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---------------------- 2703 - PHYSICS - Uncertainty and Exclusions principles?
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- Quantum Mechanics describes the behavior of matter on the microscopic scale. Isaac Newton’s physics worked great in our everyday lives. But, the math did not work at very high velocities or at very high fields of gravity. It does not work on very small scales either.
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- Albert Einstein’s physics did work on the extreme velocity cases and it also was the same as Newton’s math on ordinary cases. Quantum Mechanics math had to be invented to work on the atomic scales. Much of this Quantum Mechanic’s math relied on two principles, the Uncertainty Principle and the Exclusion Principle.
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- In 1924 Louis de Broglie defined all mass as having wavelengths as well. Everything has a particle-wave duality but it only becomes noticeable at the microscopic level, at the Quantum Mechanic’s level.
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- He said that the product wavelength * momentum is always a constant = to:
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----------------------------- Planck’s Constant of Action = 6.626*10^-34 joule*seconds.
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----------------------------- Momentum,”p” = mass * velocity, “m” * “v”:
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----------------------------- wavelength * momentum = Planck’s Constant
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----------------------------- w*p = 6.6*10^-34
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----------------------------- w * m * v = 6.6*10^-34
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- If the product is a constant that means that if mass increases, wavelength decreases. If mass decreases, wavelength must increase. All matter has waves, but, only very small stuff has waves big enough to notice.
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- Let’s take a particle of dust that is very small and has a mass of 10^-9 kilograms. It is not moving very fast, say 10 meters / second. The calculation for its wavelength is 6.6^-26 meters. This wavelength is so small as not to be noticeable.
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------------------- wavelength = 6.6*10^-34 / mass * velocity
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------------------- w = 6.6*10^-34 / 10^-9 * 10
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-------------------- w = 6.6 * 10^-26 meters
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- However, let’s take something much smaller than a dust particle, say an electron. It has a mass of 9.1*10^-31 kilograms. If we get down to the atomic scale these small waves become noticeable. The electron’s velocity is usually 10^6 meters / second. Its wavelength is therefore 0.7 nanometers.
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- The diameter of an atom is 0.1 nanometers. Now the wavelength is extremely noticeable. It is 7 times bigger than the atom
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-------------------- w = 6.6*10^-34 / mass * velocity
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------------------- w = 6.6*10^-34 / 9.1*10^-31 * 10^6
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-------------------- w = 0.7 *10^-9 meters
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- Neutrons in a gas cloud having a temperature of 35 Kelvin will have an average velocity of 1500 meters / second. The mass of a neutron is 1.6*10^-27 kilograms. Its wavelength is 0.28 nanometers.
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- As a result of these neutrons having a wavelength it can be shown that they have an interference pattern just like other waves , or, like photons of light. Passing a stream of neutrons with 2 nanometers wavelength through a double-slit that is 20,000 nanometers wide separated by 104,000 nanometers creates an interference pattern of circular bands on the screen similar to a light beam. Similar to a water wave.
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- The electron has a mass of 10^-30 kilograms. Its orbital speed inside an atom is 10^6 meters per second. Its momentum is therefore 10^-25 kilogram*meters/second. The diameter of the atom is 10^-10 meters.
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- Can we pin down the location of an electron to within 10% of the atom’s size, therefore to within 10^-11 meters? The “Uncertainty Principle” says there is a trade-off between knowing the electrons position and its velocity.
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- Any attempt to localize a particle to within a distance of some Uncertainty of Location necessarily limit’s a simultaneous determination of that component of the particle’s momentum to an Uncertainty of Momentum. The product of these two uncertainties must always be greater than Planck’s Constant of Action.
---- Uncertainly of Location * Uncertainty of Momentum > Planck’s constant / 2*Pi
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---- Uncertainly of Location * Uncertainty of Mass * Velocity > Planck’s constant / 2*Pi
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---- Uncertainly of Location * Uncertainty of Mass * Velocity > 1.05*10^-34 joule*seconds.
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--------------------- dx * m*dv > 1.05*10^-34
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- This same Uncertainty Principle goes on to relate Time and Energy. If an energy measurement is to be carried out in time, Uncertainty in Time, and the accuracy, in Uncertainty in Energy, in which the energy can be measured in this time interval is limited to :
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---- Uncertainly in Energy * Uncertainty in Time = > 1.05*10^-34 joule*seconds.
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------------------------ dE * dt > 1.05*10^-34
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- This uncertainty quantity of 10^-34 is very, very small. We need not concern ourselves with it when dealing with macroscopic things like rockets, baseballs or even dust particles. However, when we get to the size of electrons this Constant of Action is huge. If we try to pin down the location of an electron to within 10% of the size of atom. The momentum is so huge relative to the atom that we can not be certain that the electron can remain inside the atom. To deal with this situation we must begin using calculations with probabilities from Quantum Mechanics.
