Disclaimer: A mere mortal’s attempt to understand the cosmos.
I will strive to explain this in a lexicon I am most comfortable with, as this subject has both perplexed and enlightened me simultaneously. My journey here has been driven by a lifelong sense of adventure, a quest to understand the tale of the unknown, a symphony written in the language of mathematics.
My goal is not to simplify or complicate, but to articulate this subject as effectively as possible, drawing upon knowledge gained through academia, library research, and countless discussions with a dear mathematician friend from Caltech. Lastly, before we begin, I will also incorporate the story behind the math, as it was this narrative that personally hooked me to the math behind this story.
Thread I: The Ultraviolet Catastrophe
This discussion begins with a black body and its radiation, known as 'Black-Body Radiation.' A black body is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence."
The Sun, while not a perfect black body, is a very good approximation. Its emission spectrum closely resembles that of a black body, which is a continuous spectrum. In this spectrum, the spectral radiance (the radiant power per solid angle per area per wavelength) is a function of the body's temperature.
In 1900, Lord Rayleigh derived a law, using classical physics arguments, to describe the spectral radiance of a black body at a particular temperature. Sir James Jeans introduced further derivations in 1905, and the combined result became known as the Rayleigh–Jeans Law. This law semi-accurately predicted observational outcomes at higher wavelengths (lower frequencies). However, as the wavelength decreased and the frequency increased, approaching the ultraviolet spectrum, the mathematical model failed. The predicted energy output diverged towards infinity, whereas actual measurements of black-body radiation showed that the energy radiance, after reaching a peak, decreased and remained finite.
“This breakdown in the arguments of classical physics is called the ‘Ultraviolet Catastrophe.’”
Rayleigh–Jeans Law & the UV Catastrophe
Thread II: Plank's Law & Constant
And so begins the introduction of Max Planck, a German theoretical physicist whose area of expertise was thermodynamics. In 1894, he was commissioned to research how to increase the efficiency of electric bulbs, which had been introduced a couple of decades prior. Over time, his attention became fixated on black-body radiation. He deeply studied the Rayleigh–Jeans law and, in an act of desperation (as he described it), resorted to the then-controversial theory of statistical mechanics, the probabilistic models bridging the gap between microscopic and macroscopic states, pioneered by Ludwig Boltzmann.
This led to Planck's Law, which accurately described the ultraviolet catastrophe and explained why the energy radiance decreases and remains finite after reaching a peak. Planck theorized that energy was not transferred continuously, but in discrete chunks, later known as 'quanta,' given by the equation E = nhν, where 'h' is Planck's constant, ν is the frequency of the photon, and n is a natural number. With this new condition, Planck imposed the quantization of energy, stating, in his own words:
“A purely formal assumption... actually I did not think much about it.’”
Planck’s Radiation Law
Thread III: Photoelectric Effect
Jump to 1905, Albert Einstein publishes a paper that advanced the theoretical idea of energy 'quantization,' building upon Planck's work, through his explanation of the Photoelectric Effect. This experiment demonstrated that electrons are emitted from a thin metal sheet only when the light's frequency exceeds a certain threshold, regardless of its intensity, implying that energy is delivered in discrete packets (photons). The accuracy of Einstein's model was further reinforced by Robert A. Millikan's 1914 experiments, which yielded precise values for Planck's constant using the photoelectric effect.
Thus began the Age of Quantum Mechanics. A journey initiated by questioning classical mechanics and analyzing phenomena through statistical mechanics. With a hunch and a desperation to explain the unknown, it mysteriously led itself to its intended destination.
Thread IV: Note to Future Self:
None that I can think of, at this time.