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University Quantum Mechanics I · Finite Square Wells · 7.09

Solving the well numerically by shooting

Integrate outward from the centre at a trial energy and watch the far tail: as E crosses an eigenvalue the divergence of that tail flips sign, so the roots can be bracketed and then bisected.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

Integrate outward from the centre at a trial energy and watch the far tail: as E crosses an eigenvalue the divergence of that tail flips sign, so the roots can be bracketed and then bisected.

A strong response uses well profiles with the bound levels drawn in and states where the model stops being reliable.

Reasoning checklist

Evidence, assumptions and limits

01

Assumptions to state

State the system, observable, approximation, and conditions held fixed before using a model.

02

Evidence to collect

Do graphical roots, a bracketed root-finder, and outward shooting return the same bound-state energies for one well, and where do they part?

Read the complete note

Do graphical roots, a bracketed root-finder, and outward shooting return the same bound-state energies for one well, and where do they part? Useful evidence includes the z0 circle plot with its intersections read off, roots from a bracketed solver, eigenvalues from shooting, the three sets tabulated against the floor(2 z0/π) + 1 prediction, and the tolerance at which the methods disagree..

03

Limits to state

This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus.

Read the complete note

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.

01 · Physical claimSolving the well numerically by shooting

Integrate outward from the centre at a trial energy and watch the far tail: as E crosses an eigenvalue the divergence of that tail flips sign, so the roots can be bracketed and then bisected.

Read the complete note

Integrate outward from the centre at a trial energy and watch the far tail: as E crosses an eigenvalue the divergence of that tail flips sign, so the roots can be bracketed and then bisected. The method works for wells with no closed form at all, but the growing solution contaminates a long outward integration.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

Which response about Solving the well numerically by shooting is most defensible?

04 · Evidence & boundaryDecide, then qualify

Do graphical roots, a bracketed root-finder, and outward shooting return the same bound-state energies for one well, and where do they part?

Read the complete note

Do graphical roots, a bracketed root-finder, and outward shooting return the same bound-state energies for one well, and where do they part? Useful evidence includes the z0 circle plot with its intersections read off, roots from a bracketed solver, eigenvalues from shooting, the three sets tabulated against the floor(2 z0/π) + 1 prediction, and the tolerance at which the methods disagree..

Interactive diagram for Solving the well numerically by shooting: follow the physical claim through its representation, proposed test, evidence, and model boundary.

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.

Diagram check

Which response about Solving the well numerically by shooting is most defensible?

Quick check

Test the reasoning, not recall

Which response about Solving the well numerically by shooting is most defensible?

Choose an answer to test the model.