The Quantization of Energy
According to 19th-century physics, the atoms that make up your body should not exist. The laws that predicted planetary orbits and electromagnetic waves also predicted that every atom should collapse in less than a billionth of a second. The fact that you are here to read this sentence is, by classical physics, impossible.
The solution to this paradox required rethinking how energy works at the atomic scale. That solution, the quantization of energy, is the idea that separates modern chemistry from classical physics.
The Staircase and the Ramp
Consider the difference between a staircase and a ramp.
On a ramp, you can stand at any height. You can be at 1 meter, 1.5 meters, 1.0001 meters, or any value you choose. The ramp is continuous. This is how 19th-century physicists believed energy worked. An electron orbiting an atom could have any energy value, just as a ball rolling on a ramp could stop at any height.
On a staircase, you can only stand on the steps. You can be on step 1, step 2, or step 3, but you cannot stand between steps. The staircase is discrete. This is how energy actually works at the atomic scale. Electrons in an atom can only occupy specific energy levels, like standing on specific steps of a staircase.
Why the Staircase?
The staircase is not arbitrary. It emerges from something fundamental about how waves behave when confined to a small space.
Think of a guitar string. When you pluck it, the string vibrates, but not just any vibration is possible. The ends of the string are fixed to the guitar, so the vibration pattern must have zero motion at both ends. This constraint limits which wavelengths can exist on the string. The string can vibrate with one arch, two arches, three arches, but never one and a half arches. The number of arches must be a whole number.
These stable vibration patterns are called standing waves. They are the only patterns that “fit” between the fixed boundaries. All other patterns would require the string to move at the ends, which it cannot do. The allowed patterns are discrete, not continuous.
Electrons in atoms behave like waves, and they face a similar constraint. An electron is confined to the region around the nucleus. Just as a guitar string can only sustain certain vibration patterns, an electron wave can only exist in certain configurations that fit within the atom. Each allowed configuration corresponds to a specific energy. The electron cannot have an energy between these allowed values for the same reason a guitar string cannot vibrate with one and a half arches: that pattern simply does not fit.
This is the origin of the staircase. The steps are not imposed by some external rule. They emerge naturally from the wave nature of matter combined with confinement. Erwin Schrödinger and Werner Heisenberg worked out the full mathematical description in the 1920s. But the essential insight, that energy comes in discrete amounts, was Planck’s contribution in 1900.
The Failures of Classical Physics
In the late 1800s, physicists had two powerful theories: Newton’s mechanics and Maxwell’s electromagnetism. These worked well for everything from cannonballs to radio waves. But when applied to atoms, the predictions were wrong.
The Collapsing Atom
When physicists discovered that atoms contain electrons orbiting a nucleus, they immediately saw a problem. An orbiting electron constantly changes direction, which means it is accelerating. Maxwell’s equations state that any accelerating charge must radiate energy. So an orbiting electron should continuously lose energy, spiraling into the nucleus in about 10−11 seconds.
In the ramp model, this makes sense. The electron can smoothly slide down the energy ramp, radiating energy continuously until it crashes into the nucleus.
But atoms do not collapse. Matter is stable. The ramp model is wrong.
In the staircase model, an electron on the lowest step has nowhere lower to go. It cannot continuously radiate energy because there are no “in-between” positions to slide through. The electron is stuck on the bottom step, and atoms are stable.
The Ultraviolet Catastrophe
When objects are heated, they glow. A warm stove glows red; a hotter furnace glows orange; the sun glows white. Physicists wanted to predict exactly how much light of each color a hot object emits.
Using the ramp model, they calculated that as wavelength gets shorter, the amount of radiated energy should increase without limit. A hot object should emit infinite energy in the ultraviolet, X-ray, and gamma-ray portions of the spectrum. Every warm object should be a lethal radiation source.
This prediction, called the ultraviolet catastrophe, was obviously wrong. Hot objects emit most of their radiation at a specific wavelength that depends on temperature, then the emission drops off at shorter wavelengths.
The Mystery of Atomic Spectra
When gases are heated or electrified, they glow. But unlike hot solids, which emit a continuous rainbow of colors, gases emit light only at specific, discrete wavelengths. Hydrogen, for example, emits a particular red, a particular blue-green, and a particular violet, but nothing in between.
The ramp model predicts a continuous smear of colors as electrons spiral inward, losing energy smoothly. It cannot explain why atoms have specific “color fingerprints.”
