Quantum That Clicks ── Series Contents

The backbone is one sentence ── ℏ is the "grain of action," and the exchange rate that links particle and wave.
Energy and frequency are joined by E＝ℏω, momentum and wavelength by p＝ℏk (just as relativity's c joined time and space, ℏ joins particle and wave). The phase of the wave function is S/ℏ (Episode 2); nature adds the phases of all paths, and for S≫ℏ the classical world of least action emerges (Episode 3). The standing waves in a box make energy come in discrete steps (Episode 4, Schrödinger), and because it is a wave, Δx·Δp≥ℏ/2 (Episode 5). And everyday life looks classical because S/ℏ is astronomically large (Episode 6). Quantities with units are stage machinery; all that counts is the ratio S/ℏ.

Download all published files at once The button below bundles the published episodes into a single ZIP (the set grows as more episodes are added).

Main Episodes

Episode 1interactive figure

ℏ, the grain size ── particle and wave

ℏ is nature's "smallest grain of action." It is the exchange rate linking energy with frequency and momentum with wavelength, and particle and wave are two faces of the same thing. The action divided by ℏ, S/ℏ, decides whether something is classical (≫1) or quantum (∼1). E=ℏω, p=ℏk / S/ℏ

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Episode 2interactive figure

The wave function ── phase is S/ℏ, probability is |ψ|²

A particle is represented by a complex wave ψ＝|ψ|e^(iS/ℏ). The phase is exactly "how many ℏ's of action" there are. Only the probability |ψ|² is observable. Superposition and interference both come out of adding phases. ψ = |ψ| e^(iS/ℏ)

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Episode 3interactive figure

Summing over all paths ── least action and the emergence of the classical

Nature adds up the phase e^(iS/ℏ) of every possible path (Feynman). For S≫ℏ, only the path where the action is stationary (least action) survives, and the classical trajectory appears. Classical mechanics is quantum theory in the limit S/ℏ→∞. amplitude = Σ e^(iS/ℏ)

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Episode 4interactive figure

The Schrödinger equation and quantization

The rule the wave ψ obeys is the Schrödinger equation. A wave confined to a box is only allowed to be a standing wave with nodes at both ends, so its energy comes in "discrete steps" ── the same logic as the overtones of a string. Quantization is a consequence of being a wave. iℏ ∂ψ/∂t = Ĥψ

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Episode 5interactive figure

Uncertainty ── the smallest area ℏ spans

Position and momentum cannot both be pinned down at once: Δx·Δp≥ℏ/2. Not because measurement is clumsy, but because a wave cannot be both "narrow in place" and "of definite wavelength." Phase space has a smallest grain of area, ℏ. Δx·Δp ≥ ℏ/2

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Episode 6interactive figureFinale

Superposition and measurement ── why is everyday life classical?

A quantum holds several possibilities at once. So why does the cat look either alive or dead? Entanglement with the environment (decoherence) wipes out the interference of the superposition, and everyday life, with its enormous S/ℏ, looks classical. The close of the story of ℏ. superposition → decoherence

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Bonus

Bonusinteractive figure

Quantum theory and the Fourier transform ── a record and a CD are the same song

Position ψ(x) and momentum φ(p) are the front and back of a Fourier transform with ℏ as its kernel (two guises of the same state). Take the bandwidth theorem, "a narrow waveform has a broad spectrum," multiply by p=ℏk, and uncertainty falls out automatically. A deeper look at the duality between the continuous and its components. ψ(x) ⇄ φ(p) (Fourier pair)

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Collected Works Capstone

Collectioninteractive figure

The Physics Cube ── the tetralogy was one single cube all along

Along the three axes c, ℏ, G, relativity, quantum theory, fields, and gravity sit at the corners of a cube. Switch on one more constant and one more theory appears. A bird's-eye map of the whole collection, letting you toggle to see which corner each series occupies. the cube of c / ℏ / G

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