SCIENCE · INVESTIGATE AND EXPLAIN

Evolve interference while retaining fixed energy probabilities

Prepare equal coherent amplitudes of the first two stationary infinite-well states. After removing an irrelevant overall phase, psi=[phi1+exp(i theta)phi2]/sqrt(2), where theta=theta0−(E2−E1)t/hbar. Because E2=4E1, the relative angular frequency is 3E1/hbar, not E1/hbar.

Learning outcome: Predict local probability at phases 0,π/2,π before revealing it. Integrate the cross term over the entire well to verify zero and over a half-well to see why it survives. Compare a coherent preparation with an incoherent mixture that has the same energy populations.

Starting recap

Review definite integrals, complex amplitudes, probability moments, elementary differential equations and energy-momentum units. State boundary and normalization conditions, and distinguish calibration from unused validation.

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Try

At theta=0, left-half probability is 1/2+4/(3π)≈0.924413182 and mean position is about 0.319873451 L. At theta=π, the left-half probability is about 0.075586818 and the mean is reflected to about 0.680126549 L. In both states the two energy probabilities remain 1/2.

Open the experiment

Predict

What remains constant while relative phase changes in this isolated two-state preparation?

Your prediction and reason

Experiment

Predict local probability at phases 0,π/2,π before revealing it. Integrate the cross term over the entire well to verify zero and over a half-well to see why it survives. Compare a coherent preparation with an incoherent mixture that has the same energy populations.

Evolve local probability without inventing a trajectory

Worked teaching model · synthetic values · Equal-amplitude coherent superposition of n1 and n2 infinite-well states. Relative phase theta evolves as theta0−(E2−E1)t/hbar; controls sample theta, not a classical particle path. Scaled coordinate u=x/L; energy probabilities remain one half each while interference changes local probability.

Relative phase divided by pi: 0 ratio Upper interval position divided by length: 0.5 ratio

Reset keeps your prediction, captures and practice history.

Relative phase moves probability, not a drawn particle

Graph axes rescale independently; read units and values, not screen angles. 1: coherent equal-weight state (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Relative phase moves probability, not a drawn particle

coherent equal-weight state

u = x/L	density per unit u
0	0
0.01666667	0.02460645
0.03333333	0.09761885
0.05	0.2166451
0.06666667	0.3777926
0.08333333	0.5758063
0.1	0.8042543
0.1166667	1.055753
0.1333333	1.322227
0.15	1.595188
0.1666667	1.866025
0.1833333	2.126302
0.2	2.368034
0.2166667	2.583953
0.2333333	2.76774
0.25	2.914214
0.2666667	3.019486
0.2833333	3.081055
0.3	3.097859
0.3166667	3.070261
0.3333333	3
0.35	2.89008
0.3666667	2.744623
0.3833333	2.568683
0.4	2.368034
0.4166667	2.148939
0.4333333	1.917904
0.45	1.681445
0.4666667	1.445847
0.4833333	1.216958
0.5	1
0.5166667	0.7994167
0.5333333	0.6187556
0.55	0.4605948
0.5666667	0.3265105
0.5833333	0.2170869
0.6	0.131966
0.6166667	0.06993371
0.6333333	0.02903638
0.65	0.006722283
0.6666667	4.930381e-32
0.6833333	0.005606249
0.7	0.02017523
0.7166667	0.04040166
0.7333333	0.06319043
0.75	0.08578644
0.7666667	0.1058795
0.7833333	0.1216804
0.8	0.131966
0.8166667	0.1360919
0.8333333	0.1339746
0.85	0.1260438
0.8666667	0.1131705
0.8833333	0.09657343
0.9	0.07771174
0.9166667	0.05816825
0.9333333	0.03953136
0.95	0.02328143
0.9666667	0.01068809
0.9833333	0.002724058
1	0

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
Probability in selected left interval	0.9244132	ratio
Mean position divided by length	0.3198735	ratio
Energy-one probability	0.5	ratio
Energy-two probability	0.5	ratio
Mean energy divided by ground energy	2.5	ratio
Full probability	1	ratio

Local interference can evolve while energy probabilities and total normalization remain fixed.

For equal coefficients, density is sin²(pi u)+sin²(2pi u)+2cos(theta)sin(pi u)sin(2pi u). Integrating the cross term gives sin(pi u)/pi−sin(3pi u)/(3pi). Orthogonality removes its full-domain integral but not a local one. The mean coordinate is1/2−16cos(theta)/(9pi²). The relative phase evolves with energy difference3E1, not the global phase or a classical oscillating particle position.

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

Local interference can evolve while energy probabilities and total normalization remain fixed.

Observation to attach to a capture

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

Understand

Define the state, source and evidence

Derive the quantitative relation

With u=x/L, density per unit u is sin²(πu)+sin²(2πu)+2cos(theta)sin(πu)sin(2πu). The cross term integrates to zero over the full well by orthogonality, but not over a partial interval. Its integral to w is cos(theta)[sin(πw)/π−sin(3πw)/(3π)]. Add this to the average of the two stationary cumulative probabilities.

Check normalization, limits and inference

The left-half probability is 1/2+4cos(theta)/(3π). Integrating u times the density gives mean(u)=1/2−16cos(theta)/(9π²). Meanwhile an energy measurement returns E1 or E2 with probability one half each, so mean energy stays 2.5E1. Time-dependent local probability does not require changing those populations.

Compare a changed premise

An incoherent equal mixture has the same two energy probabilities but no cross term. A local position measurement sensitive to phase can distinguish that mixture from the coherent preparation. A moving density or expectation does not specify an individual particle trajectory; the workbench samples relative phase without simulating a measurement history.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

At theta=π/2, find probability in the left half of the well.

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 derivations and virtual evidence checked 31 October 2026 against the primary explanation. All questions, numerical packets and diagrams are original. These models do not establish actual beam, radiation, vacuum, optical or electrical laboratory performance. Public feedback is practice, not authenticated independent mastery. Preserve first reasoning, criterion critique, revision and a genuinely new assigned packet in the course evidence workspace.

Apply

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

Which state, calibration or boundary premise is independently justified, and what unused observation could distinguish an alternative?

Your explanation and limitations

Continue

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

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