物理 · Physics

The Quantum Revolution

Day 18 · 2026 · Phase E Quantum Mechanics
Energy comes in packets not because the world runs out of smallness, but because it is confined — what is bound has rungs; what is free stays continuous.
In October 1900 Planck had two formulas: one that worked at long wavelengths, one that worked at short. He stitched them into a single expression and it fitted the whole curve. To find a reason for the stitched formula, two months later he was forced to assume that energy could only be emitted in packets — his own phrase for it was an "act of desperation", and for the next decade and a half he kept trying to get rid of the assumption. Textbooks say he was rescuing physics from the "ultraviolet catastrophe"; in fact Ehrenfest only coined that phrase in 1911, and Planck was simply chasing an experimental curve. The person who took the packet seriously was Einstein.

An Act of Desperation

Black-body radiation · 1900
Intuition Everything with a temperature glows: iron at 700 ℃ glows dull red, at 1500 ℃ white, and you are glowing in the infrared right now. Classical theory should have been able to compute that spectrum — the electromagnetic waves inside a hot cavity are a collection of standing waves, and equipartition hands every way of vibrating (every mode) the same share of energy. The trouble is that the higher the frequency, the more modes fit inside; their number grows as the square of the frequency, and each still gets an equal share. The total is infinite. On that arithmetic, opening the oven door would blast you with X-rays.
Mechanism Planck changed exactly one thing: a mode of frequency ν cannot hold an arbitrary amount of energy, only a whole number of some smallest packet.
E = n h ν
ν is the Greek letter nu (say "new", not the Latin letter v) and stands for frequency — vibrations per second; h is Planck's constant, 6.626×10⁻³⁴ joule·seconds; n is a whole number 1, 2, 3… — how many packets. h is absurdly small, so on everyday scales these steps are far too fine to notice.
Once there is a smallest packet, the high-frequency ones become "expensive": the energy a heat bath can hand out casually is about kT (k is Boltzmann's constant, T the temperature), and when hν greatly exceeds kT that mode cannot even afford its first packet and is exponentially suppressed. The curve therefore rises to a peak and comes back down — exactly the shape measured.
Frequency ν → Radiated intensity Classical prediction (Rayleigh–Jeans) → ∞ too many high-ν modes, each given kT Planck: energy comes in hν packets the high-ν packet is unaffordable → suppressed peak position set only by T
One furnace, two calculations. The classical curve runs off the top of the frame; once energy comes in packets, the curve peaks and falls — which is what the instruments see.
The counterintuitive part What deserves your attention is the dimension of h: energy × time, which is exactly the action of "The Principle of Least Action". What nature cuts into packets is neither energy nor length but action; h is this universe's unit of action, small enough that three centuries of classical physics never bumped into it.
Cross-disciplinary reading · Engineering / Cosmology / Method
  • Engineering: the position of the peak depends only on temperature (Wien's displacement law), so colour alone measures temperature — the pyrometer on a steel furnace, the infrared thermometer at your forehead, and the surface temperature astronomers read off a star's colour are all the same curve.
  • Cosmology: the cosmic microwave background is a black-body spectrum at 2.725 K, and COBE found deviations below a few parts in ten thousand — the most perfect black body ever measured. The early universe really was a hot cavity.
  • Method: Planck treated h as a fitting parameter and spent over a decade trying to push it back into the continuum. An assumption introduced purely to fit data turned out to be the structure of the world — there is no line drawn in advance between "just a mathematical trick" and "this is real".
In one line: h is not the smallest unit of energy but the smallest unit of action.
Think: The classical failure came from "too many high-frequency modes". The specific heat of a solid also collapses toward zero at low temperature — same mechanism?
Yes. Atomic vibrations in a solid are also a set of modes (phonons). At low temperature kT falls below the packet hν of the high-frequency modes, so they are never excited, contribute nothing to heat absorption, and the specific heat drops toward zero — which is how Einstein in 1907 and Debye in 1912 solved the low-temperature specific-heat puzzle.

