The Quantization of Light
By the late 1890s, physicists faced an embarrassing problem. They could predict planetary orbits, design electrical circuits, and explain magnetism. But they couldn’t explain why a heated piece of metal glows red, then orange, then white-hot. More precisely, they could predict it using classical physics. The problem was that their predictions were catastrophically wrong.
The crisis wasn’t just academic. Steel workers had long judged furnace temperatures by the color of heated metal. Edison and his competitors were racing to improve lightbulb efficiency by understanding how heated filaments radiate. But when physicists tried to calculate how a hot object should glow using their best theories, they got absurd answers. Classical physics predicted that any warm object should emit infinite amounts of deadly ultraviolet radiation. Obviously, this doesn’t happen.
Resolving this crisis required abandoning 200 years of physics and accepting a radical new idea: energy is quantized, coming in discrete packets rather than continuous amounts. This chapter tells the story of that revolution through two key discoveries. First, Max Planck’s 1900 solution to the black-body radiation problem introduced a new fundamental constant of nature. Five years later, Albert Einstein used that same constant to explain why light can knock electrons out of metal surfaces, proving that quantization wasn’t just mathematical curve-fitting but a real feature of the universe.
The Black-Body Problem
Experimentalists focused on heated objects that behaved as ideal black bodies, objects that absorb all electromagnetic radiation that hits them. When heated, these bodies emit radiation with a characteristic spectrum that depends only on temperature, not on the material’s composition. This universality made black-body radiation an ideal test case for theory.
A black body is called “black” because if it were cold, it would appear perfectly black, absorbing all light that hits it without reflecting any. In reality, true black bodies don’t exist, but soot-covered objects and cavities with small openings come close. The key property is that black bodies are perfect absorbers and therefore (by Kirchhoff’s law of thermal radiation) perfect emitters. Because their emission depends only on temperature and not on material composition, they serve as an ideal test case for radiation theory.
To understand what experiments showed, we need to define what they measured. Physicists measured the spectral radiance L(λ,T), which describes how the radiated intensity (power per unit area) is distributed across different wavelengths at a given temperature. For a black body, spectral radiance follows a characteristic curve: low at long wavelengths, rising to a peak at some wavelength λmax, then decreasing at shorter wavelengths. As temperature increases, the peak shifts to shorter wavelengths and the overall radiance increases.
The emitted radiation comes from oscillating charged particles (such as electrons) in the material. These particles vibrate at various frequencies, and their kinetic energy increases with temperature. Each oscillating charged particle acts as an oscillator that can emit electromagnetic radiation.
What Classical Physics Predicted
Classical physics approached the black-body problem by modeling the heated object as a cavity filled with electromagnetic standing waves. Each possible standing wave pattern represents a vibrational mode, one specific way the electromagnetic field in the cavity can oscillate. Each mode corresponds to a particular frequency of oscillation. The charged particles in the cavity walls act as oscillators that can exchange energy with these modes, emitting and absorbing electromagnetic radiation.
Why model a heated object as a cavity? When a material is heated, charged particles (electrons) inside oscillate due to thermal energy. These oscillating charges create electromagnetic waves that bounce around inside the material, like sound waves in a room. Standing wave patterns form when waves reflect off boundaries. This cavity model captures the essential physics: heated matter contains electromagnetic radiation in equilibrium with oscillating charges.
According to the equipartition theorem from classical thermodynamics, each oscillator should have an average energy of kBT, where kB is the Boltzmann constant and T is temperature. This seems reasonable. But the critical problem emerges when you count how many vibrational modes exist at different wavelengths.
Think of standing waves in a cavity. At long wavelengths, only a few wave patterns can fit. At shorter wavelengths, many more wave patterns can squeeze into the same space. The mathematics of counting these standing wave patterns (derived from electromagnetic boundary conditions in a cavity) shows that the number of available modes scales as 1/λ4. Going from 400 nm to 200 nm doesn’t just double the number of modes; it increases them by a factor of 24 = 16.
