Thinking in Waves Episode 22 / Part IX — Information, as Waves
Settling the homework left in Episode 4
Measurement as a Wave —
To observe is to choose which basis to project onto
Measure the same wave by time or by frequency — that choice is all the observer gets.
And only the allocation is choosable; the total \(\Delta t\,\Delta\omega\ge1/2\) will not move.
★ One thing fell out along the way — a wavelet is a Fourier transform on the logarithmic frequency axis, the same family as Episodes 13 and 20.
Episode 21 looked at information as waves. This time we look at measuring itself as a wave.
And we settle the homework left over from Episode 4 — the limitation that "Fourier assumes stationarity."
01To "measure" is to choose a basis to project onto
Since Episode 1 we have written a wave two ways — as a sequence in time \(x(t)\) and as a sequence in frequency \(X(\omega)\). Two displays of the same information, and you may choose which to measure in.
| Basis | What you see | Time resolution | Frequency resolution | Instrument |
|---|---|---|---|---|
| delta function \(\delta(t-t_0)\) | the instant | perfect | none | oscilloscope |
| sinusoid \(e^{i\omega t}\) | the frequency | none | perfect | spectrum analyser |
| windowed sinusoid | both | finite | finite | ★ short-time Fourier transform |
| wavelet | time and log frequency | \(\propto\omega\) | \(\propto\omega\) | ★ constant-Q analysis |
The point of §01
"To observe" turned out to be the operation of picking a row in this table.
And Episode 9's bandwidth theorem decides how far the choice can go.
02How resolution moves when you change the window
| window width \(\sigma\) [ms] | \(\Delta t\) [ms] | \(\Delta\omega\) [rad/s] | \(\Delta t\cdot\Delta\omega\) | \(\Delta f\) [Hz] |
|---|---|---|---|---|
| 0.5 | 0.3536 | 1414.21 | 0.500000 | 225.08 |
| 1.0 | 0.7071 | 707.11 | 0.500000 | 112.54 |
| 2.0 | 1.4142 | 353.55 | 0.500000 | 56.27 |
| 10.0 | 7.0711 | 70.71 | 0.500000 | 11.25 |
| 20.0 | 14.1421 | 35.36 | 0.500000 | 5.63 |
The conclusion of §02
The product is always 0.5 — sitting exactly on Episode 9's Gabor limit.
★ "Widen the window and frequency sharpens while time blurs" is now a number.
── the observer chooses only where to allocate it; the total will not move.
03★ A wavelet is a Fourier transform on the log-frequency axis
| centre frequency [Hz] | \(\sigma_t\) [ms] | \(\Delta t\) [ms] | \(\Delta\omega\) [rad/s] | \(\Delta\omega/\omega\) | \(\Delta t\cdot\Delta\omega\) |
|---|---|---|---|---|---|
| 55 | 48.664 | 34.411 | 14.53 | 0.042047 | 0.500000 |
| 110 | 24.332 | 17.205 | 29.06 | 0.042047 | 0.500000 |
| 440 | 6.083 | 4.301 | 116.24 | 0.042047 | 0.500000 |
| 880 | 3.042 | 2.151 | 232.49 | 0.042047 | 0.500000 |
| 1760 | 1.521 | 1.075 | 464.97 | 0.042047 | 0.500000 |
The point of §03 — the best find of this episode
\(\Delta\omega/\omega\) is constant, and \(\Delta t\cdot\Delta\omega\) is constant too (\(=0.5\)).
★ So a wavelet is an analysis with uniform resolution on the \(\ln\omega\) axis.
Which connects to Episodes 13 and 20 —
Episode 13: the phase of an imaginary-order derivative is proportional to \(\ln\omega\) (the Mellin transform)
Episode 20: in the \(c\cdot t=\)const coordinate system, phase is proportional to \(\ln t\)
Here: a wavelet is evenly spaced in \(\ln\omega\)
★★ All three are tools of "a world where the logarithmic axis is natural."
If Fourier is the instrument of time translation, a wavelet is the instrument of scale transformation.
Which is why it suits self-similar signals — 1/f noise, turbulence, seismic waves — the world of exponents lined up in Episode 14 was the wavelet's world.
★ Which is why music is analysed at constant Q — the human ear also hears on a logarithmic axis
04Settling the homework from Episode 4
Episode 4 said: "Fourier assumes stationarity. A one-off jump spreads thinly across all frequencies and never touches any band's limit."
So what if we switch to a time-localized basis? We build a sinusoid whose amplitude jumps once by 20 % and vary the window width to detect the jump.
| window \(\sigma\) [ms] | \(\Delta f\) [Hz] | transition width [ms] | \(\Delta t\) [ms] | ratio | width × \(\Delta\omega\) |
|---|---|---|---|---|---|
| 1 | 112.54 | 2.480 | 0.707 | 3.507 | 1.754 |
| 3 | 37.51 | 7.680 | 2.121 | 3.620 | 1.810 |
| 10 | 11.25 | 25.600 | 7.071 | 3.620 | 1.810 |
| 30 | 3.75 | 76.800 | 21.213 | 3.620 | 1.810 |
| 100 | 1.13 | 258.000 | 70.711 | 3.649 | 1.824 |
The point of §04 — the answer to Episode 4
The transition width is cleanly proportional to \(\Delta t\) (the ratio is a constant 3.62). Narrow the window and "when it jumped" comes out sharply — but \(\Delta f\) widens by a factor of 100.