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- Quantum Mechanics is different from any of Newton’s theories. It does not make any predictions about the outcome of a single event. It makes predictions only about the probabilities of different outcomes. Quantum Mechanic’s math can not be used to predict the future behavior of a system, only the probability of a set of possible behaviors.
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------------------- The Exclusion Principle
- Quantum Mechanics provides math that allows us to predict with great accuracy the properties of matter. There are two fundamental laws used in Quantum Mechanics:
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----------------- The Werner Heisenberg (1901 - 1976) Uncertainty Principle
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---------------- The Wolfgang Pauli (1900 - 1958) Exclusion Principle
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- The Uncertainty Principle says that the Uncertainty of Location, “dx”, times the Uncertainty of Momentum, “dp”, must always be greater than Planck’s Constant / 2*Pi = 1.05*10^-34 joule-seconds.
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-------------------------- Momentum is mass * velocity, p=m*v.
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------------------------- dx * m*dv > 1.05*10^-34 joule-seconds.
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- 10^-34 is a very small number so this uncertainty in location or in the velocity of a baseball would not be noticeable. However, with a particle the size of an electron this uncertainty is a very big deal.
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- Let’s illustrate this with an imaginary baseball game except instead of pitching a baseball, we pitch an electron. Due to the uncertainty principle the batter will find that hitting the electron is nearly impossible. The batter may see the electron coming towards him but it is impossible to know which direction it is going. He does not know whether to swing up, down, or sideways. Due to the uncertainty the batter can never tell both where an electron is and where it is going.
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- The second Uncertainty Principle says you can never know the energy of the electron, that is how fast it is going, and the time it takes to get there. Therefore, the batter never knows when to swing. The Uncertainty of Energy, “dE”, times the Uncertainty of Time must always be greater than Planck’s Constant.
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------------------------------ dE * dt > 1.05*10^-34 joule-seconds.
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- Either way you look at it our chance of hitting the electron is completely random. However, using Quantum Mechanics we can calculate the “probabilities” of all these factors very accurately.
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- It turns out that the Uncertainty Principle determines the stability of all atoms. It determines the energy levels of all orbiting electrons and assures that the ground state of an atom is 13.6 electron-volts and never zero electron volts, which would make the atom unstable.
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- The Exclusion Principle limits how many electrons can be in each orbit. It therefore defines all the elements that each have a different number of electrons corresponding to the number of protons in the nucleus.
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- The Exclusion Principle says that in an atom no two particles called Fermions can share the same energy state. What is an energy state? Well let’s explain using an energy state for body as you are sitting there.
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- You can measure your energy state using several different instruments, like a digital thermometer, an asphygmanometer, a calorimeter, a stethoscope, a watch, etc. Let’s say your current velocity in that easy chair is zero. Your hear rate is 65 beats per minute, your blood pressure is 120 over 70, breathing rate is 12 breaths per minute. body temperature is 37 degrees C. Your metabolic rate is 200 calories per hour .
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- All of those numbers could illustrate your particular energy state. And, if people obeyed the Exclusion Principle no other person could share your exact energy state. The energy state for a “particle” is also very complicated. And, particles called Fermions do obey the Exclusion Principle.
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- The energy state for a particle can be a particular location, momentum, orbital angular momentum, and spin. Each of these states are quantified. That is they can only have particular values and no other values in between.
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- The Exclusion Principle only applies to Fermions. These include the familiar protons, neutrons, and electrons. The Exclusion Principle does not apply to Bosons. Bosons are the force carriers the most familiar being photons.
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- Photons have no problem overlapping and sharing the same energy state. But, for Fermions the rule is that two Fermions of the same type cannot occupy the same quantum state at the same time. Period.
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- This does not sound too profound but it has tremendous impact to how the world works. In chemistry this principle dictates how many electrons occupy the various energy levels in atoms. This in turn, defines all the elements in the Periodic Table.
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- For every atom only two electrons can occupy the lowest energy level. One electron is spin-up and the other electron is spin-down. Electrons can only have two levels of spin, therefore, a third electron can not occupy the ground state orbit of any atom. The third and more electrons must occupy higher energy states in the atom creating different elements.
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- Together the Uncertainty Principle and the Exclusion Principle determine the size of all atoms. These rules also apply to the size of the nucleus with its protons and neutrons. In fact, these two principles determine the life spans of all the stars in the Universe. For this understanding see the next Review 1028 “ Quantum Mechanics to Astronomy”
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--------------- Other reviews on these subjects;
- 1028 - How Quantum Mechanics applies to Astronomy. Stars and Blackholes.
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- 1032 - The atom’s stability with the Uncertainty Principle. Atom’s ground energy state calculations using the Conservation of Energy.
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- April 9, 2020 1026 1027 2703
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--------------------- Thursday, April 9, 2020 -------------------------
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