SCIENCE · INVESTIGATE AND EXPLAIN

Interpret localization as a distribution, not a trajectory

A prepared quantum state can produce a distribution of positions even if an ideal instrument has no additive bias. The supplied Gaussian density is exp[−(x−μ)²/(2σx²)]/[σx sqrt(2π)]. Its horizontal variable is position and its vertical unit is inverse length. A density larger than one per nanometre is not a probability larger than one: probabilities refer to areas over intervals. The whole-line integral is one, while the finite plotted range omits small tails.

Learning outcome: Compare widths 0.1,0.2,0.4 nm at the same center, then translate the center at fixed width. Predict height, spread and product separately. Explain why estimating a mean from more events does not create a state with a smaller intrinsic σx.

Starting recap

Separate per-event energy, expectation rate and measurement indication. Check particle identities, conservation, threshold branches and calibration roles before making a source or detector claim.

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Try

With σx=0.2 nm, the minimum momentum standard deviation is ℏ/(2σx)≈2.63642954×10⁻²⁵ kg m/s. Halving σx doubles that value. Moving μ from 0 to 0.5 nm shifts the graph but leaves the uncertainty product equal to ℏ/2.

Open the experiment

Predict

For the supplied minimum-uncertainty packet, halving σx while keeping the preparation class unchanged makes σp:

Your prediction and reason

Experiment

Compare widths 0.1,0.2,0.4 nm at the same center, then translate the center at fixed width. Predict height, spread and product separately. Explain why estimating a mean from more events does not create a state with a smaller intrinsic σx.

Separate quantum preparation spread from measurement bias

Worked teaching model · synthetic values · Supplied normalized Gaussian position density, prepared as a minimum-uncertainty packet with zero mean momentum. sigma_x sigma_p=hbar/2. Spatial density shown over five standard deviations is not a particle trajectory; the exact normalization refers to the whole real axis.

Position standard deviation: 0.2 nm Packet center: 0 nm

Reset keeps your prediction, captures and practice history.

Normalized prepared position density, not a trajectory

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: Gaussian position density (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Normalized prepared position density, not a trajectory

Gaussian position density

position (nm)	probability density (nm⁻¹)
-1	0.0000074336
-0.975	0.0000137797
-0.95	0.0000251475
-0.925	0.0000451819
-0.9	0.0000799187
-0.875	0.00013917
-0.85	0.000238593
-0.825	0.000402702
-0.8	0.000669151
-0.775	0.00109466
-0.75	0.00176298
-0.725	0.00279531
-0.7	0.00436341
-0.675	0.00670559
-0.65	0.0101452
-0.625	0.0151113
-0.6	0.0221592
-0.575	0.0319906
-0.55	0.0454678
-0.525	0.0636209
-0.5	0.0876415
-0.475	0.11886
-0.45	0.158698
-0.425	0.208605
-0.4	0.269955
-0.375	0.343931
-0.35	0.431387
-0.325	0.532691
-0.3	0.647588
-0.275	0.775061
-0.25	0.913245
-0.225	1.05938
-0.2	1.20985
-0.175	1.36027
-0.15	1.50569
-0.125	1.6408
-0.1	1.76033
-0.075	1.85928
-0.05	1.93334
-0.025	1.97919
0	1.99471
0.025	1.97919
0.05	1.93334
0.075	1.85928
0.1	1.76033
0.125	1.6408
0.15	1.50569
0.175	1.36027
0.2	1.20985
0.225	1.05938
0.25	0.913245
0.275	0.775061
0.3	0.647588
0.325	0.532691
0.35	0.431387
0.375	0.343931
0.4	0.269955
0.425	0.208605
0.45	0.158698
0.475	0.11886
0.5	0.0876415
0.525	0.0636209
0.55	0.0454678
0.575	0.0319906
0.6	0.0221592
0.625	0.0151113
0.65	0.0101452
0.675	0.00670559
0.7	0.00436341
0.725	0.00279531
0.75	0.00176298
0.775	0.00109466
0.8	0.000669151
0.825	0.000402702
0.85	0.000238593
0.875	0.00013917
0.9	0.0000799187
0.925	0.0000451819
0.95	0.0000251475
0.975	0.0000137797
1	0.0000074336

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Position standard deviation	0.2	nm
Mean position	0	nm
Minimum momentum standard deviation	2.63643e-25	kg m/s
Uncertainty product divided by hbar	0.5	ratio
Supplied probability within one standard deviation	0.682689	ratio
Supplied probability within two standard deviations	0.9545	ratio

Narrower prepared position spread requires larger momentum spread; translating the packet does neither.

The Gaussian density is exp[−(x−mu)²/(2sigma²)]/[sigma sqrt(2pi)]. Its integral over the whole line is one; height is density with inverse-length units, not the probability at a single point. The minimum-uncertainty preparation saturates sigma_x sigma_p=hbar/2; other states can have a larger product. This is distinct from instrumental offset or uncertainty in the sample mean. Repeated independent preparations have a distribution; the curve is not a simultaneous position-momentum trajectory.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Narrower prepared position spread requires larger momentum spread; translating the packet does neither.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

State the event and measurement model

Reconcile the quantitative evidence

For this specifically chosen minimum-uncertainty state, σxσp=ℏ/2. Reducing position standard deviation therefore increases momentum standard deviation. General states satisfy a greater-than-or-equal relation, so equality is a property of the supplied preparation, not a universal rule for arbitrary states. Translating μ moves the density without changing either standard deviation. The displayed density does not specify a simultaneous exact position and momentum for each particle.

Preserve limits and alternative explanations

For a Gaussian, the supplied probabilities within one and two standard deviations are about 0.682689 and0.954500. These are properties of repeated independent preparations, not a promise that one observation lies within either interval. Increasing the number of trials can estimate the distribution more precisely but does not shrink the intrinsic spread of the identically prepared state. Calibration offset, uncertainty of an estimated mean, and quantum preparation spread must be reported as different concepts rather than all being labeled measurement error.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

If σx=0.1 nm, find σp in units of 10⁻²⁵ kg m/s, using ℏ=1.054571817×10⁻³⁴ J s.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original cases checked 31 October 2026 against the primary explanation. These are bounded virtual models, not instructions to illuminate, accelerate, collect or expose real material. No real laboratory participation, detector certification, personal medical conclusion or independent mastery is claimed. Keep first work, critique, revision and genuinely new assigned evidence separately from public practice.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which independent observation would reject the selected model or calibration premise, and which quantity is not identified by these records?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

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