University Quantum Mechanics I · The Infinite Square Well · 6.06

Expanding an arbitrary state on the eigenbasis

Any square-integrable Ψ(x,0) on [0, L] expands as sum cₙ ψₙ(x), with cₙ = ∫ ψₙ* Ψ(x,0) dx extracted by orthogonality and sum |cₙ|² = 1 once normalised.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

Any square-integrable Ψ(x,0) on [0, L] expands as sum cₙ ψₙ(x), with cₙ = ∫ ψₙ* Ψ(x,0) dx extracted by orthogonality and sum |cₙ|² = 1 once normalised.

A strong response uses ψₙ and |ψₙ|² stacked against x 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 a two-term superposition oscillate at angular frequency (E₂ - E₁)/ℏ, and does a many-term packet return at the revival time 4mL²/(π ℏ)?

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Does a two-term superposition oscillate at angular frequency (E₂ - E₁)/ℏ, and does a many-term packet return at the revival time 4mL²/(π ℏ)? Useful evidence includes snapshots of |Ψ(x, t)|², ⟨x⟩(t) with its fitted period set against 2 pi ℏ/(E₂ - E₁), |cₙ|² and ⟨H⟩ shown numerically constant, and the revival traced to level gaps that are integer multiples of E₁.

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.

01 · Physical claimExpanding an arbitrary state on the eigenbasis

Any square-integrable Ψ(x,0) on [0, L] expands as sum cₙ ψₙ(x), with cₙ = ∫ ψₙ* Ψ(x,0) dx extracted by orthogonality and sum |cₙ|² = 1 once normalised.

Read the complete note

Any square-integrable Ψ(x,0) on [0, L] expands as sum cₙ ψₙ(x), with cₙ = ∫ ψₙ* Ψ(x,0) dx extracted by orthogonality and sum |cₙ|² = 1 once normalised. The star is invisible here only because these ψₙ are real; keep it, because later bases are not. A physical state also vanishes at the walls, but the expansion itself converges even for a trial function that does not, such as a constant, which is where the overshoot lives.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

Which response about Expanding an arbitrary state on the eigenbasis is most defensible?

04 · Evidence & boundaryDecide, then qualify

Does a two-term superposition oscillate at angular frequency (E₂ - E₁)/ℏ, and does a many-term packet return at the revival time 4mL²/(π ℏ)?

Read the complete note

Interactive diagram for Expanding an arbitrary state on the eigenbasis: 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 Expanding an arbitrary state on the eigenbasis is most defensible?

Quick check

Test the reasoning, not recall

Which response about Expanding an arbitrary state on the eigenbasis is most defensible?

Choose an answer to test the model.