The Photoelectric Effect: One Electron, One Packet

Einstein · 1905
Intuition Shine light on a metal and electrons come out. On the wave picture light is energy flowing in continuously: brighter light should push harder, and even feeble light should do the job if you wait long enough. Every result is the opposite — brightness decides only how many electrons come out, never how fast the fastest one moves; speed is set by colour. Light redder than some threshold colour ejects nothing at all no matter how bright the lamp, while light above the threshold produces current the instant you switch it on, with no waiting time. Throw fine sand at a window all day and nothing breaks; throw one stone and it shatters.
Mechanism Einstein's step: the energy of light is itself absorbed in packets of size hν (later called photons), and one electron swallows exactly one packet.
Emax = hν − W
Emax is the largest kinetic energy an ejected electron can have; hν is the entire energy of the packet it swallowed; W is the work function, the minimum toll for dragging one electron out of that particular metal. So when hν is less than W the right-hand side is negative — nothing comes out, and no amount of intensity changes that.
This is a straight line: frequency across, maximum kinetic energy up, slope exactly h, intercept −W. Change the metal and the line slides up or down without the slope budging. Millikan spent a decade trying to disprove the relation; his 1916 experiment instead pinned the line down to high precision.
Light frequency ν (colour) Max kinetic energy of ejected electron ν₀ (metal A) ν₀ (metal B) metal A (small work function) metal B (large work function) below ν₀ : no electrons, however bright −W (A) −W (B) Strictly parallel: slope = h, the same for every metal
Changing the metal moves the intercept, never the slope. The slope of this line is Planck's constant — a number that first crawled out of a furnace, turning up again in a completely different experiment.
The counterintuitive part Einstein's 1921 Nobel Prize was for this work, not for relativity. And here is the part usually left out: the photoelectric effect alone does not prove that photons exist. A semiclassical model — atoms treated quantum mechanically, the electromagnetic field still treated as a classical wave — reproduces both the frequency threshold and the instant response. What really forced light quanta was Compton scattering in 1923 (light bouncing off an electron and changing wavelength, like two balls colliding) and, later, antibunching experiments with single-photon sources. The photoelectric effect opened the door; it did not close it.
Cross-disciplinary reading · Biology / Energy
  • Biology: a rod cell in your retina responds to a single photon (Hecht's threshold experiment in 1942, confirmed directly in 2016). You cannot see infrared not because it is weak but because one infrared photon carries too little energy to flip the molecule inside rhodopsin into its other shape — the retina has a work function of its own.
  • Energy: a solar cell's band gap is its work function. Photons below the gap pass straight through; the excess energy of photons above it turns into waste heat. That leak-plus-waste is the main reason a single-junction silicon cell is capped near 33% efficiency (the Shockley–Queisser limit).
In one line: intensity governs how many, frequency governs whether at all.
Think: Could two beams of red light, each below threshold, "club together" to eject one electron?
Not at ordinary intensities — two photons will essentially never hit the same electron at once. But push the instantaneous intensity up by a dozen orders of magnitude with an ultrashort laser pulse and two-photon ionisation does happen; it is routine in strong-field physics. So the "no" is a matter of probability, not of law, which is precisely what you expect if the threshold comes from eating one packet at a time.

