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---------------------------- - 2266 - The Birth of Quantum Mechanics
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- Everyone knows that the color of a hot poker iron changes as the iron heats up. First it gets red hot, then it turns yellow, and if the temperature rises even higher it turns “white hot”. Color is the same as frequency of electromagnetic radiation. Color just happens to be in the frequency range of visible light.
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- There is a device called an optical pyrometer that measures temperature by measuring the color, or frequency, of the electromagnetic radiation emitted by a hot object.
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- If you take an object at a particular temperature, and measure the intensity of radiation across increasing frequencies you get a graph, or curve, that has a particular shape. The energy, or intensity of radiation increases as frequency increases which makes perfect sense. The higher the frequency the higher the energy.
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- But, keep measuring at even higher frequencies and something happens that does not make sense. The energy always peaks at a maximum and then decreases with increasing frequency. Also, an objects peak or maximum frequency shifts to shorter wavelengths or higher frequency as it gets hotter. A hotter object emits energy, radiation, at shorter wavelengths.
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- If the object absorbs or radiates 100% of its energy it is know as a “ blackbody”. The curve that you get measuring radiation intensity with frequency is always the same shape, increasing rapidly with frequency, reaching a maximum, and then decreasing more slowly.
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- Plotted on log-log paper the data always plots out this predictable “blackbody curve“. This is not what scientists expect. Higher frequency should mean higher energy, the curve should just continue to increase up and to the right. In the year 1900 no scientists could come up with an explanation. No theory existed as to why blackbody radiation behaved this way.
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- A lot of experiments were made with different materials, different temperatures but the results were always the same. In 1900 Max Planck had lots of data but was frustrated at deriving an explanation. Finally, he just decided to derive a formula that fit the data, since all the curves had the same shape. He ended up adding a fudge factor that became known as Planck’s constant in order to fit the data exactly. This is the formula he came up with:
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- Blackbody radiation as a function of frequency and temperature =
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- 8*pi*h*f^3
--------------------------- Energy = ----------------
- c^3 * (e^hf/kT - 1)
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- Wow! How did he come up with that?
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--------------------------- T is temperature in degrees Kelvin
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--------------------------- f is frequency in cycles per second
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--------------------------- c is the speed of light, 3*10^8 meters / second, or the speed of electromagnetic waves at every frequency.
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-------------------------- k is Boltzmann’s constant = 1.381 * 10^-23 joules / Kelvin, which is the constant that provides the bridge between temperature and energy.
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--------------------------- h is Planck’s constant. This is something new. Planck inserted it as a fudge factor to make the formula fit the data. In order to get a perfect fit the constant had to equal = 6.625 * 10^-34 joule * seconds
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--------------------------- hf/kT is an exponent so it must be dimensionless. Therefore, hf and kT must have the same dimensions so they cancel out.
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--------------------------- hf = kT = joule * seconds * cycles / second = joule * Kelvin / Kelvin
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--------------------------- Joule = Joule Ok, so the exponent is dimensionless.
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- If an exponent is very small then (e^x) can be approximated by (1+x). That is if “hf” is much less than “kT“, then e^hf/kT = 1 + hf/kT -1
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--------------------------Substituting, the energy function =
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- 8*pi*h*f^3 *kT
--------------- Energy = ----------------
- c^3 * hf
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- 8*pi*f^2 *kT
------------ Energy = ----------------
- c^3
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- Planck’s constant cancels out of the equation. At lower frequencies, where hf < kT Planck’s constant disappears. The energy of radiation increases with frequency, f^2, and linearly with temperature, T. This all made sense to the scientists. But, why did it change when the frequencies got even higher? It reaches a maximum and then energy decreases with frequency.
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- What happens to Max Planck’s formula when frequencies get higher? When hf > kT.
