University Quantum Mechanics I · Potential Barriers and Quantum Tunneling · 8.06
Barriers that are not rectangular: stacking slabs into an integral
Slice a smooth V(x) into thin rectangles and multiply their exponentials: T ~ exp(−2 times the integral of κ(x) dx) between the classical turning points.
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
Slice a smooth V(x) into thin rectangles and multiply their exponentials: T ~ exp(−2 times the integral of κ(x) dx) between the classical turning points.A strong response uses real and imaginary parts of psi across faces and states where the model stops being reliable.
Reasoning checklist
Evidence, assumptions and limits
Assumptions to state
State the system, observable, approximation, and conditions held fixed before using a model.
Evidence to collect
Does microwave power crossing an air gap between two paraffin prisms fall exponentially with gap width, and does the fitted decay constant match the evanescent prediction?
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Does microwave power crossing an air gap between two paraffin prisms fall exponentially with gap width, and does the fitted decay constant match the evanescent prediction? Useful evidence includes transmitted power against gap width at fixed angle, a log-linear fit giving the decay constant with uncertainty, that constant against the value predicted from angle and index, standing waves named as a systematic, and a written statement that this is a classical electromagnetic analogue of barrier penetration - the same exponential mathematics, not a quantum measurement..
Limits to state
This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus.
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This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus. It is a first dedicated quantum-mechanics course, distinct from University Physics V, which covers quantum mechanics alongside atomic, nuclear and particle physics in twenty units: this course is narrower, slower, and teaches the linear algebra it needs rather than assuming it. Hydrogen appears here as an introduction, with the full radial derivation belonging to a second course. A midterm examination is assumed around week 8. Follow your institution's published scope, notation, laboratory programme, and assessment rules. A result should be checked against units, signs, limiting cases, and the conditions under which its model was derived.
Diagram & examples
Work the claim before choosing an equation
Interactive concept map
Follow the model from claim to evidence.
Slice a smooth V(x) into thin rectangles and multiply their exponentials: T ~ exp(−2 times the integral of κ(x) dx) between the classical turning points. It is WKB's result reached without WKB, and it fails where kappa varies fast - at the turning points.
Model, evidence, and boundary
What is the strongest test of a claim about Barriers that are not rectangular: stacking slabs into an integral?
modelModel, evidence, and boundary turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
What is the strongest test of a claim about Barriers that are not rectangular: stacking slabs into an integral?
Quick check
Test the reasoning, not recall
What is the strongest test of a claim about Barriers that are not rectangular: stacking slabs into an integral?
Choose an answer to test the model.