Question of the Day: How does a quantum state predict measurement outcomes?
Learning objectives
Notebook 01 showed that diffraction builds from individual detections, but it did not predict where the next detection would land. Here we prepare a state and ask what it predicts for one measurement and what becomes visible only after many repeated measurements.
We have seen diffraction patterns build up from individual detections. The question is not whether particles arrive, but what a prepared quantum state lets us predict about where and how often. This notebook introduces the state as a probability predictor and asks how repeated statistics become stable while single outcomes remain unpredictable.
Prediction. Before running Experiment 1, consider amplitudes that give probabilities $P=(0.5,0.3,0.2)$ for outcomes $a=(1,2,3)$. For $N=20$ trials, would you expect exactly 10, 6, 4 counts? For $N=2000$, how close should the relative frequencies be to the model probabilities?
Statistics refresh. Arithmetic mean of $N$ values: $\bar{x}=\frac{1}{N}\sum x_i$. Weighted mean with probabilities $p_n$: $\langle A\rangle=\sum p_n a_n$. Relative frequency: $f_n = \text{counts}_n / N$. Variance: $\sigma_x^2 = \langle x^2\rangle - \langle x\rangle^2$, standard deviation $\sigma_x = \sqrt{\sigma_x^2}$.
Physical question: How does a prepared state predict the statistics of repeated measurements?
An ideal three-channel analyser receives identically prepared particles one at a time. Each run exactly one channel fires. Controls are the prepared-state preset and the number of trials. The state is written through amplitudes $c_n$ for three outcomes $a_n=(1,2,3)$ with $\sum |c_n|^2=1$.
A single run is unpredictable, yet after many trials the relative frequencies settle near the model probabilities. The expectation value $\langle A\rangle$ sits among the outcomes, weighted toward high-probability channels, and is not itself a single-run result. Finite-sample fluctuation shrinks as $N$ grows and is ordinary sampling, not quantum uncertainty.
Mathematical Window — Discrete probabilities and expectation
The prepared state is
$$ \psi = c_1\varphi_1 + c_2\varphi_2 + c_3\varphi_3 $$
with
$$ \sum_n |c_n|^2 = 1 $$
and
$$ P(a_n)=|c_n|^2 $$
and
$$ \langle A\rangle = \sum_n |c_n|^2 a_n $$
The expectation value is the theoretical ensemble mean predicted by the state; the sample mean is the arithmetic mean of actual finite detector records. They approach each other as $N$ grows but remain conceptually distinct. Controlled: amplitudes $c_n$ and trial count $N$. Detected: channel fires. Inferred: $P(a_n)$ and $\langle A\rangle$.
The controls are the prepared amplitudes $c_n$ and trial count $N$. On each run, the ideal three-channel analyser records which single channel fires; over repeated trials it accumulates counts per channel. From those records we calculate relative frequencies $f_n$, a histogram, and the finite-sample mean $\bar{A}$. A relative frequency is calculated from counts, not a raw detector reading; a histogram summarizes many records, not one individual event; and the sample mean is not the state’s expectation value.
The normalized amplitudes predict outcome probabilities $P(a_n)=|c_n|^2$ and the state’s expectation $\langle A\rangle$ for the distribution, not which channel will fire on a particular run. This ideal analyser assumes perfectly distinguished discrete channels and has no separate detector-response parameter. Counts and sample statistics fluctuate at finite $N$: that sampling variation is ordinary statistics, not intrinsic quantum uncertainty.
Mathematical Window — Probability density and normalization
The position state is described by the wave function $\psi(x)$. The probability density for finding the particle near $x$ is
$$ P(x)=|\psi(x)|^2 $$
with
$$ \int_{-\infty}^{\infty} |\psi(x)|^2\,dx = 1 $$
$|\psi(x)|^2$ is model prediction, not data; the detector records positions. The total area under the density is one. Do not place $\psi$ and $|\psi|^2$ on the same dimensional footing: $\psi$ has units $\text{m}^{-1/2}$, density has $\text{m}^{-1}$.
The same idea now lives on a continuous axis. A localized Gaussian packet predicts where position detections will cluster. The smooth $P(x)=|\psi(x)|^2$ curve is the model; individual recorded positions are data. The next figure shows $\psi(x)$ and $|\psi(x)|^2$ for the same deliberately real reference state.
Physical question: How do individual position detections relate to $|\psi(x)|^2$?
A Gaussian packet with centre $x_0$ and intrinsic spread $\sigma_x$ is sent to a position-sensitive screen. Controls are $x_0$, $\sigma_x$, number of runs, detector-response width $\sigma_{\text{det}}$, and optional acquisition noise $\sigma_{\text{noise}}$. Intrinsic quantum spread $\neq$ detector-response width $\neq$ acquisition noise $\neq$ finite-sample fluctuation.