★ And width × \(\Delta\omega\) is constant too. All we did was reallocate.
The stationarity assumption can be dropped. The uncertainty cannot.
A step-like change is caught by a wavelet. But "when" and "at what frequency" cannot both be made fine.
★ Limitation ① of Episode 4 (stationarity) can be overcome; the limitation of Episode 9 (the bandwidth theorem) remains — one of the framework's two holes is now plugged.
05Why measuring destroys interference
| phase scatter \(\sigma_\varphi\) [rad] | visibility \(V\) | state |
|---|---|---|
| 0.0 | 1.000 | almost perfect interference |
| 0.5 | 0.883 | interference visible |
| 1.0 | 0.607 | ★ the dividing line |
| 2.0 | 0.135 | largely gone |
| 3.0 | 0.0111 | essentially gone |
| 5.0 | \(3.7\times10^{-6}\) | ★ classical |
The point of §05
Past about 1 rad of \(\sigma_\varphi\), interference vanishes rapidly.
"It was observed" means the environment stirred the phase by more than 1 rad.
★ This connects to Episodes 19 and 21 —
\(V\) decreases monotonically. It does not oscillate. So decoherence is a phenomenon on the imaginary axis. Which is why it does not come back.
06Why large things do not interfere
★ A superposition wider than the thermal de Broglie wavelength collapses instantly
| Object | mass [kg] | \(\lambda_{\rm dB}\) [m] (300 K) | \(\tau_D/\tau_{\rm relax}\) at \(\Delta x=1\) µm |
|---|---|---|---|
| electron | \(9.11\times10^{-31}\) | \(1.21\times10^{-9}\) | \(1.47\times10^{-6}\) |
| hydrogen atom | \(1.67\times10^{-27}\) | \(2.83\times10^{-11}\) | \(8.02\times10^{-10}\) |
| \(\mathrm{C_{60}}\) molecule | \(1.20\times10^{-24}\) | \(1.06\times10^{-12}\) | \(1.12\times10^{-12}\) |
| dust grain (1 µm) | \(1\times10^{-15}\) | \(3.66\times10^{-17}\) | \(1.34\times10^{-21}\) |
| sand grain (1 mm) | \(1\times10^{-6}\) | \(1.16\times10^{-21}\) | \(1.34\times10^{-30}\) |
The point of §06
Electron and dust grain differ by 15 orders of magnitude.
★ "Large things do not interfere" is not because quantum mechanics stops applying, but because decoherence is too fast to see.
| Interference actually demonstrated | mass [kg] | speed [m/s] | de Broglie wavelength [m] |
|---|---|---|---|
| electron (1927) | \(9.11\times10^{-31}\) | \(10^{7}\) | \(7.3\times10^{-11}\) |
| neutron (1970s) | \(1.68\times10^{-27}\) | 2000 | \(2.0\times10^{-10}\) |
| \(\mathrm{C_{60}}\) molecule (1999) | \(1.20\times10^{-24}\) | 200 | \(2.8\times10^{-12}\) |
| large molecules (2019) | \(4.15\times10^{-23}\) | 100 | \(1.6\times10^{-13}\) |
★ \(\mathrm{C_{60}}\)'s wavelength is 2.8 pm — one 360th of the molecule itself (1 nm across). It interferes with itself using a wavelength 360 times smaller than it is.
07The limit of measurement itself — the standard quantum limit
It comes from Episode 9 — measure position precisely and you disturb the momentum, which blurs the position over τ
| System | mass [kg] | time \(\tau\) [s] | SQL [m] |
|---|---|---|---|
| LIGO mirror (100 Hz) | 40 | \(1.59\times10^{-3}\) | \(6.5\times10^{-20}\) |
| LIGO mirror (1 kHz) | 40 | \(1.59\times10^{-4}\) | \(2.0\times10^{-20}\) |
| 1 g oscillator | \(10^{-3}\) | \(10^{-3}\) | \(1.0\times10^{-17}\) |
| single atom (1 ms) | \(1.45\times10^{-25}\) | \(10^{-3}\) | \(8.5\times10^{-7}\) |
§07 — Episode 9's "the allocation is choosable," used by a real instrument
The SQL for LIGO's 40 kg mirror at 100 Hz is \(6.5\times10^{-20}\) m — smaller than a ten-thousandth of a proton.
★ And LIGO actually pushes past it — with squeezed light, compressing one quadrature's fluctuation and pushing it into the other.
Episode 9's "the product will not move, but the allocation is choosable," implemented in hardware.
6 dB of squeezing multiplies the noise by 0.50, doubling the sensitivity. The observable volume of the universe grows by \(2^3=8\).
