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Planck's Constant: Motion and Rest

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Dr. Eric Gaze
Director of the QR Program
Bowdoin College

Numb:Planck's constant is given by h = 6.55 x 10-27erg sec.

Number:The energy of electromagnetic radiation is proportional to the frequency with the constant of proportionality given by Planck's constant: h = 6.55 x 10-27erg sec.

Num-best: The energy of electromagnetic radiation is proportional to the frequency with the constant of proportionality given by Planck's constant: h = 6.55 x 10-27erg sec . This fundamental constant of nature was first hypothesized in 1900 along with the assumption that the energy exists only in discrete bundles or "quanta", leading to the quantum theory.

The 20th century may best be remembered for the discovery of the quantum theory, which overturned the classical Newtonian world-view which had held sway for two hundred years. Quantum theory is popularly known for such provocative statements as: "There is no objectively defined reality out there, the universe seems to require us to interact with it before it can come into being." The origins of this theory lie in a paper written by Max Planck in 1900 on the energy of a heated radiating body in which he surmises the existence of a proportional relationship between the vibrational frequency of the heated elements and the resulting energy given off. The resulting constant of proportionality, h, has since attained the exalted status of one of the fundamental constants of nature; right up there with π, c, e, etc. Wow!

Physicists have long struggled with the concepts of motion and rest. The ancient Greek philosphers, who gave us the word physics from "physis" meaning the essence of being, had differing views on the subject. Heraclitus taught that all is change, a constant interplay of dynamic opposites, while his contemporary Parmenides argued that change was an illusion of the senses and all reality emanated from an unchanging ground of being. Heraclitus was in essence a Greek Taoist and interestingly was alive at roughly the same time as Lao Tzu was teaching about the Yin and Yang in China. Parmenides is perhaps best known for his disciple Zeno, whose paradoxes sought to illustrate the impossibility of motion and are still a crowd favorite in Calculus classes. Aristotle taught that the natural state of an object was at rest and thus motion was the result of "pushing," which makes sense for a cart being pushed but not so helpful for a rock thrown in the air. It would take 2000 years before physicists would begin to rethink these ideas.

In 1666, an ominous year for numerologists, Isaac Newton demonstrated that visible light was actually a spectrum of many colors from blue to red by shining light through a glass prism. In doing so he ironically sowed the seeds that would lead to the eventual downfall of his own theory.

Electromagnetic Spectrum

At this time he hypothesized that light was comprised of particles which he called "corpuscles." If light is made of particles then shadows should have very sharp edges. The simple fact that shadows have fuzzy edges led Huygens to offer the alternate hypothesis that light is actually a wave, since water waves when hitting a object don't sharply break off at the edge of the object but overlap into smaller (fuzzier) ripples. This wave theory of light gradually held sway as more experiments confirmed that light exhibits other wave-like properties. Thus the spectrum of light came to have a horizontal axis of units giving the frequency of the wave associated to each color. Each wave has a wavelength, the distance between two successive peaks of the wave, and the frequency is the number of wavelengths that pass by a fixed point per second.

The spectrum of light was extended to matter in the 1800's when scientists began heating elements and analyzing the light spectra given off. Interestingly the spectrum of light for each element has unique "signature" of alternating light and dark lines just like a bar code on items you purchase.

Helium Spectrum

This unique signature allowed scientists to analyze the light from our sun and other stars and determine what elements were emitting the light.

In 1900 Max Planck was extending this idea to the study of "black body radiation." By painting a metal box black on the inside and outside, drilling 1 small hole into the box, and then heating it, you can study the continuous spectrum of light given off through the small hole. For a given temperature the intensity or brightness of light along the color spectrum had a tell tale curve, and Planck was attempting to derive an equation for this curve, Intensity as a function of frequency, ν, or wavelength, λ and temperature, T.

Planck's Law

In order to fit his equation to the experimentally observed data Planck had to make an assumption about the "vibrational energy," U, of the heated black body:

"Moreover, it is necessary to interpret U not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts."

He referred to each such part as an "energy element, ," and it was only later that scientists gave the attribute "quanta" leading to the "quantum theory."

Planck used his hypothesized energy element, ε, to then derive an equation for the entropy of the black body:

Planck Equation1

The entropy he proceeded to show was a function solely of
Planck Equation 2
and when compared to the first equation for entropy he concludes:

"we find that the energy element ε must be proportional to the frequency ν, thus: ε = hν

The constant of proportionality, h, between the energy element and the frequency is what we know as "Planck's constant." Like most physical constants, units are important, Planck computed h = 6.55 x 10-27erg sec , a very small number with units of energy times time or "action."

Using the proportional relationship he was then able to derive an equation for "energy density, u," now known as "Planck's Law":

Planck Equation 5

Note that this is an equation of the with only 2 variables, temperature, T, and frequency, . All the other letters are fundamental constants. Holding temperature constant this equation is essentially of the form:
Planck Equation 6
a family of curves we can easily plot on our graphing calculators. These graphs are perfect fits to the experimentally determined energy intensity curves shown above, and the reason why Planck's assumption about energy quanta was not ridiculed. Planck himself, however, did not seem to think much about his assumption instead referring to it as a "mathematical necessity" to make the equation come out right.

The true physical significance would have to wait another 5 years for Einstein. In 1905 the photoelectric effect was a mystery. Physicists could shine a light on a metal and induce an electric current, but only when shining light near the blue end of the spectrum. Infrared light did not induce a current! To add to the confusion, increasing the brightness of the ultraviolet light did not increase the voltage as was expected; while making the light bluer did increase the energy of the ejected electrons. Einstein unraveled the mystery (while working at the patent office) by going back to Planck's constant and the energy quanta. Einstein hypothesized that light traveled in discrete packets, or energy bundles, with energy proportional to the frequency of light as in Planck's constant. Thus blue light with a higher frequency has enough energy to dislodge electrons. Red light with its long wavelength and low frequency does not have enough energy to knock out electrons from the metal.

Thus we come full circle to Newton's light "corpuscles," now called photons, and the quantum theory was born. Both Planck and Einstein would receive Nobel prizes for their role in this story, Planck for his energy quanta idea in 1918, and Einstein for the photoelectric effect in 1923. Planck's constant would eventually find its way into one of the most celebrated equations of physics, Heisenberg's Uncertainty Principle, in 1927. This principle states that it is impossible to measure the exact position and velocity of a particle at the same time. Physicists had guessed that if wave-like light can exhibit particle-like behavior then maybe particles could exhibit wave-like behavior. The velocity of the particle is associated to the wavelength, but the position is associated to a sharp peak in the wave which will then have indeterminate wavelength. Thus isolating the position of a particle "washes out" the wavelength, while measuring the wavelength requires no sharp localized peaks. It is physically impossible to measure both at once with perfect accuracy, leading to the inequality

Planck Equation 7

.

The product of the uncertainty in position and momentum must be greater than or equal to Planck's constant over 2π. Thus motion and rest are fundamentally inseparable with one being the Yin to the other's Yang. Heraclitus would be pleased.

Sapere Aude!



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