A problem of momentum representation

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Discussion Overview

The discussion revolves around a proof involving the momentum representation in quantum mechanics, specifically the relationship between the position operator \(X\) and the momentum operator \(p\) as expressed through their commutator. Participants are examining the validity of using the commutator in this context and the implications for distributional extensions.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the relevance of the commutator in proving the relationship between \(\) and the delta function, expressing confusion about its necessity.
  • Another participant asserts that the commutator defines the relationship between \(p\) and \(X\) and emphasizes that the proof should rely solely on the commutator without explicit realizations of the operators.
  • Some participants argue that the commutator is only defined for Hilbert space operators, suggesting that the proof pertains to their distributional extensions, which complicates the argument.
  • A later reply suggests that it is possible to derive the equation using the commutation relation and certain assumptions, although this is met with skepticism from others.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of using the commutator for the proof, with some asserting it is inappropriate for distributional extensions while others believe it can be applied under certain conditions. No consensus is reached on the correct approach.

Contextual Notes

There are unresolved issues regarding the assumptions necessary for applying the commutator to distributional extensions, as well as the definitions and normalization of the states involved.

fish830617
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Given
[x,p] = i * h-bar,
prove that
<p|X|p'> = [i * h-bar / (p' - p)] * δ(p - p').

I don't understand why commutator matters with this proof?
 
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The commutator tells you what the relationship between p and x is. You are not supposed to use explicit realizations of x or p, but only the commutator.

Cheers,

Jazz
 
This is nonsense. The commutator is defined for the H-space operators, the thing you got to prove is for their distributional extensions.
 
And what's written is not even true for the distributional extensions.
 
fish830617 said:
Given
[x,p] = i * h-bar,
prove that
<p|X|p'> = [i * h-bar / (p' - p)] * δ(p - p').

I don't understand why commutator matters with this proof?

Hint: try evaluating <p|[X,p]|p'>, and use the fact (not given, but based on the result this is how |p> is normalized) that <p|p'>= δ(p - p').
 
dextercioby said:
This is nonsense. The commutator is defined for the H-space operators, the thing you got to prove is for their distributional extensions.

I don't understand how is this a nonsense? I mean we can arrive at second equation (with minor correction) starting from commutation relation and certain assumptions. Can't we?
 

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