08Measurement in the language of information
| Stage | State | Phase | Reversible? |
|---|---|---|---|
| before measurement | superposition | definite | reversible |
| during measurement | correlated with the environment | leaking away | ★ this is where it joins the imaginary axis |
| after measurement | a definite value | lost | irreversible |
The conclusion of §08
Measurement is irreversible because the phase is thrown away into the environment. The same structure as Episode 21's Landauer — heat appears in proportion to what was discarded.
To be precise, though — measurement itself can in principle be free (reversible measurement).
What must be paid for is erasing the record: at room temperature, \(2.87\times10^{-21}\) J per bit \(=0.0179\) eV.
Exercises
- Why are wavelets (constant Q) the right tool for analysing music?
Show answer
Because pitch is defined by ratios. An octave up is a factor of 2; a semitone is a factor of \(2^{1/12}\) — a multiplicative world, not an additive one.
The short-time Fourier transform has constant \(\Delta f\), so it cannot separate semitones in the bass and is needlessly fine in the treble.
★ At constant Q, \(\Delta f/f\) is constant — semitones are separated equally well in every octave (requiring \(Q=16.8\)). The human ear already hears logarithmically; the tool was simply matched to it. - Is "large objects do not obey quantum mechanics" correct?
Show answer
No. They do obey it; decoherence is simply too fast for the interference to be seen.
As the table in §06 shows, a 1 µm dust grain decoheres 15 orders faster than an electron. Quantum mechanics does not switch off; the coupling to the environment is orders of magnitude stronger.
★ The evidence is experimental — even \(\mathrm{C_{60}}\) (720 atoms) interferes once the environment is sufficiently isolated. In 2019 it was done with molecules of 25000 u. - Is the SQL an unbreakable limit?
Show answer
It is breakable. The SQL is only the limit "when position and momentum fluctuations are allocated evenly."
With squeezed light one quadrature can be compressed, so for the quantity you care about you can go below the SQL (at the cost of the one you do not measure).
★ What is unbreakable is \(\Delta x\,\Delta p\ge\hbar/2\) — Episode 9's "the product will not move, but the allocation is choosable," again.
LIGO has used squeezed light routinely since 2019.
Summary — only the allocation is choosable
"To observe" was to choose which basis to project onto — time, frequency, or something in between. Measured with a Gaussian window, \(\Delta t\cdot\Delta\omega\) is always 0.5, exactly Episode 9's Gabor limit.
★ And a wavelet is an analysis with uniform resolution on the \(\ln\omega\) axis (constant \(\Delta\omega/\omega\), confirmed numerically). The same family as Episode 13's imaginary order and Episode 20's logarithmic time — if Fourier is the instrument of time translation, a wavelet is the instrument of scale transformation.
Episode 4's homework is settled too — a step-like change can be caught by a wavelet (transition width proportional to \(\Delta t\), ratio a constant 3.62). But width × \(\Delta\omega\) is constant too. ★ The stationarity assumption can be dropped. The uncertainty cannot.
Interference dies as \(V=e^{-\sigma_\varphi^2/2}\) — stir the phase by 1 rad and it is over. And \(V\) falls monotonically, so — ★ decoherence is a phenomenon of the imaginary axis. Which is why it does not come back.
Electron and dust grain decohere at rates 15 orders apart. "Large things do not interfere" is not quantum mechanics switching off but decoherence being too fast — \(\mathrm{C_{60}}\) interferes with itself at one 360th of its own size.
★ Only the allocation is choosable; the total \(\Delta t\,\Delta\omega\ge1/2\) will not move.
And measurement is irreversible because the phase is discarded onto the imaginary axis.
Next time we break that assumption head-on. What happens when things go nonlinear — frequencies mix, new frequencies are born, and superposition stops working.
★ That is where we first see the place at which "the language of waves" genuinely fails.
This document is Episode 22 of the "Thinking in Waves" series. Computations are in kenshou/hakaru.js. On the numbers here: resolution is defined as in Episode 9, as an intensity-weighted standard deviation; change the definition and the numbers change. The wavelet in §03 is assumed to be Morlet-type (a complex sinusoid under a Gaussian window); other wavelets (Haar, Daubechies and so on) give \(\Delta t\cdot\Delta\omega\) larger than 0.5. The "width/\(\Delta t\) = 3.62" of §04 depends on the 1.02–1.18 threshold chosen; what matters is only that the ratio is constant. The \(V=e^{-\sigma_\varphi^2/2}\) of §05 is exact for Gaussian phase noise, and real environments need not be Gaussian. The decoherence-time formula in §06 is a high-temperature, long-wavelength approximation, and \(\tau_{\rm relax}\) varies enormously between environments — the table shows the dependence on mass and separation, not absolute times. The interference figures are representative literature values (the 2019 25000 u result is due to Fein et al.). The SQL expression is the standard free-mass form; LIGO's real noise budget combines radiation pressure, shot noise and thermal noise, and this article estimates only one term of it. — To print, use your browser's Print and "Save as PDF."