When you combine the energy per mode (kBT) with this rapidly increasing number of high-frequency modes, classical physics predicted that spectral radiance should be inversely proportional to the fourth power of wavelength. This relationship became known as the Rayleigh-Jeans law:
\[L(\lambda, T) \propto \dfrac{1}{\lambda^4}\]
The Rayleigh-Jeans law for a given wavelength (λ) as a function of temperature (T) is written as
\[L(\lambda, T) = \dfrac{2ck_{\mathrm{B}}T}{\lambda^4}\]
where
- L(λ,T) is the spectral radiance (power emitted per unit emitting area, per steradian, per unit wavelength)
- c is the speed of light
- kB is the Boltzmann constant
- T is temperature (in K)
Spectral radiance increases linearly with increasing temperature and scales as 1/λ4 (increases as wavelength decreases to the fourth power).
The Boltzmann constant (kB) provides a bridge between temperature and energy in statistical mechanics and thermodynamics. It relates the average energy of particles in a system to the system’s temperature expressed as
\[E_{\mathrm{avg}} = \dfrac{3}{2}k_{\mathrm{B}}T\]
The Boltzmann constant is expressed in J K–1 and defined as
\[k_{\mathrm{B}} = 1.380~649 \times 10^{-23}\]
As wavelength decreases, spectral radiance increases as the fourth power without bound. This description implied that an infinite amount of energy would be emitted in the ultraviolet region of the electromagnetic spectrum.
The Ultraviolet Catastrophe
According to classical physics, everything around you should be emitting unbounded amounts of deadly ultraviolet and X-ray radiation. Your desk, your coffee mug, even your own body at 37 °C should be glowing with infinite high-energy radiation. Obviously, this doesn’t happen. Room-temperature objects emit primarily infrared radiation. Heated metal glows red, then white-hot, but never emits unbounded UV.
Experimental measurements told a different story. The spectral radiance rises to a peak at λmax, then decreases toward zero at shorter wavelengths. The radiation remains bounded. Classical physics had no explanation.
The figure below shows what experiments actually measured. The curves show black-body spectra at different temperatures. Notice how each curve peaks at a specific wavelength and then decreases at shorter wavelengths. Classical physics (the Rayleigh-Jeans law) predicted that radiance should keep increasing as 1/λ4 at short wavelengths, diverging to infinity. The experimental reality is bounded.
The ultraviolet catastrophe: The colored curves show experimental black-body spectra at different temperatures. The dashed gray curve shows the Rayleigh-Jeans prediction at 5000 K: spectral radiance increases as 1/λ4 at short wavelengths, diverging to infinity. The experimental 5000 K curve (red) peaks and then decreases, remaining bounded.
Planck’s Desperate Solution
In 1900, German physicist Max Planck solved the catastrophe. But the solution came in two stages: first empirical success, then theoretical understanding.
Max Karl Ernst Ludwig Planck: A German theoretical physicist (1858–1947) who won the Nobel Prize in Physics (1918) for his work on quantum theory. (Source: Wikipedia)
Planck approached the problem mathematically, trying to find a formula that fit the experimental curves. After much work, he found one that worked perfectly. His formula, now called Planck’s law, matched the data at all wavelengths both long and short. Classical physics (Rayleigh-Jeans) had predicted infinite energy at short wavelengths. Planck’s formula predicted a peak followed by a decline. The formula contained a new constant that Planck called h.