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-------------------- e^hf/kT - 1 approximates e^hf/kT
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- 8*pi*h*f^3
----------------------------- Energy = ----------------
- c^3 * e^hf/kT
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- Now the exponential function in the denominator falls off much faster then the polynomial factor, f^3, grows in the numerator, therefore the radiation energy decreases with increasing frequency.
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- Planck’s constant now makes the difference needed for the formula to fit the data. We have increasing energy with frequency, then decreasing energy with higher frequency, we must have gone through a peak or maximum energy. This maximum occurs when hf = kT.
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- At the time, 1900, Max Planck and the other scientists had no idea of the significance of this discovery. In fact, they had discovered quantum mechanics which would end up turning all of physics on its head.
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- Einstein would later come up with the needed explanation that electromagnetic radiation came in quantum or bundles. These bundles, or quantum of radiation of light he called “photons“. Radiation was no longer waves, but it also was also particles. All radiation was in the form of quantum and all matter had a wave - particle duality. This concept of quantum mechanics launched a whole new set of mysteries for the world. But, let’s not go there, let’s try to use what we have already learned.
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- If we observe our Sun the maximum intensity of radiation occurs at 1500 nanometers wavelength, or 2 * 10^14 hertz frequency.
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- The Sun is almost a perfect “blackbody“, meaning a perfect emitter of radiation, so the radiation follows the “blackbody curve“. And, according to Planck’s formula at the maximum or peak intensity:
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- hf = kT
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- So, now we can solve for the temperature of the Sun, T = hf/k. We can measure a star’s surface temperature from the location of the peak of its intensity curve. The intensity of each wavelength emitted determined solely by its temperature.
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- T = 6.625*10^-34 joule * seconds * 2 * 10^14 cycles/second
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- 1.381 * 10^-23 joules/Kelvin
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- T = 9594 Kelvin
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- So, the surface of the Sun is about 10,000 degrees Kelvin.
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- A more accurate approach can be obtained using maximum wavelength rather than maximum frequency for energy density, or radiation intensity.
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- Deriving the temperature where the peak wavelength occurs gives us the formula:
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- T = 2.9 * 10^-3 / max. wavelength
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- The Sun’s peak wavelength occurs at 500 nanometers, in the color blue-green.
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- T = 2.9 * 10^-3 meters*Kelvin / 500 * 10^-9 meters
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- T = 5,800 Kelvin
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- The surface of the Sun is about 6,000 degrees Kelvin. This is closer to the accepted right answer. However, if the peak radiation occurs at 500 nanometers, blue-green in color, why isn’t the Sun blue-green in color. Will it is.
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- What happens is that as the blue-green light comes into Earth’s atmosphere the shorter wavelengths of blue light get scattered turning the sky blue, the longer wavelengths of green, yellow, red get through the atmosphere to your eyes making the Sun look yellow to you.
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- In the early mornings or late evenings when the Sun is low on the horizon the blue-green light has to go through even deeper atmosphere to reach you eyes, more greens and yellows get scattered and the Sun begins to look more red.
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- Planck’s blackbody curve formula can be used for cosmic background radiation released after the Big Bang birth of our Universe. About 100,000 years after the Big Bang electromagnetic radiation was released at 3000 degrees Kelvin. Today this same radiation has moved down the blackbody curve and has cooled down to 3 degrees Kelvin.
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- When satellites measure the intensity of background radiation from 60*10^9 hertz to 600*10^9 hertz, from .5 millimeters to .05 millimeters wavelength, the intensity curve fits Planck’s blackbody curve exactly at a temperature of 2.735 Kelvin.
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- Planck’s fudge factor turned out to be pretty remarkable. So, if you can’t figure out the theory, just make up a formula that fit’s the data, then try to figure a theory as to why the formula works. If it works there must be something to it.
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- Again, it took Einstein to figure out the answer. Radiation comes in bundles, quantum particles, not just waves like everyone at the time believed. Photons were born and so was the birth of quantum mechanics.
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- February 10, 2019. 539 - 2-4-05
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