University Quantum Mechanics I · Potential Barriers and Quantum Tunneling · 8.05
Dependence on barrier width, particle energy, and mass
Width enters the exponent linearly while mass and V0 - E enter only under a square root, so doubling a squares the suppression but doubling m stretches the exponent by 1.41.
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
Width enters the exponent linearly while mass and V0 - E enter only under a square root, so doubling a squares the suppression but doubling m stretches the exponent by 1.41.A strong response uses potential-energy diagrams with e marked 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
Over what range of kappa a and E/V0 does T ~ exp(−2 κ a) reproduce the exact rectangular-barrier transmission to within a factor of two?
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Over what range of kappa a and E/V0 does T ~ exp(−2 κ a) reproduce the exact rectangular-barrier transmission to within a factor of two? Useful evidence includes exact T(E) and both approximate forms on a log axis for several widths, the ratio of estimate to exact against kappa a and E/V0, the kappa a where they agree to a stated tolerance, and the E → 0 and E → V0 failures..
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.
Width enters the exponent linearly while mass and V0 - E enter only under a square root, so doubling a squares the suppression but doubling m stretches the exponent by 1.41. Read as orders of magnitude: a ten per cent error in kappa a is a whole factor in T.
Model, evidence, and boundary
What must be stated before selecting an equation for Dependence on barrier width, particle energy, and mass?
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 must be stated before selecting an equation for Dependence on barrier width, particle energy, and mass?
Quick check
Test the reasoning, not recall
What must be stated before selecting an equation for Dependence on barrier width, particle energy, and mass?
Choose an answer to test the model.