Planck’s law can be expressed in many different ways. When written as a function of wavelength and temperature, we get
\[L(\lambda, T) = \dfrac{2hc^2}{\lambda^5} \left ( \dfrac{1}{e^{\frac{hc}{\left(\lambda k_{\mathrm{B}}T\right )}} - 1}\right )\]
Similarity to Rayleigh-Jeans law at long wavelength
At long wavelengths (λ), the exponential term in Planck’s law approaches 1. This term can then be approximated using the Taylor expansion
\[e^x \approx 1 + x\]
giving
\[e^{\dfrac{hc}{\lambda k_{\mathrm{B}}T}} \approx 1 + \dfrac{hc}{\lambda k_{\mathrm{B}}T}\]
Substituting this approximation, at long wavelenghts, into Planck’s law gives
\[L(\lambda, T) = \dfrac{2hc^2}{\lambda^5} \left ( \dfrac{1}{\left ( 1 + \dfrac{hc}{\lambda k_{\mathrm{B}}T} \right ) - 1}\right )\]
Simplifying, the 1 terms cancel out
\[ \begin{align*} L(\lambda, T) &= \dfrac{2hc^2}{\lambda^5} \left ( \dfrac{1}{\left ( \dfrac{hc}{\lambda k_{\mathrm{B}}T} \right )}\right ) \\[1.5ex] &= \dfrac{2ck_{\mathrm{B}}T}{\lambda^4} \end{align*} \]
which agrees with the Rayleigh-Jeans law (at long wavelength) given as
\[L(\lambda, T) = \dfrac{2ck_{\mathrm{B}}T}{\lambda^4}\]
Behavior at short wavelength
At shorter wavelength (high frequency), the exponential term in Planck’s Law becomes large (much greater than 1).
\[e^{\dfrac{hc}{\lambda k_{\mathrm{B}}T}} >\!> 1\]
Thus, the denominator term is approximately equal to the exponential term.
\[e^{\dfrac{hc}{\lambda k_{\mathrm{B}}T}} -1 ~~\approx~~ e^{\dfrac{hc}{\lambda k_{\mathrm{B}}T}}\]
Planck’s Law simplifies to
\[L(\lambda, T) = \dfrac{2hc^2}{\lambda^5} \left ( \dfrac{1}{e^{\frac{hc}{\left(\lambda k_{\mathrm{B}}T\right )}}}\right )\]
As wavelength continues to decrease, the exponential term grows rapidly, causing the spectral radiance to decrease sharply. The combination of the exponential factor in the denominator and the 1/λ5 term creates a rapid decline in radiance at short wavelengths. Planck’s law imposed a bound on spectral radiance at short wavelengths, whereas the Rayleigh-Jeans law incorrectly predicted unbounded growth.
Planck’s formula worked. It fit the experimental data precisely. But Planck wasn’t satisfied with just having a formula. He wanted to understand why it worked. He wanted to derive it from the fundamental principles of thermodynamics and statistical mechanics.
When Planck tried to derive his formula theoretically, he hit a wall. Using classical physics assumptions (that oscillators can exchange any amount of energy with radiation), he couldn’t reproduce his empirical formula. The math didn’t work out. Something had to give.
Planck realized he could derive his empirical formula if he made a radical assumption. What if oscillators can only emit or absorb energy in discrete packets, not continuously? What if the energy of each packet is proportional to the frequency?
He called this proportionality constant h, the same constant that appeared in his empirical formula. The relationship is now called Planck’s Equation:
\[E = h\nu = \dfrac{hc}{\lambda}\]
With this assumption, the statistical mechanics worked. The mathematics led directly to his empirical formula. The quantization assumption explained why his formula had worked.
Planck introduced this assumption reluctantly. He later called it an “act of desperation,” something he did purely to make the theoretical derivation match the empirical formula. He didn’t believe energy was actually quantized. He viewed it as a mathematical trick, not a statement about physical reality. For years afterward, he tried to find a way to derive his formula using classical physics, without assuming quantization. He never succeeded.
In classical physics, energy was continuous. You could have any amount you wanted, like water flowing from a faucet. But Planck’s equation said energy is quantized, coming in discrete lumps. An oscillator can emit hν or 2hν or 3hν, but not 1.37hν.
At higher frequency, the energy packet hν becomes larger. A low-frequency infrared oscillator might emit a packet worth 0.1 eV. A high-frequency UV oscillator must emit packets worth several eV each.
