QM: Operator in momentum representation

[Niles]

Homework Statement

Hi guys

As we have discussed earlier, we can represent some operator in an arbitrary basis by the use of the 1-operator:

<br /> T = \hat{1} T \hat{1} = \sum\limits_{\sigma_a,\sigma_b } {\left| {\psi _{\sigma_a} \left( {r_i } \right)} \right\rangle T_{\sigma_a\sigma_b \right\rangle \left\langle {\psi_{\sigma_b} \left( {r_i } \right)} \right|}<br />

However, in my book they represent the kinetic energy operator in momentum space by the following (disregarding spin)

<br /> \left\langle {{\bf{k}}&#039;} \right|T\left| {\bf{k}} \right\rangle \propto k^2 \delta _{{\bf{k}},{\bf{k}}&#039;}.<br />

I cannot seem to connect these two methods of representing operators in some basis. How can one realize that the book's way of transforming is the same as ours with 1-operators?Niles.

 

[fzero]

The expression

<br /> \left\langle {{\bf{k}}&#039;} \right|T\left| {\bf{k}} \right\rangle \propto k^2 \delta _{{\bf{k}},{\bf{k}}&#039;}<br />

is for the matrix elements of T. The corresponding operator could be written as

\hat{T} = \sum_k c k^2 |k\rangle\langle k|,

where c is the proportionality constant (probably 1/(2m)).

 

[Niles]

The expression

<br /> \left\langle {{\bf{k}}&#039;} \right|T\left| {\bf{k}} \right\rangle \propto k^2 \delta _{{\bf{k}},{\bf{k}}&#039;}<br />

is for the matrix elements of T. The corresponding operator could be written as

\hat{T} = \sum_k c k^2 |k\rangle\langle k|,

where c is the proportionality constant (probably 1/(2m)).

Thanks, but how do we know what I have highlighted above? I can see that you have inserted the 1-operator.

 

[fzero]

All I did was substitute the matrix elements into the corresponding version of the equation <br /> <br /> T = \sum\limits_{\sigma_a,\sigma_b } {\left| {\psi _{\sigma_a} \left( {r_i } \right)} \right\rangle T_{\sigma_a\sigma_b \right\rangle \left\langle {\psi_{\sigma_b} \left( {r_i } \right)} \right|}<br /> <br />

 

[Niles]

Thanks!