The staircase model explains this. When an electron jumps from a higher step to a lower step, it releases exactly the energy difference between those two steps. That specific energy corresponds to a specific color of light. Different elements have different step spacings, so they emit different colors.
The Path to a Solution
Before the ultraviolet catastrophe was resolved, physicists developed several approaches to understanding blackbody radiation, the light emitted by an idealized perfect absorber and emitter of radiation.
Wilhelm Wien developed an approximation that accurately described the radiation at short wavelengths (high frequencies) but failed at long wavelengths. Lord Rayleigh and James Jeans derived a formula from classical physics that worked at long wavelengths but diverged to infinity at short wavelengths, producing the ultraviolet catastrophe.
The figure below shows why this was such a crisis. The Rayleigh-Jeans law, derived from classical physics, predicts that radiated energy should increase without bound as wavelength decreases. Wien’s approximation captures the short-wavelength behavior but fails at long wavelengths. Neither could describe the full spectrum.
Spectral radiance measures how much light a surface emits at each wavelength. The units MW·m−2·sr−1·nm−1 break down as:
- MW·m−2: megawatts per square meter of emitting surface
- sr−1: per steradian (a unit of solid angle, like a 3D slice of directions)
- nm−1: per nanometer of wavelength
The “per nanometer” part is what makes it spectral: it tells you the intensity at each specific wavelength, not the total across all wavelengths.
Comparison of three approaches to blackbody radiation at 5800 K (approximately the Sun's surface temperature). The Rayleigh-Jeans law (classical physics) diverges catastrophically at short wavelengths. Wien's approximation works at short wavelengths but fails at long wavelengths. Planck's law correctly describes the full spectrum.
Planck’s Revolution
In 1900, Max Planck found a formula that worked. His equation reduced to Wien’s approximation at short wavelengths and to the Rayleigh-Jeans law at long wavelengths, while remaining finite everywhere. But to derive it, he had to make a radical assumption: energy is not emitted continuously like water from a hose, but in discrete packets, which he called quanta (singular: quantum).
The energy of each quantum is determined by a simple equation:
\[ E = h\nu \]
where E is the energy of one quantum, ν (the Greek letter nu) is the frequency of the radiation, and h is a fundamental constant now called Planck’s constant:
\[ h = 6.626\,070\,15 \times 10^{-34}~\mathrm{J \cdot s} \]
This constant sets the “step height” of the energy staircase. It is very small, which is why we do not notice quantization in everyday life. A baseball’s energy steps are so tiny and numerous that they appear continuous, like a ramp with trillions of microscopic stairs. But at the atomic scale, where energies are small, the steps become significant.
Planck initially proposed this as a mathematical trick to make his equations work. In 1905, Albert Einstein showed that light itself is made of these energy packets, later named photons. By 1913, Niels Bohr had applied the idea to atoms, proposing that electrons can only exist in specific allowed orbits, like steps on a staircase. What Planck proposed as a mathematical trick proved to be physically real.
Seeing Quantization: Emission Spectra
The spectrum of light emitted by a source reveals how that light is produced. There are three types of emission spectra, and the distinctions come directly from the physical structure of the emitting material.
Continuous spectra come from hot, dense objects like the filament in an incandescent bulb or the surface of the sun. These objects emit thermal radiation across all wavelengths, producing a smooth curve that depends only on temperature. This is blackbody radiation, the same phenomenon Planck explained in 1900. The spectrum contains every color, blending smoothly from one wavelength to the next.
Line spectra come from isolated atoms making quantum jumps between discrete energy levels. When electrons drop from higher steps to lower steps on the energy staircase, they emit photons with specific energies corresponding to specific wavelengths. Hydrogen discharge tubes, neon signs, and sodium vapor lamps produce true line spectra: sharp, narrow lines at exact wavelengths characteristic of the element.
Band spectra come from solid-state materials like the phosphors in fluorescent bulbs or the semiconductors in LEDs. Unlike isolated atoms, which have discrete energy levels, solids have continuous energy bands. Electron transitions within these bands produce broad peaks rather than sharp lines. The peaks in LED and fluorescent spectra are not quantum lines from atomic transitions but bands resulting from the solid-state electronic structure of the emitting material.