Why Quantization Solves the Catastrophe
Planck’s quantization hypothesis resolves the ultraviolet catastrophe through a fundamental principle from statistical mechanics: the Boltzmann distribution. When energy can only come in discrete packets of size E = hν, the probability that a mode at frequency ν is excited depends on the Boltzmann factor:
\[\text{Probability} \propto e^{-E/(k_{\mathrm{B}}T)} = e^{-h\nu/(k_{\mathrm{B}}T)}\]
For low-frequency modes (long wavelengths), hν is small compared to kBT, so the exponential is close to 1 and these modes are easily excited. However, for high-frequency modes (short wavelengths), hν becomes much larger than kBT. The exponential factor e−hν/(kBT) becomes vanishingly small, meaning these modes are almost never excited.
This exponential suppression of high-frequency modes is what prevents the catastrophe. Even though there are infinitely many high-frequency modes available (the 1/λ4 factor from counting modes), the Boltzmann factor ensures that almost none of them actually get excited at any given temperature. The energy doesn’t blow up at short wavelengths because the system cannot access those high-energy states with any significant probability.
In classical physics, all modes get kBT energy regardless of frequency, leading to infinite total energy. With quantization, high-frequency modes require energy hν which is much larger than kBT, so they contribute negligibly to the total energy.
The Boltzmann distribution is a fundamental result from statistical mechanics that describes how energy is distributed among particles in thermal equilibrium. It says that the probability of finding a system in a state with energy E is proportional to e−E/(kBT).
Why this exponential form? Higher energy states are less probable because thermal energy is distributed randomly among many particles and modes. Getting a large amount of energy concentrated in one place (one high-energy mode) by chance is exponentially unlikely. The exponential form arises from counting how many ways energy can be distributed among many particles. This counting is covered in statistical mechanics.
At room temperature (kBT ≈ 0.025 eV), a mode requiring 3 eV to excite (UV range) has probability ~ e−3/0.025 = e−120 ≈ 10−52. It essentially never happens.
Room Temperature (~300 K)
\[E_{\mathrm{thermal}} \approx k_{\mathrm{B}} \times 300~\mathrm{K} = 4.14\times 10^{-21}~\mathrm{J} ~~~(\mathrm{or~0.025~eV})\]
Boiling Point of Water (~373 K)
\[E_{\mathrm{thermal}} \approx k_{\mathrm{B}} \times 373~\mathrm{K} = 5.17\times 10^{-21}~\mathrm{J} ~~~(\mathrm{or~0.032~eV})\]
Sun’s Surface Temperature (~5800 K)
\[E_{\mathrm{thermal}} \approx k_{\mathrm{B}} \times 5~800~\mathrm{K} = 8.01\times 10^{-20}~\mathrm{J} ~~~(\mathrm{or~0.50~eV})\]
Note: The electronvolt (also written as electron-volt or electron volt), is a measure of the kinetic energy gained by a single electron accelerating through an electric potential difference of one volt in a vacuum. The eV is related to the Joule given as
\[1~\mathrm{eV} = 1.60218\times 10^{-19}~\mathrm{J}\]
At room temperature, the available thermal energy is about 0.025 eV. This amount of energy is enough to excite oscillators emitting infrared radiation (longer wavelengths, lower frequencies), but not enough to excite oscillators that would emit UV radiation because UV photons require much more energy (several electron volts).
The figure below puts these energies in perspective, comparing thermal energies at different temperatures to photon energies across the electromagnetic spectrum.
Planck’s law encodes the Boltzmann factor directly. The mathematical form, with its 1/(ehν/kBT − 1) term, captures exactly this physics: at high frequencies, the exponential becomes enormous, driving the radiance to zero and preventing the catastrophe.
