-26 / Wavelength.
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-  This formula tells us that the smaller the wavelength ( or the higher the frequency) the greater the energy.  However, there is a limit.  When the wavelength gets so small and the energy gets so extremely concentrated it collapses into a black hole.  This happens at the wavelength of 10^-35 meters.
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-   Light takes 10^-35 seconds to traverse this distance.  That is where physics breaks down because nobody knows what goes on inside a black hole.  A proton is 10^-15 meters diameter. This wavelength is 10^19 times smaller than a proton.
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-   The quantum of energy within this tiny wavelength each carry as much energy as the rest-mass of 10^19 protons.  E = 10^9.  This energy is in joules, in electron-volts, it is equivalent to 10^28 eV.  Our best particle accelerators are Terra-electron-volts, TeV.  That is 10^12 eV so the energy needed to create these baby black holes has to grow by 10^16, which is 10 times a quadrillion fold.
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-  These distance scales that we are working with are smaller than atoms by just as much as atoms are smaller than stars.
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-  Within orders of magnitude we can get to these numbers using Newton’s formulas and not bother with the accuracies we get from more complex Einstein relativistic formulas.  A Black Hole has so much mass and therefore so much gravity that the escape velocity needed to leave it exceeds the speed of light. 
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-  It is the point where the Kinetic Energy of the velocity of light equals the Potential Energy of mass being pulled by gravity at radius, “r” ,  Kinetic Energy = Potential Energy
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-----------------------------------  KE  = PE
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-----------------------------------  ½ *m*v^2   =  G * m * M / r       , where the velocity, v , of the speed of light is “c”.
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-----------------------------------   “r”  =  2 * G* M / c^2
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-----------------------------------   “G”  =  the gravitational constant = 6.7*10^-11
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-----------------------------------   “M” = mass of 10^19 protons,     m  = 1.17*10^-27 kilograms each.
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-----------------------------------   “r”  =  10^-11 * 10^-8  /  10^16
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-----------------------------------   “r”  = 10^-35 meters.
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-  This radius is the smallest dimension.  10^-35 is the smallest unit of length.  You cannot get any smaller than that because it turns into a black hole.
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-  So, In conclusion, we have defined all the lumps, the smallest quantities we can have in our Quantum World:
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 -----------------------------------  , in space, 10^-35 meters.
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-----------------------------------  , in time, 10^-43 seconds,
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-----------------------------------  , in mass, 10^-8 kilograms with a radius of 10^-35 meters,
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-----------------------------------  , in energy, 10^9 joules, which is 10^28 electron-volts,
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-----------------------------------  , in action, 10^-34 kilograms / meters^2 / second, which is Max Planck’s Constant.
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-  It does not get any smaller than that.
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(1)  #630  Neutrinos - The Little Neutral Ones
(2)  # 631  Mass, Momentum, and Inertia.  What is mass?
(3)  #632  Muons and Taus -  The Heavy Electrons
(4)  #633  The Force Carriers - Gluons, Bosons, and Photons
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-  October 18, 2018.       724 , 725
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 ---------------------   Thursday, October 18, 2018     -------------------------
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