Quick question on Fermi Golden Rule

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Discussion Overview

The discussion revolves around the Fermi Golden Rule and its application in a specific problem involving massless particles B and C. Participants explore the relationship between momentum and energy, particularly in the context of conservation laws.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions the relationship between momentum and energy, suggesting that ##\frac{dp}{dE} = \frac{E}{\sqrt{E^2 - m^2}}## should hold true.
  • Another participant agrees with the initial claim but notes that the problem specifies particles B and C are massless, leading to the conclusion that their momenta equal their energies, resulting in ##dp_B/dE=1/2##.
  • A later reply reiterates the massless condition and the conservation of energy, confirming the earlier calculations regarding momenta and energy distributions.
  • There is a brief mention of the speed of light being set to 1, indicating a potential assumption in the context of the discussion.

Areas of Agreement / Disagreement

Participants generally agree on the implications of masslessness for particles B and C and the resulting calculations, but the initial relationship proposed by the first participant is not fully resolved, as it is contingent on the definitions used.

Contextual Notes

The discussion assumes massless particles without explicitly defining the implications of this assumption on the broader context of the Fermi Golden Rule. The relationship between momentum and energy is also dependent on the definitions and conditions set by the problem.

unscientific
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Adopted from my lecture notes, found it a little fishy:

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Shouldn't ##\frac{dp}{dE} = \frac{E}{p}## given that ##p = \sqrt{E^2 - m^2}##. Then the relation should be instead:

\frac{dp}{dE} = \frac{E}{p} = \frac{E}{\sqrt{E^2 - m^2}}
 
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Yes, you are right but the problem tells you that B and C are massless. This means that the magnitude of their momenta is equal to their energies. In particular, conservation of energy tells you that ##E_B=E_C=E_A/2## and so ##p_B=p_C=E_A/2##, giving ##dp_B/dE=1/2##.
 
Einj said:
Yes, you are right but the problem tells you that B and C are massless. This means that the magnitude of their momenta is equal to their energies. In particular, conservation of energy tells you that ##E_B=E_C=E_A/2## and so ##p_B=p_C=E_A/2##, giving ##dp_B/dE=1/2##.
And I suppose ##c=1##?
 
Exactly.
 
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