The key to making this work was the constant h that sets the scale of energy quantization. Planck discovered this constant empirically by fitting his equation to experimental black-body spectra. Planck determined the value of h that gave the best match. This constant is now recognized as one of the fundamental physical constants and is given as
\[ \begin{align*} h &= 6.626~070~15 \times 10^{-34}~\mathrm{J~s} \\[1.5ex] &\approx 6.626\times 10^{-34}~\mathrm{J~s} \end{align*} \]
This very small number (about 10-34 J·s) sets the scale at which quantum effects become important. For everyday objects with macroscopic energies, the graininess imposed by h is imperceptible. But at the atomic scale, where energies are measured in electron volts, quantization becomes dominant.
Practice
Amplitude modulation (AM) radio at 1 100 kHz frequency has a wavelength of 272.5 m and frequency modulation (FM) radio at 105.9 MHz frequency has a wavelength of 2.831 m.
What is the energy (in electronvolts, eV) of these two waves?
\[1~\mathrm{eV} = 1.602~18\times 10^{-19}~\mathrm{J}\]
Solution
Recall that wavelength and frequency are related through the speed of light.
\[c = \lambda \nu\]
AM Radio at 1 100 kHz
\[ \begin{align*} E &= h\nu \\[1.5ex] &= h \left ( \dfrac{c}{\lambda} \right ) \\[1.5ex] &= 6.626\times 10^{-34}~\mathrm{J~s} \left ( \dfrac{2.998\times 10^{8}~\mathrm{m~s^{-1}}}{272.5~\mathrm{m}} \right ) \left ( \dfrac{1~\mathrm{eV}}{1.60218\times 10^{-19}~\mathrm{J}} \right ) \\[1.5ex] &= 4.54\bar{9}9\times 10^{-9}~\mathrm{eV} \\[1.5ex] &= 4.550\times 10^{-9}~\mathrm{eV} \\[1.5ex] \end{align*} \]
FM Radio at 105.9 MHz
\[ \begin{align*} E &= h\nu \\[1.5ex] &= h \left ( \dfrac{c}{\lambda} \right ) \\[1.5ex] &= 6.626\times 10^{-34}~\mathrm{J~s} \left ( \dfrac{2.998\times 10^{8}~\mathrm{m~s^{-1}}}{2.831~\mathrm{m}} \right ) \left ( \dfrac{1~\mathrm{eV}}{1.60218\times 10^{-19}~\mathrm{J}} \right ) \\[1.5ex] &= 4.37\bar{9}5\times 10^{-7}~\mathrm{eV} \\[1.5ex] &= 4.380\times 10^{-7}~\mathrm{eV} \end{align*} \]
FM radio has a shorter wavelength.
Einstein and the Photoelectric Effect
Planck’s quantization solved black-body radiation, but was it real physics or just mathematical curve-fitting? In 1905, Albert Einstein answered that question by applying Planck’s idea to a different phenomenon: the photoelectric effect, the emission of electrons from a material caused by electromagnetic radiation.
Albert Einstein: A German theoretical physicist (1879–1955) who won the Nobel Prize in Physics (1921) for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect. (Source: Wikipedia)
Einstein went further than Planck. While Planck proposed energy is emitted in discrete packets, Einstein proposed light itself travels as discrete packets (photons). Light existed as particles of energy even while propagating through space. Even Planck rejected this idea at first.
Shining light on a metal surface can cause electrons to be ejected from the metal.
Light at low frequency (below the threshold frequency for a given material) did not eject any electrons. However, if the frequency of light was increased enough (past the threshold frequency), electrons were ejected immediately without a time delay. The more intense the light, the more electrons were ejected.
The maximum velocity (vmax) of the ejected electron is related to the kinetic energy of the ejected electron (Telectron). This kinetic energy is calculated as the difference in energy of the electron (Ephoton) and the work function (φ) of a material (i.e. the minimum thermodynamic work, or energy, needed to remove an electron from a solid to a point in a vacuum). Work functions are typically reported in electronvolts (eV). Common metals have work functions between 2 and 5 eV: cesium (2.1 eV), sodium (2.3 eV), copper (4.7 eV).
\[T_{\mathrm{electron}} = E_{\mathrm{photon}} - \phi\]
Practice
What is the velocity (in m s–1) of an electron that is ejected from the surface of a metal by 550.0 nm light? The work function of the metal is φ = 2.00 eV.
