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← University Quantum Mechanics I

University Quantum Mechanics I · The Quantum Harmonic Oscillator · 9.04

Eₙ = (n + 1/2) ℏ omega and the energy that will not go away

Every gap is ℏ omega wide, where the infinite well's widen as 2n + 1, and the half-quantum left at n = 0 is what minimising ⟨H⟩ = ⟨p²⟩/2m + (1/2) m ω² ⟨x²⟩ subject to δ-x δ-p ≥ ℏ/2 returns.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

Every gap is ℏ omega wide, where the infinite well's widen as 2n + 1, and the half-quantum left at n = 0 is what minimising ⟨H⟩ = ⟨p²⟩/2m + (1/2) m ω² ⟨x²⟩ subject to δ-x δ-p ≥ ℏ/2 returns.

A strong response uses classical and quantum position histograms 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 the vibrational band spacings of iodine vapour stay constant, as an exactly quadratic potential requires, or converge towards a dissociation limit?

Read the complete note

Do the vibrational band spacings of iodine vapour stay constant, as an exactly quadratic potential requires, or converge towards a dissociation limit? Useful evidence includes calibrated band positions, spacings plotted against quantum number, a Birge-Sponer extrapolation to the dissociation energy, an anharmonicity constant with uncertainty, and the level where equal spacing stops fitting..

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 claimEₙ = (n + 1/2) ℏ omega and the energy that will not go away

Every gap is ℏ omega wide, where the infinite well's widen as 2n + 1, and the half-quantum left at n = 0 is what minimising ⟨H⟩ = ⟨p²⟩/2m + (1/2) m ω² ⟨x²⟩ subject to δ-x δ-p ≥ ℏ/2 returns.

Read the complete note

Every gap is ℏ omega wide, where the infinite well's widen as 2n + 1, and the half-quantum left at n = 0 is what minimising ⟨H⟩ = ⟨p²⟩/2m + (1/2) m ω² ⟨x²⟩ subject to δ-x δ-p ≥ ℏ/2 returns. The bound is a property of non-commuting x and p, not a limit on apparatus. Equal spacing lasts only while V stays exactly quadratic.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

What must be stated before selecting an equation for Eₙ = (n + 1/2) ℏ omega and the energy that will not go away?

04 · Evidence & boundaryDecide, then qualify

Do the vibrational band spacings of iodine vapour stay constant, as an exactly quadratic potential requires, or converge towards a dissociation limit?

Read the complete note

Do the vibrational band spacings of iodine vapour stay constant, as an exactly quadratic potential requires, or converge towards a dissociation limit? Useful evidence includes calibrated band positions, spacings plotted against quantum number, a Birge-Sponer extrapolation to the dissociation energy, an anharmonicity constant with uncertainty, and the level where equal spacing stops fitting..

Interactive diagram for Eₙ = (n + 1/2) ℏ omega and the energy that will not go away: 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

What must be stated before selecting an equation for Eₙ = (n + 1/2) ℏ omega and the energy that will not go away?

Quick check

Test the reasoning, not recall

What must be stated before selecting an equation for Eₙ = (n + 1/2) ℏ omega and the energy that will not go away?

Choose an answer to test the model.