A problem of momentum representation

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SUMMARY

The discussion centers on the proof of the equation = [i * h-bar / (p' - p)] * δ(p - p') using the commutation relation [x,p] = i * h-bar. Participants debate the relevance of the commutator in this context, with one user asserting that the proof should focus on distributional extensions rather than H-space operators. The importance of evaluating and the normalization condition = δ(p - p') is emphasized as critical to understanding the proof.

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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?
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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