Solution
Find the energy (in J) of the photon
\[\begin{align*} E_{\mathrm{photon}} &= \dfrac{hc}{\lambda} \\[1.5ex] &= \dfrac{6.626\times 10^{-34}~\mathrm{J~s} \left ( 2.998\times 10^{8}~\mathrm{m~s^{-1}} \right )} {550.0~\mathrm{nm} \left ( \dfrac{\mathrm{m}}{10^9~\mathrm{nm}} \right )} \\[1.5ex] &= 3.61\bar{1}7\times 10^{-19}~\mathrm{J} \end{align*}\]
Find the kinetic energy (in J) of the ejected electron
\[\begin{align*} T_{\mathrm{electron}} &= E_{\mathrm{photon}} - \phi \\[1.5ex] &= 3.61\bar{1}7\times 10^{-19}~\mathrm{J} - \left [ 2.00~\mathrm{eV} \left ( \dfrac{1.602~08\times 10^{-19}~\mathrm{J}}{\mathrm{eV}} \right )\right ] \\[1.5ex] &= 4.0\bar{7}5\times 10^{-20}~\mathrm{J} \end{align*}\]
Find the velocity (in m s–1) of the electron
Recall that the mass of an electron, me, is 9.109 383 7 × 10–31 kg.
\[\begin{align*} T &= \dfrac{1}{2}mv^2 \longrightarrow \\[1.5ex] v &= \sqrt{\dfrac{2T}{m_e}} \\[1.5ex] &= \sqrt{\dfrac{2~(4.0\bar{7}5\times 10^{-20}~\mathrm{J})} {9.109~383~7~\times 10^{-31}~\mathrm{kg}}} \\[1.5ex] &= 2.9\bar{9}1\times 10^{5}~\mathrm{m~s^{-1}} \\[1.5ex] &= 2.99\times 10^{5}~\mathrm{m~s^{-1}} \end{align*}\]
In classical electromagnetic theory, the intensity of light is related to the electric field amplitude (E0) of the electromagnetic wave:
\[I \propto E_0^2\]
Doubling the amplitude increases intensity by a factor of four. Classical physicists therefore expected that brighter light (higher amplitude, higher intensity) would deliver more energy to electrons.
Classical physics predicted that light intensity (brightness, or the amount of power per unit area hitting the surface) would be the deciding factor in ejecting electrons. According to classical wave theory, with bright enough light of any frequency, electrons should eventually accumulate enough energy to escape, given sufficient time. It was presumed that light, viewed as a continuous electromagnetic wave, would continuously deliver energy to the metal surface, allowing electrons to gradually “build up” enough energy before finally being ejected.
Think of breaking a window. To shatter a window, you need to hit it hard enough in one impact. Tapping it gently a thousand times won’t break it. Why? Because between each tap, the energy dissipates as sound, heat, and elastic deformation. The glass doesn’t “remember” previous taps. Each impact must deliver enough energy at once to overcome the glass’s strength.
Similarly, each photon must carry enough energy in one interaction to overcome the work function and eject an electron. Below the threshold frequency, each photon is like a gentle tap. It doesn’t matter how many low-energy photons hit the metal or how long you wait. The electron cannot accumulate energy from multiple photons. Only when a single photon carries enough energy (threshold frequency or higher) does the electron get ejected immediately.
(Two-photon absorption is possible but requires extremely intense light from lasers, which weren’t available in 1905. Under normal conditions, electrons interact with one photon at a time.)
However, the experimental observations demonstrated that it was the frequency (and hence, energy) of the light that dictated the ejection of an electron as well as the electron’s kinetic energy. A frequency at or beyond the threshold frequency resulted in an ejection of electrons. Increasing light intensity at or beyond the threshold frequency resulted in more electrons being ejected. Below the threshold frequency, no amount of waiting or intensity increase produced any electrons. Only frequency mattered.