The figure below compares emission spectra from several common light sources. The incandescent bulb (a) shows the smooth, continuous curve of thermal emission. Fluorescent and LED bulbs (b-f) show the broad band spectra characteristic of solid-state phosphors and semiconductors. Sunlight (g-i) is primarily thermal but shows absorption lines where atmospheric gases have absorbed specific wavelengths. Panel (j) compares how sunlight’s spectrum shifts throughout the day as it passes through different amounts of atmosphere.
Source: Abdel-Rahman et al.1
Absorption Spectra: The Reverse Process
Emission spectra show light given off when electrons fall from higher energy levels to lower ones. Absorption spectra reveal the opposite process: electrons absorbing light and jumping up to higher levels.
When white light passes through a cool gas, the atoms absorb photons that match their energy level spacings. The absorbed wavelengths are removed from the light, leaving dark lines in an otherwise continuous spectrum. These dark lines appear at exactly the same wavelengths as the bright lines in that element’s emission spectrum. The staircase has the same step spacing whether electrons are climbing up or falling down.
This phenomenon was first observed in sunlight. In 1814, Joseph von Fraunhofer catalogued hundreds of dark lines in the solar spectrum. These Fraunhofer lines puzzled scientists for decades until Gustav Kirchhoff and Robert Bunsen demonstrated in 1859 that they corresponded to known elements. The dark lines are caused by atoms in the sun’s cooler outer atmosphere absorbing specific wavelengths from the continuous thermal radiation below.
Panels (g–i) in the figure above show these absorption features in sunlight. The dips in intensity at specific wavelengths reveal which elements are present in the sun’s atmosphere: hydrogen, helium, sodium, calcium, iron, and dozens of others. The same technique identifies elements in stars billions of light-years away. Absorption spectroscopy has become one of the most powerful analytical tools in chemistry, used to detect trace contaminants in water, monitor atmospheric pollutants, and identify unknown compounds.
Why Quantization Matters for Chemistry
Quantization is the foundation of modern chemistry.
Electron shells and subshells are quantized energy levels. When you learn that electrons fill the 1s, 2s, 2p, 3s orbitals in a specific order, you are learning about the steps of the atomic energy staircase. The shapes of s, p, d, and f orbitals arise from the quantum mechanical description of these energy levels.
The periodic table is organized by electron configuration, which is determined by quantized energy levels. Elements in the same column have similar properties because they have similar arrangements of electrons on their energy staircases.
Spectroscopy works because of quantization. When you identify an element by the colors of light it emits or absorbs, you are measuring the spacing between its energy steps. Each element has a unique staircase and therefore a unique spectral fingerprint.
Chemical bonding occurs when atoms share or transfer electrons to achieve more stable (lower-energy) configurations. The driving force is the tendency of electrons to occupy the lowest available steps on the energy staircase.
The Modern World Built on Staircases
The story of quantization is the story of a paradox and its resolution. Classical physics predicted that atoms should be unstable, that hot objects should emit infinite radiation, and that atomic spectra should be continuous smears. None of this matched reality. In 1900, Max Planck proposed that energy might come in discrete packets, initially as nothing more than a mathematical device to make his equations work. But the mathematics turned out to reflect something real about nature.
The staircase is not a convenient teaching analogy. It is how the physical world actually operates at the atomic scale. Every laser produces light when electrons in countless atoms jump down the same energy step simultaneously, releasing photons of identical wavelength. Every LED and computer chip depends on engineers controlling the exact spacing of energy levels in semiconductors. Medical imaging technologies like MRI manipulate the quantized spin states of hydrogen nuclei to map the interior of your body.
This idea, that energy at the atomic scale is discrete rather than continuous, connects directly to the chemistry you will learn in the textbook. Electron configurations, orbital shapes, the structure of the periodic table, and the behavior of chemical bonds all follow from the mathematics of the quantum staircase.
At the beginning of this article, I noted that according to 19th-century physics, the atoms in your body should not exist. They should have collapsed in a fraction of a second. The resolution is the staircase. Once an electron reaches the lowest step, there is nowhere lower to fall. The atom is stable. You exist because energy is quantized.
References
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Fawzia Abdel-Rahman, F. A., Bethel Okeremgbo; Saleh, M. A. Caenorhabditis Elegans as a Model to Study the Impact of Exposure to Light Emitting Diode (LED) Domestic Lighting. J Environ Sci Health A 2017, 52 (5), 433–439. https://doi.org/10.1080/10934529.2016.1270676.