120 years ago, on the fateful day of Dec 14, 1900, Max Planck presented one of the most revolutionary papers in the history of humankind, the paper that gave birth to the theory of quantum mechanics and changed our perception of the observed universe at the microscale in the most unexpected manner. Max Planck presented his paper titled Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum, explaining the theory of the Law of Energy Distribution in Normal Spectrum at a meeting of the German Physical Society in Berlin.

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Max Karl Planck, the father of quantum mechanics

But what was exactly there in the paper? Was it just the work of a single mind? Did the revolutionary idea take its form overnight? Well, to answer all these questions, let’s travel back in time!

By the end of the 19th century, we had pretty much figured out everything. We had Kepler’s law to explain the motion of planets. We could explain the waves on the strings. The ripples in the ponds had an explanation, and we even had developed the relationships between electricity and magnetism. But whenever we think that we have an explanation for everything, nature throws new problems at us. And this is exactly what happened. In the second half of the nineteenth century, humanity had started its voyage to understand the interaction between matter and radiation.

What is a Blackbody and the Blackbody Spectrum?

According to Kirchhoff’s law, every object absorbs the radiations that are falling on it. Then, it also emits radiation. The amount of radiation emitted and absorbed depends upon the type of the object. An object that absorbs all the radiation falling on it and then emits it when heated is termed as a black body. Perfect blackbodies do not exist in nature – we only have approximations. The Sun is a blackbody, stars are black bodies, and even a light bulb can be considered a blackbody.

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Black body curve (Image: esfsciencenew)

The graphical representation of the intensity of emitted radiation of varying energies or wavelengths emitted by a black body at some constant temperatures is known as the black body spectrum. In other words, the blackbody spectrum is a curve between the energy density (energy emitted per unit time per unit volume) of the radiation and its frequency or wavelength. So, now the question was to provide a mathematical explanation of the blackbody spectrum. This is where things got complicated, as classical physics could not explain the blackbody spectrum’s shape.

Experimental laws obtained from black body spectrum:

While observing the blackbody spectrum, we came across two experimental laws associated with a blackbody. It was observed that the wavelength at which the spectrum showed a peak varied with the blackbody’s temperature. At lower temperatures, the curve showed a peak at far higher wavelengths, whereas, at higher temperatures, the peak occurred at lower wavelengths. In other words, the temperature at which the energy density of blackbody radiation peaks up is inversely proportional to the wavelength. This law is known as Wein’s displacement law. The fact that hotter stars appear to be blue while the cooler ones appear to be reddish is well explained by this law.

The second law deduced from the graph is Stefan’s law. It states that the total energy output of a blackbody or the total luminosity is proportional to the blackbody’s fourth power of temperature. Stefan’s law has many applications in astronomy.

You can find the complete explanation with mathematical steps here.

Attempts to explain Blackbody spectrum Classically:

Theoretically, to explain the observed blackbody spectrum, we needed a function between the energy density, wavelength, and temperature of the blackbody that could fit the blackbody spectrum at all the temperatures and frequencies or wavelengths. But, the job wasn’t just a simple curve-fitting exercise. It wasn’t as straightforward as expected.

Wein’s distrubution law:

Wilhelm Wein made the first attempt to explain the spectrum. He gave an empirical formula known as Wein’s distribution law to explain the blackbody curve. Wien considered the wavelength of black body radiation and then combined it with the Maxwell–Boltzmann distribution for atoms to obtain his relation. Although Wein’s relation did a commendable job in explaining the blackbody curve at high frequencies (low wavelengths), unfortunately, it failed to explain the spectrum’s behavior at low frequencies (high wavelengths).

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Wein’s Distribution Law

Rayleigh and Jeans law:

After Wein’s partially successful attempt, the next pair to land in the battleground was Rayleigh and Jeans. Rayleigh and Jeans came up with a brilliant idea to explain the curve. They considered the blackbody radiation to be made of standing waves inside a cubical box. Using classical statistical mechanics, they calculated the number of vibrations’ modes or the number of standing waves inside a frequency interval. They associated average energy to each of the standing waves as kT where k is the Boltzmann’s constant and T is the temperature.

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By multiplying this average energy with the number of modes, they got the energy density in terms of temperature and frequency, an empirical relation known as the Rayleigh-Jeans law. However, the Rayleigh-Jeans law suffered a major setback. Contrary to Wein’s distribution law explaining the curve at high frequencies only, the Rayleigh-Jeans law could only explain the black body spectrum at low frequencies. It blew up at high frequencies, a shortcoming also known as the Ultraviolet Catastrophe.

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Rayleigh Jeans Law

The final call: The Planck’s Law:

After the partial successful attempts made by Wein, Rayleigh, and Jeans, we needed something that could explain the blackbody spectrum completely at all the frequencies, without any shortcomings. And this is where Max Planck’s brilliance came to the rescue. Planck extended Rayleigh and Jeans’ idea. However, he assumed that the standing waves could only have a particular energy, in the multiples of hv where h is the Planck’s constant and v is the frequency.

Using this assumption, Max Planck mathematically derived and came up with Planck’s radiation law that could easily explain the shape of the black body spectrum in all the regimes. Moreover, we could also derive the Rayleigh-Jeans Law, Wein’s law, and Stefan’s law from Planck’s radiation law, which further confirmed Planck’s law’s accuracy.

All the mathematical steps to derive Planck’s law are given in the video below:

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Planck’s Law

In this way, the black body spectrum was finally explained by Max Planck using quantum physics, and this is what was there in Planck’s paper presented on December 14, 1900. Planck’s theory held that radiant energy is made up of particle-like components, known as “quantum,” and in this way, quantum mechanics came to existence! But surprisingly, Planck was himself not convinced with the prowess of the idea he had put forwarded.

In 1905, Einstein used Planck’s theory to explain the Photoelectric effect. Planck’s theory helped to resolve several previously unexplained natural phenomena, such as the behavior of heat in solids and the nature of light absorption on an atomic level. Taking inspiration from Planck’s discovery, many other breakthrough discoveries were made in the 20th century, which gave new dimensions to the microscopic world.

In 1918, Planck won the Nobel Prize in physics for his work on blackbody radiation. Although Planck made many contributions to theoretical physics, his fame as a physicist rests primarily on his role as the originator of quantum theory, which revolutionized human understanding of the atomic and subatomic process.

Learn Quantum Mechanics At Home For Free

On our YouTube channel, we have started the ‘Introduction to Quantum Mechanics’ series. It starts from the very definition of quantum mechanics and includes the mathematics required to understand the concepts. Watch the first video below.

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