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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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----- Comments appreciated and Pass it on to
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--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
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------
jamesdetrick@comcast.net
------ “Jim Detrick” -----------
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--------------------- Thursday, October 18, 2018 -------------------------
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