University Quantum Mechanics I · Introduction to Three-Dimensional Quantum Mechanics · 14.02
Additive Cartesian potentials and the three-dimensional box
With V = Vx + Vy + Vz the product X(x)Y(y)Z(z) splits the equation into three solved one-dimensional ones whose energies add, so a cubical box gives E = (π² ℏ² / 2mL²)(nx² + ny² + nz²).
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
With V = Vx + Vy + Vz the product X(x)Y(y)Z(z) splits the equation into three solved one-dimensional ones whose energies add, so a cubical box gives E = (π² ℏ² / 2mL²)(nx² + ny² + nz²).A strong response uses radial functions r(r) beside u = rr 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
Do the resonances of a circular membrane sit at ratios of the Bessel zeros, and do the visible nodal circles and diameters match the two indices of the separated solution?
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Do the resonances of a circular membrane sit at ratios of the Bessel zeros, and do the visible nodal circles and diameters match the two indices of the separated solution? Useful evidence includes driven frequencies with nodal patterns recorded, modes labelled by nodal circles and diameters, ratios against Bessel zeros, split degenerate pairs traced to tension asymmetry, and the ideal-membrane assumption named.
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.
Model, evidence, and boundary
When should a model used for Additive Cartesian potentials and the three-dimensional box be revised?
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.
When should a model used for Additive Cartesian potentials and the three-dimensional box be revised?
Quick check
Test the reasoning, not recall
When should a model used for Additive Cartesian potentials and the three-dimensional box be revised?
Choose an answer to test the model.