Rungs Inside the Atom

Bohr · 1913
Intuition Two facts classical physics cannot account for. First, an electron circling a nucleus is an accelerating charge, and electromagnetism says it must radiate continuously; it would spiral in within about ten nanoseconds. Atoms should collapse instantly, yet they have been stable for over thirteen billion years. Second, heat a tube of hydrogen and it emits not a continuous rainbow but a handful of sharp bright lines, fixed precisely enough to serve as an identity card.
Mechanism Bohr's assumption: an electron may only sit at certain energies (stationary states), and while sitting there it does not radiate; light is emitted only in the instant it drops from one rung to another, spitting out a photon whose energy is exactly the gap. For hydrogen the rungs come out as
En = − 13.6 eVn2
n is the rung number 1, 2, 3…; eV means "electron volt", the convenient energy unit at atomic scale, 1 eV ≈ 1.6×10⁻¹⁹ joules. The minus sign says this is a bound state: by convention an electron dragged infinitely far away and fully free has energy zero, so anything still inside the atom lies below it; larger n means closer to zero and more loosely bound.
Every transition landing on n = 2 falls in the visible band — the Balmer series of hydrogen at 656, 486, 434 and 410 nanometres: red, cyan, blue, violet. The higher rungs crowd together, and so do the lines toward the violet end.
n → ∞ : energy = zero, electron is free n = 4, 5, 6… (ever denser) n = 3  −1.51 eV n = 2  −3.40 eV n = 1  −13.6 eV (ground state) four drops to n = 2 → visible light drops to n = 1 → ultraviolet drops to n = 3 → infrared Balmer series of hydrogen 400 500 600 700 wavelength / nm 410 434 486 656
The size of the gap on the left fixes where the line glows on the right. The four positions are not fitted; they drop straight out of the energy formula.
The counterintuitive part Bohr's model gets hydrogen's levels right while being a semiclassical patchwork: it copies classical orbits, then rules by hand which ones are "allowed", with no account of why, and it fails as soon as you hand it helium. The real explanation needs the electron treated as a wave — energy has rungs not because the scale is small but because the thing is confined: a string clamped at both ends can only vibrate in whole numbers of half-wavelengths (the standing-wave modes of "Oscillation & Waves"), and a bound electron likewise has only certain shapes that close on themselves. A free electron, by contrast, has a perfectly continuous energy — quantum mechanics has never meant "discrete everywhere".
Cross-disciplinary reading · Astronomy / Metrology
  • Astronomy: every element's set of lines is unique. Helium was identified in the solar spectrum during an eclipse in 1868 and only found on Earth more than twenty years later; and how far the whole set of lines has slid toward the red is exactly a galaxy's redshift.
  • Metrology: the second is defined as 9,192,631,770 periods of the radiation from the transition between two hyperfine levels of caesium-133. The international standard of time rests on one rung of a ladder.
In one line: an atom is not a little solar system but an instrument with fixed pitches.
Think: If a spectral line corresponds to an exact energy gap, why does every real line have a width?
Three sources: an excited state has a finite lifetime, and the shorter it is the blurrier the energy (natural linewidth); the atoms are moving, so their differing Doppler shifts smear the line (wider the hotter it is); and collisions interrupt the radiation (wider the higher the pressure). Astronomers read temperature and turbulent velocity straight out of that width.

Wave–Particle Duality: Two Old Words, Neither Sufficient

de Broglie · 1924
Intuition De Broglie turned the question around: if light, always treated as a wave, is absorbed in particle-like packets, might the electron, always treated as a particle, have a wavelength? The relation he wrote down is absurdly simple.
λ = hp
λ is the Greek letter lambda and means wavelength (not the digit 1 or the letter l); p is momentum, at everyday speeds just mass times velocity; h is still Planck's constant. More momentum, shorter wavelength.
Three years later Davisson and Germer fired electrons at a nickel crystal and got the same diffraction pattern X-rays give, with the wavelength the formula predicts. Since then the relation has been pushed to heavier and heavier things — neutrons, atoms, a football-shaped molecule of sixty carbon atoms (1999), organic molecules beyond 25,000 daltons (2019) — all of them producing interference fringes.
de Broglie wavelength (each step = ×10) ↓ atomic spacing 10⁻¹⁰ m: the finest "grating" we can build electron (100 eV) 1.2×10⁻¹⁰ m neutron at 300 K 1.8×10⁻¹⁰ m helium atom at 300 K 0.7×10⁻¹⁰ m electron in an EM (100 kV) 4×10⁻¹² m C₆₀ molecule 2.5×10⁻¹² m baseball (145 g, 40 m/s) 1.1×10⁻³⁴ m  19 orders below a nucleus 10⁻³⁵ 10⁻³⁰ 10⁻²⁵ 10⁻²⁰ 10⁻¹⁵ 10⁻¹⁰ wavelength / m
Five objects that do produce interference sit within one step of each other; the baseball is thrown twenty-odd orders of magnitude away. The dividing line is not "microscopic versus macroscopic" but wavelength: it has to be comparable to the slit.
The counterintuitive part "Sometimes a wave, sometimes a particle" is a surrender dressed as an explanation. The honest version: an electron is neither a tiny ball nor a ripple on water, it is a third kind of thing; "wave" and "particle" are two old words borrowed from the large-scale world, each useful for one class of experiment. Feynman's line about nobody understanding quantum mechanics is about exactly this — what is not understood is not the mathematics, which is accurate to a dozen significant figures; what is not understood is which picture to hold.
One more thing to be clear about: duality has nothing to do with whether anyone is watching. What decides which face shows is the apparatus itself. "Measurement & Interpretations" will deal with that piece of folklore head-on.
Cross-disciplinary reading · Engineering / Philosophy
  • Engineering: an optical microscope is stuck at a few hundred nanometres by the wavelength of visible light. An electron microscope accelerates electrons through 100 kV, giving a wavelength near 0.004 nanometres and a resolution orders of magnitude better; cryo-EM now resolves individual atoms in a protein (2017 Nobel Prize in Chemistry). That machine is λ = h/p industrialised.
  • Philosophy: when two ready-made concepts are each half right, the correct move is neither to pick one nor to declare the world contradictory, but to admit that both words were borrowed from elsewhere and go build a third.
In one line: the electron is not fickle; our vocabulary is old.
Think: If everything has a wavelength, why not build a "baseball microscope" and use baseball waves to see smaller things?
Shorter wavelength does mean better resolution in principle, and a baseball's is preposterously short, so it sounds ideal. The other end is what stops you: to show interference the slit spacing has to be comparable to the wavelength, around 10⁻³⁴ metres, far smaller than any structure known — and any single collision with the environment scrambles the phase anyway.

