A problem of momentum representation

[fish830617]

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?

 

[Jazzdude]

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

 

[dextercioby]

This is nonsense. The commutator is defined for the H-space operators, the thing you got to prove is for their distributional extensions.

 

[Avodyne]

And what's written is not even true for the distributional extensions.

 

[kaplan]

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').

 

[Ravi Mohan]

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?