Rules for transforming operators

[aaaa202]

The attached picture shows a representation of a general operator, which I found quite weird. The matrix elements are calculated in the position basis as far as I can tell, but I am not sure how. Do they do something like?

<klTlk'> = ∫ dx dx' <klx><xlTlx'><x'lk'>

In that case what happens to the double integral?

 

[Simon Bridge]

<klTlk'> = ∫ dx dx' <klx><xlTlx'><x'lk'>

$$<k|T|k'>=\int \psi_k^\star(x) \hat T \psi_{k'}(x) dx$$

... then expand $\psi_{k'}$ in terms of the eigenstates of $\hat T$.

It will probably help you understand the representation is you consider the case for only two particles (if I've read that correctly). You should also make explicit what each of the indexes mean ... the small number of articles will allow you to expand out the sums.

 

[aaaa202]

No I mean according to my calculation;

<klVlk'> = ∫∫dr dr' $\psi$_k(x) V(r,r') $\psi$_k'(x')

But your integral is a single integral. What have you done to achieve that?

 

[Simon Bridge]

Only one dimension ... I was attempting to illustrate what I meant about being careful about the definitions.
It looks to me like you don't quite understand what the different terms are for - but I cannot be sure because you don't seem to want to talk about it.