Going deeper

Why is the everyday world free of quantum effects?
Two reasons usually blurred into one. The first is scale: h is only 10⁻³⁴ joule·seconds, so an everyday object's wavelength is far too short to show. The second matters more — decoherence: any interaction with the environment at all, a collision with an air molecule, the emission of one thermal photon, carries away the phase relations interference depends on. A dust grain in air stays coherent for less than 10⁻³¹ seconds. So the classical look of the large-scale world is not quantum mechanics stopping; it is quantum mechanics applied to a system that is never isolated. Molecular interference experiments have to run in high vacuum and at low temperature: the problem is not that the molecules are big, it is that the environment is loud.
Does "quantised" mean the world is pixellated at the bottom?
No, and that is the commonest misreading of the word. Quantum means "a portion", not "tiny". Which quantities are discrete depends on the situation: energies of bound states are discrete (the electron in an atom), energies of free particles are perfectly continuous; angular momentum and spin do take only discrete values; while position, time and the values of the electromagnetic field itself are not discrete in standard quantum mechanics. Whether space has a smallest unit is a question inside conjectures such as loop quantum gravity, with no experimental support to date. Read "quantised" as "the universe is a pixel image" and it will mislead you at every step afterwards.
If the old quantum theory got hydrogen right, why did it have to go?
Because it was stitched together: keep the classical orbits, then add a rule by hand that picks out the "allowed" ones, with nothing behind the rule. The cost showed up fast — it cannot give the relative brightness of the lines, cannot say when a transition happens, and fails on helium, which has exactly one more electron. In 1925 Heisenberg started from "speak only of observables" and in 1926 Schrödinger from matter waves; two entirely different routes produced the same self-consistent theory, and the old quantum theory was replaced wholesale. The lesson is a hard one: a model that fits the main data can still be a completely wrong framework.
Einstein introduced the photon — so why did he end up opposing quantum mechanics?
What he opposed was never quantisation; he pushed that further than anyone. What he would not accept was probability as a basic layer of the world ("God does not play dice" is about this) and the kind of long-range correlation the theory permits. In 1935, with Podolsky and Rosen, he wrote the EPR argument, meant to show quantum mechanics was incomplete: either there are hidden variables, or you must accept nonlocality. Thirty years later Bell turned the dispute into an inequality experiments could rule on, and experiment has sided with quantum mechanics every time ("Entanglement & Nonlocality" covers this). The objection was valuable precisely because it forced out new measurable questions.

Further reading