Single positions scatter, yet the empirical histogram traces the smooth $|\psi|^2$ prediction. The observed width combines contributions: intrinsic $\sigma_x$ (state/model), detector-response $\sigma_{\text{det}}$ (apparatus), acquisition noise $\sigma_{\text{noise}}$ (measurement chain), and finite-sample fluctuation (sampling). The detector-broadened prediction is $\sigma_{\text{detector}}=\sqrt{\sigma_x^2+\sigma_{\text{det}}^2}$; the full recorded prediction is $\sqrt{\sigma_x^2+\sigma_{\text{det}}^2+\sigma_{\text{noise}}^2}$. When $\sigma_{\text{det}}=\sigma_{\text{noise}}=0$, recorded positions equal intrinsic positions.
The controls are the prepared packet centre $x_0$, intrinsic width $\sigma_x$, number of runs, detector-response width $\sigma_{\mathrm{det}}$, and optional acquisition noise $\sigma_{\mathrm{noise}}$. The direct record from each run is one final detected position; the collection of final positions is the accumulated data. A histogram, recorded sample mean, and recorded sample standard deviation are statistics calculated from those records. Intrinsic positions hidden inside the model are latent, not detector readings.
The state predicts the intrinsic density $|\psi(x)|^2$ and intrinsic spread $\sigma_x$. Detector response (resolution) and acquisition noise are separate apparatus/measurement-chain contributions; detector resolution is not intrinsic quantum spread. Broadened curves and widths are apparatus-aware model predictions, not raw records. Variation from a finite number of recorded positions is sampling fluctuation, not quantum uncertainty.
Physical question: What happens to momentum spread when position spread shrinks?
One prepared Gaussian state gives both a position distribution and a momentum distribution. No detector or noise concepts belong here; the trade-off is intrinsic to the state. Narrow the position state and observe momentum.
SI quantum momenta are numerically tiny, so the explorer uses $\hbar\,\mathrm{nm}^{-1}$ as a convenient natural display unit. Atomic units go one step further: they use the Bohr radius $a_0$ and momentum unit $\hbar/a_0$. No atomic-unit calculation is required in this notebook.
Narrowing the position distribution broadens the momentum distribution and vice versa. The displayed product $\Delta x\Delta p$ stays at $\hbar/2$ for this family.
The general quantum relation is
$$ \Delta x\,\Delta p \ge \frac{\hbar}{2} $$
For the Gaussian minimum-uncertainty family explored here, $\Delta x\Delta p = \hbar/2$. This Gaussian family reaches equality; the general relation remains $\ge$.
Mathematical Window — Variance and uncertainty
Variance:
$$ \sigma_x^2 = \langle x^2\rangle - \langle x\rangle^2 $$
with $\Delta x = \sigma_x$ and $\Delta p = \sigma_p$. The intrinsic spreads satisfy
$$ \Delta x\,\Delta p \ge \frac{\hbar}{2} $$
For the Gaussian minimum-uncertainty family explored here, $\Delta x\Delta p = \hbar/2$. No Fourier or commutator derivation is required; a commutator may be named only as later formal explanation.
Single-molecule localization microscopy (PALM/STORM) detects individual photons around an emitting molecule, seen as a finite optical spot (point-spread function). Many detections allow statistical inference of emitter position, and many inferred localizations reconstruct labelled structure.
A reconstructed localization map is not the quantum position probability density $|\psi|^2$ of a single fluorophore. It is a statistical reconstruction from many photon detections across many molecules.
The optical point-spread function is not a quantum wave function. PET uses related counting statistics, but its physical measurement process is different from the idealized analyser used earlier.
Synthesis across three experiments: individual outcomes are unpredictable; accumulated records give stable frequencies that trace model densities; intrinsic spreads obey an inverse trade-off.
Notebook 02 — Measurement and Uncertainty — How does a quantum state predict measurement outcomes?
A prepared state predicts probabilities: $P(a_n)=|c_n|^2$ for discrete outcomes and $P(x)=|\psi(x)|^2$ for positions, both normalized ($\sum|c_n|^2=1$, $\int|\psi|^2dx=1$). Expectation value $\langle A\rangle$ is theoretical; sample mean $\bar{A}$ is finite-sample. Position and momentum intrinsic spreads satisfy $\Delta x\Delta p \ge \hbar/2$, with equality for the Gaussian family explored here.
Measurement Statistics showed discrete unpredictability becoming stable frequencies. Position Detection showed intrinsic quantum spread distinct from detector response $\sigma_{\text{det}}$, acquisition noise $\sigma_{\text{noise}}$, and finite-sample fluctuation. Uncertainty Explorer showed the intrinsic trade-off without apparatus.
Expectation value versus sample mean: expectation is model prediction; sample mean is statistic from recorded positions. Intrinsic sigma versus detector sigma: one is state property, the other is apparatus.
So far, the spatial state has been allowed to vary along an open line. If boundaries confine the particle, they restrict which spatial patterns are allowed. That points toward standing-wave states and discrete energies. If we confine our particle so it cannot wander, the boundary conditions select which standing-wave patterns nature allows; the eigenfunctions and energy formulas are deferred to Notebook 03.
Why does confinement produce discrete energies?