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---------------------------- - 2358 - Life is Uncertain
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- No really. It is.
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- It is a physical thing. Right down to the atomic level. Cause and effect can not be certain. In fact, there is a mathematical relationship that describes this degree of uncertainty. It is called the Heisenberg Uncertainty Principle.
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- This uncertainty is not intuitive. We think that we can measure mass and velocity with as much precision as our measurement instruments will allow. However, there is a physical limit, or at least a “physics” limit.
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- We can only measure mass and velocity as precise as nature will allow. When you get down to the smallest dimensions the measurement itself always disturbs the system. Ultimately the measurement becomes part of the system and can not be separated.
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- We say we can measure mass to be so many grams plus or minus a certain amount of uncertainty in grams. The same with measuring velocity in meters per second. Distance in meters and time in seconds can only be measured with a certain amount of accuracy, or uncertainty, depending on the precision of the instruments used in the measurement. To get a more accurate measurement you simply need a more accurate instrument.
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- At the atomic level of measurement you get down to the photon as the carrier of any measurement information. If you are trying to measure an electron the photon will bump into the electron and bounce back telling you the electron is in one place but it just bumped it into a different place and you no longer know exactly where the electron is or how fast it is moving.
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- If you are trying to measure the electron’s velocity when the photon bumps into it the velocity will change. What the photon tells you when it comes back will not be the right answer.
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- We can not measure more precisely than the wavelength of the radiation we use to locate it. Visible light has a wavelength of 0.0005 millimeters. Sound has a wavelength of one meter.
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- So how can a bat catch a bug if his measurement error is one meter? That is why the bat uses a high pitched squeak. Bats use 35,000 cycles per second in the search mode, giving them 1 centimeter wavelength resolution. When they go into the kill mode they go up to 90,000 cycles per second, which gets them to 4 millimeters of their target.
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- There is a formula that describes the uncertainty trade off being made between knowing the momentum, which is mass times velocity, and the position of an atomic particle. Position is a distance measurement. An action is momentum moving over some distance. Therefore, Planck’s Constant becomes the minimum possible uncertainty of any atomic action:
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------------------- Momentum Uncertainty * Position Uncertainty >= Planck’s Constant
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------------------- Momentum Uncertainty * Position Uncertainty >= h / 4 * pi
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------------------- A photon’s momentum = Planck’s Constant / Wavelength
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------------------- h is Planck’s constant. It is a very small number.
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-------------------- h = 4.136^10-15 electron-volt-seconds
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------------------- Or, h = 6.625 * 10^-34 kg * m^2 / seconds
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------------------- Pi is 3.1416 from the geometry of a circle
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------------------- Uncertainty is measured in standard deviations
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------------------- Momentum Uncertainty * Position Uncertainty >=
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----- 0.00000000000000000000000000000000005272 kilogram * meter^2 / seconds
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- Standard deviation is the difference between observation and the true value ( or, the nearest known value). It is mathematically the square root of the average of squares of the deviations of all the measurements.
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- The uncertainties are the range of results you get when you make the measurement over and over again. For example, let’s measure the book on your desk. You get 23.6 centimeters. You measure it again and get 23.7 centimeters. Since the markings on your ruler are only 1 millimeter intervals repeated measurements will be between 23.5 and 23.7 centimeters with 23.6 as the average value after many measurements. The measurement uncertainty is plus or minus one millimeter.
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- The distribution of measurements will be a normal distribution. Sometimes this is called the Bell Curve. It is the same distribution you expect to get when you grade your tests at school. You create the bell curve of the class’s distribution of scores before you divide up the scores between A,B,C,D,F. I always wondered why the left out the E? A normal distribution is considered a random distribution.
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- Mathematically, one standard deviation about the class average encompasses 68% of the scores. Two standard deviations around the average takes in 95% of the scores and three standard deviations 99.7% of the scores. If you are not within three standard deviations of the class average you are pretty much off the curve.
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- With these definitions, let’s go back and look at the Uncertainty formula:
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------------------- Momentum Uncertainty * Position Uncertainty >= h / 4 * pi
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--------------(Momentum Uncertainty * Position Uncertainty) - (Momentum Uncertainty *
Position Uncertainty) does not equal zero, it equals Planck’s Constant
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-------------------- Position = distance measurement
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-------------------- Momentum = mass * velocity = mass * distance / time
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-------------------- Action = distance * momentum = distance * mass * distance / time
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-------------------- Photon’s momentum = Planck’s constant / wavelength
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-------------------- The Uncertainty Principle and the wave properties of light are two expressions for the same thing. But, it applies to all particles, not just light.
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- You should be able to subtract something from itself and get zero. But not, at the atomic scale. Here you always get this small constant Planck number.
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- Let’s say we measure the position of an electron with great accuracy, so the Position Uncertainty is a very small number. Since Position Uncertainty is in an inverse relationship and the Momentum Uncertainty becomes a very large uncertainty in order for their product to remain the small Planck‘s Constant.