These observations directly confirmed Einstein’s quantum hypothesis. There was no “build up” of energy that resulted in the ejection of an electron. The energy transfer process was “all-or-nothing”. This meant that light behaved, not as a continuous wave, but as a stream of discrete packets of energy called photons (where more photons equates to higher intensity). Either the electron absorbed all of a photon’s energy (to be ejected) or it didn’t absorb it at all. Each photon carries a specific amount of energy and that energy had to be large enough to free an electron from the material. If the energy of the photon was too low (low frequency), no ejection is observed, regardless of the intensity of the light.
The photoelectric effect proved Planck’s constant wasn’t a mathematical coincidence. The same h appeared in different phenomena. Black-body radiation and metal surfaces ejecting electrons seemed unrelated, yet both obeyed E = hν. When Planck introduced h in 1900, he viewed it as a mathematical trick to fit experimental data. But Einstein showed that energy quantization was universal, not just a peculiarity of hot cavities.
Light is quantized. Light doesn’t just get emitted in packets. It exists and travels as discrete quanta called photons. Each photon carries energy E = hν. If a photon lacks sufficient energy to eject an electron, its energy is reflected, absorbed as heat, or scattered.
The photoelectric effect isn’t just historical. Modern technology relies heavily on this quantum phenomenon. Digital cameras use photoelectric sensors (CCDs and CMOS chips) where incoming photons eject electrons to create an image. Solar cells convert sunlight to electricity through the same principle. Photons with sufficient energy eject electrons that flow as current. Light detectors in scientific instruments, automatic doors, and smoke detectors all exploit the photoelectric effect. In analytical chemistry, photoelectron spectroscopy uses the effect to identify elements and their chemical environments by measuring the kinetic energies of ejected electrons.
If light consists of particles (photons), why does it still exhibit wave-like behavior such as diffraction and interference? This is the famous wave-particle duality of quantum mechanics. Photons are neither classical particles nor classical waves. They are quantum objects that exhibit properties of both depending on how we observe them. Understanding how this works requires the full machinery of quantum mechanics, which goes beyond this course. For now, recognize that E = hν connects the particle property (energy E) with the wave property (frequency ν). We’ll explore this duality in more detail after examining atomic structure, where we’ll see that matter itself exhibits wave-like properties.
Planck’s constant explained more than hot objects. Chemists had long wondered why heated elements emit specific colors rather than continuous rainbows. Sodium glows yellow, hydrogen shows distinct red and blue lines. Within thirteen years, Niels Bohr applied h to the hydrogen atom, showing that electrons occupy specific energy levels separated by quantum jumps. Each spectral line represents an electron dropping between quantized levels, emitting a photon with energy E = hν.
If light has quantized energy, what about atoms themselves? Planck’s constant h appeared first in black-body radiation and again in the photoelectric effect. In the next section, we’ll see it determine atomic structure itself, explaining why atoms emit only certain wavelengths of light.
Summary
The Ultraviolet Catastrophe
Classical physics predicted that heated objects should emit infinite energy at short wavelengths. The Rayleigh-Jeans law (L ∝ 1/λ4) matched experiments at long wavelengths but diverged catastrophically in the UV.
Planck’s Quantum Hypothesis
In 1900, Planck resolved the crisis by proposing that energy is quantized:
\[E = h\nu = \dfrac{hc}{\lambda}\]
where h = 6.626 × 10−34 J·s is Planck’s constant. High-frequency modes require large energy packets (hν), making them exponentially unlikely to be excited (Boltzmann factor), which prevents the catastrophe.
The Photoelectric Effect
Einstein (1905) proved quantization is real by explaining the photoelectric effect. Light travels as discrete photons, each with energy E = hν.
- Below the threshold frequency, no electrons are ejected regardless of intensity
- Above threshold, electrons are ejected immediately with kinetic energy:
\[T_{\mathrm{electron}} = E_{\mathrm{photon}} - \phi\]
where φ is the work function (minimum energy to remove an electron from the material).