University Quantum Mechanics I · The Free Particle · 5.03

Momentum eigenstates and delta normalisation

p-hat = −i ℏ d/dx returns ℏ k on e^(ikx), so sharp momentum lives outside L2.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

p-hat = −i ℏ d/dx returns ℏ k on e^(ikx), so sharp momentum lives outside L2.

A strong response uses the ω(k) parabola and its tangent at k0 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 FFT evolution of a Gaussian packet, applying the free phase exp(−i ℏ k² t/(2m)) in k-space, reproduce σ₀ √(1 + (ℏ t/(2 m σ₀²))²), and where does the periodic grid break it?

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Does FFT evolution of a Gaussian packet, applying the free phase exp(−i ℏ k² t/(2m)) in k-space, reproduce σ₀ √(1 + (ℏ t/(2 m σ₀²))²), and where does the periodic grid break it? Useful evidence includes width fitted from |ψ|² at each step against the analytic curve, ⟨x⟩(t) against x₀ + ℏ k₀ t/m, the norm conserved to a stated tolerance, the time at which the packet reaches the grid edge and wraps, and one run started from a chirped packet that narrows before it spreads..

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 claimMomentum eigenstates and delta normalisation

p-hat = −i ℏ d/dx returns ℏ k on e^(ikx), so sharp momentum lives outside L2.

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p-hat = −i ℏ d/dx returns ℏ k on e^(ikx), so sharp momentum lives outside L2. Two standard repairs: ⟨k|k'⟩ = δ(k - k') in the 1/√(2π) convention used here, or a periodic box of length L sent to infinity. Watch the convention: writing the same statement in p as δ(p - p') requires the prefactor 1/√(2 pi ℏ). Only integrals against a weight φ(k) are physical.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

Which response about Momentum eigenstates and delta normalisation is most defensible?

04 · Evidence & boundaryDecide, then qualify

Does FFT evolution of a Gaussian packet, applying the free phase exp(−i ℏ k² t/(2m)) in k-space, reproduce σ₀ √(1 + (ℏ t/(2 m σ₀²))²), and where does the periodic grid break it?

Read the complete note

Interactive diagram for Momentum eigenstates and delta normalisation: 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 Momentum eigenstates and delta normalisation is most defensible?

Quick check

Test the reasoning, not recall

Which response about Momentum eigenstates and delta normalisation is most defensible?

Choose an answer to test the model.