QII: I, Thermal Radiation and Planck’s Postulate
This is the first post in a series of posts on Quantum Physics. Images and information come directly from a fantastic topic on the matter, called Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles by Eisberg and Resnick.
Here is a rough outline of the topics to be covered:
- Thermal Radiation and Planck’s Postulate
- Photons
- De Broglie’s Postulate
- Bohr’s Model of the Atom
- Schrodinger’s Theory of Quantum Mechanics
- Time-Independent Schrodinger Equations
- Solids — Conductors and Semiconductors
- Nuclear Decay and Nuclear Reactions
In this post, let’s examine the following sections:
- Introduction
- Thermal Radiation
- Classical Theory of Cavity Radiation
- Planck’s Theory of Cavity Radiation
- The Use of Planck’s Radiation law in Thermometry
- Planck’s Postulate and its Implications
- A Bit of Quantum History
I. Introduction
Max Planck started a revolution in physics by publishing a little-known paper in 1905 titled, “On the Theory of the Energy Distribution Law of the Normal Spectrum.”
So began old quantum theory, of which we will explore the next three blog posts on. In a nutshell, quantum physics examines the smaller dimensions of our universe. Just as in how relativity relies on the constant speed of light, quantum physics relies on the Planck’s constant.
Where does Planck’s constant come from?
Max Planck derived this constant from characterizing a certain type of energy called thermal radiation.
Why does this matter?
Max Planck allows us to see that energy is a discrete property. In classical physics, energy is considered infinitely divisible. Through quantum physics, we understand that energy holds a discrete value. For example, understanding this concept has helped astrophysicists see the origin of the universe.
Come marvel at the beauty of this quantum adventure by examining further what thermal radiation is.
II. Thermal Radiation
What is heat? Why is a cup of coffee warm?
Heat comes from the motion of molecules that are vibrating. The kinetic energy (energy from motion) turn into heat energy when bonds break as the object, say cup of coffee, seeks order.
In addition, heat leaves off a certain type of radiation called thermal radiation. All bodies such as people, dogs, cats, and coffee release thermal radiation and absorb it from their surroundings. When thermal equilibrium is achieved, an object takes in and releases out just as much thermal radiation.
When an object is very hot, they turn self-luminous, such as objects that glow in a dark room. Perhaps in this dark room, there is an iron fire poker. Sticking this fire poker into the fire, it heats up. At relatively low temperatures, the iron poker is certainly hot but it doesn’t appear so. As you leave it in, it gets hotter and hotter until you can visibly macroscopic changes. The poker has a dull red color, then a brighter red color, then an intense blue-white color.
Increasing the temperature of an object allows for the object to emit more thermal radiation and the frequency of the most intense radiation becomes higher.
How does one measure this?
Using an optical pyrometer. This is essentially a spectrometer that gives out the temperature of a star based on the color (hence frequency) of the thermal radiation that it emits.
There is an idealized object, called blackbodies, that absorb all thermal radiation incident upon them and do not reflect it back. One interesting characteristic of blackbodies is that all blackbodies at at the same temperature release the same spectrum.
How this spectrum is distributed is specified by the quantity R_t(v), which is called the spectral radiancy. This is defined so that R_t(v) is equal to the energy emitted per unit time in radiation of frequency in the interval v to v + dv from a unit area of the surface at absolute temperature T.
The above graph indicates 6 important features of a blackbody.
- When the frequency (v) is lower than 10¹⁴ Hz, there is very little power radiated
- As the frequency increases , the power increases
- For 1000 K, this increase in power maximizes at around 1.1 × 10¹⁴ Hz. At this frequency, the power radiated is most intense.
- As velocity increases past this maximum, the radiated power drops slowly but continuously — and is zero again as frequency approaches very large values
- As you increase the temperature to 1500 or even 2000 K, the frequency at which the maximum power is achieved becomes greater and greater. There is a linear relationship between temperature of a blackbody and its maximum frequency
- The total power radiated can be found through integral calculus:
One can additionally calculate the spectral radiancy by the relationship between a a constant and its temperature.
In addition, the noted relationship of temperature progressing with maximal frequency is summarized by Wein’s Displacement Law.
Experimentally, it has been shown that by drilling a cavity into a blackbody-like object, the radiation inside that cavity whose walls are at temperature T has the same character as the radiation emitted by the surface of a blackbody at temperature T.
The set up is seen below. The object, a blackbody like object, has a hole drilled into it. The walls of this blackbody absorb light that enters through the hole after it bounces around.
The flip side is this. If we heat the walls of the cavity such that they release thermal radiation, then the hole will emit thermal radiation. Since the cavity has properties similar to a blackbody surface and can also emit temperature, it is a good way to produce a blackbody spectrum by means of a cavity in a heated body with a hole to the outside.
III. Classical Theory of Cavity Radiation
A blackbody, according to the classical theory of the Rayleigh-Jeans formula, follows a proportional relationship between its energy density and the square of its frequency. As one increases its frequency, its intensity will astronomically increase, meaning that the total energy under the curve would also be infinite and it would be infinitely bright at large wavelengths such as UV light. This meant that the power emitted by blackbodies could be infinite!
This relationship couldn’t be right.
Experimental results supported the idea that the Rayleigh Jeans formula was wrong. The power emitted by blackbodies certainly couldn’t be infinite — it had to be finite for sure!
This discrepancy of experiment and theory was called the ultraviolet catastrophe.
IV. Classical Theory of Cavity Radiation
Planck fixes the ultraviolet catastrophe by showing that the energy of electromagnetic standing waves was discrete instead of a continuous quantity.
Planck’s theory showed some new ways of thinking about energy. One of his findings: as one increases the temperature, there is a decrease in the wavelength at which there is a maximum.
V. Use of Planck’s Radiation Law in Thermometry
Why does this matter?
Using blackbody radiation data, three scientists by the names of Dicke, Penzias, and Wilson in the 1950s showed that there is a spectrum of electromagnetic radiation (with a temperature of around 3K) uniformly striking on the Earth with equal intensity. This is considered strong evidence to support the Big Bang Theory.
VI. Planck’s Postulate and Its Implications
Planck postulated this (in laymen’s terms):
Energy is of a discrete value, whereby this relationship is satisfied:
VII. Quantum History
To summarize, an exploration of blackbodies allows us to see where classical physics fails.
Classical physics failed when we tried to understand energy. Experimental evidence indicates that energy is made of discrete values and that therefore, electromagnetic waves are themselves quantized.
Quantized energy levels depend on a constant called Planck’s constant, h. Plack himself was unsure of whether or not the quantum explanations of physics was correct in the sense of being more than an “act of desperation” as he put it.
Planck, in 1905, while doubting his own work, came upon Einstein’s paper on relativity and Planck became one of Einstein’s fervent supporters. This is despite the fact that Einstein would build upon Planck’s theory of quantized energy — one that Planck, at one point, doubted himself.