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- In fact, if your position measurement gets so accurate approaching zero uncertainty the Mass*Velocity Uncertainty becomes infinitely large. Once you approach infinity, momentum uncertainty becomes completely undefined.
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- The mass of an electron may remain 9.9 * 10^-31 kilograms, but you have completely lost track of its velocity, and therefore its momentum.
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- All measurements must involve some exchange of energy, some exchange of information. It is a little like measuring the tire pressure on your bicycle. When you get the pounds per square inch you lose a little air due to the use of the tire gage. So, the pressure in the tire ends up being a little less in order to determine what it is.
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- A thermometer measuring temperature warms up in order to take a measurement thus removing a little heat from what it is measuring. These measurement errors are so small you can ignore them. However, at the atomic level they have a significant affect.
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- The effect on a single electron is enormous. If we were to measure the position of a free electron to within the width of an atom, the momentum uncertainty would instantly fling the electron away at such a high speed in one second the electron could be anywhere in several miles radius.
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- When you delve into this uncertainty relationship you run into some very strange things. One result is that you can not absolutely predict where a particle will be with 100% certainty.
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- A mathematical description of position of a particle must always be given in probabilities. The electron is in this position with 99% probability. But, there is still a 1% probability that it will be somewhere else.
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- Momentum and position are a pair of trade-offs. You must always trade-off one against the other when making precise atomic measurements. This same trade-off relationship occurs between measuring energy and time, or mass and time, since Einstein showed that mass and energy were different forms of the same thing.
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------------------- Energy Uncertainty * Time Uncertainty = h / 4 * pi
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- We use probabilities to describe these trade-offs in uncertainties, and that works for modeling and describing what is actually going on. But, scientists are not certain this is how the Universe actually works.
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- We can model it with probabilities but maybe there is a deeper understanding still needed to explain how the Universe is working in this probabilistic manner. The effects of Uncertainty Principle are not apparent with large systems because Planck’s constant is extremely small. However, at the atomic scale it becomes fundamentally important in understanding the behavior of atoms.
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--------- The diameter of electron is 0.000000000000005.6 meters (5.6*10^-15 m).
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- The Planck length is 0.0000000000000000000000000000000001 meters (10^-34 m)
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- The Planck time interval is 0.000000000000000000000000000000000000000001 seconds ( 10^-42 sec.)
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- So, you see we are dealing with some really small numbers here.
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- The Uncertainty Principle is used to explain the inherent width of spectral lines ( See Review 36 “The Electromagnetic Spectrum” ). If the lifetime of a particular atom is very short in the excited state there is a large uncertainty in its energy and the spectral line is broad.
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- In contrast if the lifetime in the excited state is long the energy uncertainty is small and the spectral line is fine.
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- Stephen Hawkings used the Uncertainty Principle to show that blackholes eventually evaporate and have a finite lifetime. The principle was used to show why White Dwarfs and Neutron Stars do not collapse into a single point. The Uncertainty Principle keeps the electrons applying an outward pressure that prevents the collapse.
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- What happens to gravity in these small dimensions? There is a fifth grader somewhere that will probably be telling us the answer to this question in another 20 years.
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- Professor Richard Feynman said,” If anyone claims to know what the quantum theory is all about, they haven’t understood it.”
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- The Heisenberg Uncertainty Principle is named after its discovery by Werner Karl Heisenberg who was a German physicist born in 1901. His father was a history and humanities professor.
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- Werner got his Ph.D. from University of Munich in 1923. At that time there was much debate about the structure of the atom and whether to treat electrons and particles of waves. Physicists were studying the elements using spectral lines. No one at that time could explain how the spectral lines were created.
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- In 1927 Werner went on vacation to a North Sea island to escape the discomfort of his hay fever. While on vacation he developed the mathematics that define the wavelengths of an elements spectral lines. Shortly after that he developed the uncertainty principle which states the uncertainties of the determination of momentum and position, when multiplied yielded a value approximately that of Planck’s constant.
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- His principle of uncertainties and the need to use probabilities to explain natural events threw a big uproar in the scientific community. Tradition had taught scientists that there was a direct relationship between cause and effect. Werner’s principle was counter to that. Even Einstein had problems with using probabilities. As he said, “ God does not play dice” to show his distrust in these theories.
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- Heisenberg was awarded the Nobel Prize in Physics in 1932. He worked for the Nazi’s in WWII and was in charge of the German research on the atomic bomb.
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- In February 1940 Heisenberg had completed a full report on how to construct a workable atomic bomb. It is an amazing story how the allies prevented him from being successful ( Let me know if you want to learn it).
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- After the war he became the director of the Max Planck Institute for Physics.
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- May 8, 2019. 40
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--------------------- Wednesday, May 8, 2019 -------------------------
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