University Physics V · Conservation Laws and Particle Reactions · 19.02

Time-translation invariance, Q values, and reaction thresholds

H generates time translation through U(t) = exp(−i H t / ℏ), so energy is conserved whenever H has no explicit time dependence - and it is conserved exactly at every vertex of a Feynman diagram.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

H generates time translation through U(t) = exp(−i H t / ℏ), so energy is conserved whenever H has no explicit time dependence - and it is conserved exactly at every vertex of a Feynman diagram.

A strong response uses β-spectrum and kurie plots 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

What endpoint energy does a calibrated spectrum from a sealed beta source support, and how much energy per decay is unaccounted for if nothing but an electron leaves?

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What endpoint energy does a calibrated spectrum from a sealed beta source support, and how much energy per decay is unaccounted for if nothing but an electron leaves? Useful evidence includes an energy calibration from a conversion-electron line such as Bi-207 or Cs-137, a background-subtracted spectrum, a Kurie fit with an interval on the endpoint compared with the tabulated Q value, and detector resolution and backscatter named as the limit that blocks any neutrino-mass claim.

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 written for a four-credit course of roughly three lecture hours plus two to three laboratory or computational hours a week across fifteen weeks, and it is deliberately more mathematical than University Physics I–IV: linear algebra and differential equations are working tools here, not background. Twenty units are mapped against a suggested fifteen-week delivery, so several units share a teaching week. Departments differ widely in how much formalism they expect at this stage; 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 claimTime-translation invariance, Q values, and reaction thresholds

H generates time translation through U(t) = exp(−i H t / ℏ), so energy is conserved whenever H has no explicit time dependence - and it is conserved exactly at every vertex of a Feynman diagram.

Read the complete note

H generates time translation through U(t) = exp(−i H t / ℏ), so energy is conserved whenever H has no explicit time dependence - and it is conserved exactly at every vertex of a Feynman diagram. Q follows from tabulated masses; the threshold follows from the invariant s = (sum of p)², which must satisfy √(s) ≥ sum of final masses. An internal line is off shell, q² ≠ m² c², a statement about the propagator denominator and not about energy being borrowed: δ-E δ-t ~ ℏ is a lifetime-linewidth relation, not a commutator uncertainty, since no time operator conjugate to H exists.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

When should a model used for Time-translation invariance, Q values, and reaction thresholds be revised?

04 · Evidence & boundaryDecide, then qualify

What endpoint energy does a calibrated spectrum from a sealed beta source support, and how much energy per decay is unaccounted for if nothing but an electron leaves?

Read the complete note

Interactive diagram for Time-translation invariance, Q values, and reaction thresholds: 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

When should a model used for Time-translation invariance, Q values, and reaction thresholds be revised?

Quick check

Test the reasoning, not recall

When should a model used for Time-translation invariance, Q values, and reaction thresholds be revised?

Choose an answer to test the model.