SCIENCE · INVESTIGATE AND EXPLAIN

Use a matching solution without confusing amplitude and flux

Consider a rectangular barrier of height V0 and width L, with identical zero-potential leads on both sides. In each constant-potential region solve the stationary Schrödinger equation; impose continuity of amplitude and derivative at both interfaces. The supplied closed matching result is expressed using e=E/V0 and w=L sqrt(2mV0)/hbar.

Learning outcome: Compare below, at and above threshold at the same width, then halve the width. Derive the threshold limit before checking it. Reconstruct a supplied unequal-lead current ratio and explain why the equal-lead shortcut cannot be exported to that problem.

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

For e=1/2 and w=2, the coefficient 4e(1−e) is one, giving T=1/[1+sinh²(sqrt(2))]=sech²(sqrt(2))≈0.210771094. At the threshold e=1 with the same width, T=1/2. At w=0 both branches give T=1.

Open the experiment

Predict

For unequal lead wave numbers, which expression gives current transmission?

Your prediction and reason

Experiment

Compare below, at and above threshold at the same width, then halve the width. Derive the threshold limit before checking it. Reconstruct a supplied unequal-lead current ratio and explain why the equal-lead shortcut cannot be exported to that problem.

Keep probability current and amplitude distinct

Worked teaching model · synthetic values · Rectangular barrier with identical zero-potential leads, positive E,V0, supplied exact matching solution. Below, at and above threshold use separate analytic forms; E=V0 is a finite limit, not division by zero. Equal lead speeds permit T=|t|² here; unequal leads require their current ratio. No tunneling-time or real device-operation claim.

Particle energy divided by barrier height: 0.5 ratio Barrier width times sqrt(2mV0)/hbar: 2 ratio

Reset keeps your prediction, captures and practice history.

Finite-width matching, not a classical step rule

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

Equivalent numerical coordinates for Finite-width matching, not a classical step rule

equal-lead exact barrier model

dimensionless barrier width	current transmission
0	1
0.05	0.998751
0.1	0.9950166
0.15	0.9888338
0.2	0.9802637
0.25	0.9693897
0.3	0.9563164
0.35	0.9411669
0.4	0.924081
0.45	0.9052119
0.5	0.8847241
0.55	0.8627897
0.6	0.8395865
0.65	0.8152943
0.7	0.7900932
0.75	0.7641602
0.8	0.737668
0.85	0.7107822
0.9	0.6836605
0.95	0.6564507
1	0.6292903
1.05	0.6023053
1.1	0.57561
1.15	0.5493069
1.2	0.5234863
1.25	0.4982268
1.3	0.4735959
1.35	0.4496499
1.4	0.4264349
1.45	0.4039876
1.5	0.3823354
1.55	0.361498
1.6	0.3414874
1.65	0.3223091
1.7	0.3039628
1.75	0.2864431
1.8	0.2697398
1.85	0.2538393
1.9	0.2387245
1.95	0.2243757
2	0.2107711
2.05	0.197887
2.1	0.1856986
2.15	0.17418
2.2	0.1633048
2.25	0.153046
2.3	0.1433768
2.35	0.1342704
2.4	0.1257001
2.45	0.1176398
2.5	0.1100641
2.55	0.1029478
2.6	0.09626686
2.65	0.08999781
2.7	0.08411809
2.75	0.07860596
2.8	0.07344059
2.85	0.06860204
2.9	0.06407129
2.95	0.05983018
3	0.05586144

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
Transmitted probability-current fraction	0.2107711	ratio
Reflected probability-current fraction	0.7892289	ratio
Current conservation sum	1	ratio
Unequal-lead example transmitted amplitude squared	0.8	ratio
Unequal-lead example speed ratio	0.5	ratio
Unequal-lead example current transmission	0.4	ratio

Transmission is a probability-current ratio; equal asymptotic speeds are a premise, not a universal identity.

For e=E/V0 below1, T=[1+sinh²(w sqrt(1−e))/(4e(1−e))]^(−1). Above1 replace sinh with sin and1−e with e−1. At e=1 the continuous limit is1/(1+w²/4). At zero width T=1 exactly. The probability current j=(hbar/m)Im(psiStar psiPrime) makes T=(kRight/kLeft)|t|² for unequal leads. Nonzero transmission below the barrier does not violate total energy conservation or determine a transit time.

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

Transmission is a probability-current ratio; equal asymptotic speeds are a premise, not a universal identity.

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

For e below one, transmission is T=[1+sinh²(w sqrt(1−e))/(4e(1−e))]^−1. Above one, replace sinh by sin and the positive difference by e−1. Exactly at e=1, take the limit using sinh(z)/z or sin(z)/z approaching one: T=1/(1+w²/4). Substituting e=1 into the unsimplified denominator would manufacture division by zero.

Check normalization, limits and inference

The stationary probability current is j=(hbar/m)Im(psiStar psiPrime). Therefore transmission is (kRight/kLeft)|t|². It equals amplitude squared in this workbench only because the two lead potentials and masses match. For an independently supplied unequal-lead example with speed ratio 1/2 and amplitude squared 0.8, the current transmission is 0.4, not 0.8.

Compare a changed premise

At zero width the potential barrier disappears and T=1 exactly. At finite below-threshold width, T is positive and decreases as the evanescent region thickens; reflection plus transmission is one for this lossless stationary scattering problem. This does not provide a traversal-time measurement, a superluminal signal or permission to ignore total energy conservation.

Worked original case

For e=1/2 and w=2, the coefficient 4e(1−e) is one, giving T=1/[1+sinh²(sqrt(2))]=sech²(sqrt(2))≈0.210771094. At the threshold e=1 with the same width, T=1/2. At w=0 both branches give T=1.

Investigation

Changed-case practice

Try the changed case before feedback

At exact threshold e=1 and width parameter w=1, find T.

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.

The exact above- and below-barrier matching coefficients are also verified against Fitzpatrick’s square potential barrier derivation, equations 4.119 and 4.128. The threshold limit and unequal-lead current comparison are derived explicitly here.

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.

Skipping retains experiments and awards no mastery.

Previous: Find a bound state by matching a finite boundary and its tail Next: Compare two spectral fits with equal data budgets and a new observation

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