QM: Changing Basis | Why Use T_{σa,σb}?

[Niles]

Homework Statement

Hi

Say I have the kinetic energy operator denoted by T(r_i) for the particle i. I wish to represent it in some $\left| \sigma \right\rangle$-representation. My book says it is given by

$$T = \sum\limits_{\sigma _a ,\sigma _b } {T_{\sigma _a ,\sigma _b } \left| {\psi _{\sigma _a } \left( {r_i } \right)} \right\rangle \left\langle {\psi _{\sigma _b } \left( {r_i } \right)} \right|},$$

where the first part of the sum denotes the matrix elements of T. My question is why the author is using

$$\hat 1 = \sum\limits_{\sigma _a ,\sigma _b } {\left| {\psi _{\sigma _a } \left( {r_i } \right)} \right\rangle \left\langle {\psi _{\sigma _b } \left( {r_i } \right)} \right|} ,$$

when in fact it is given by

$$\hat 1 = \sum\limits_\sigma {\left| {\psi _\sigma \left( {r_i } \right)} \right\rangle \left\langle {\psi _\sigma \left( {r_i } \right)} \right|}$$

I hope you will shed some light on this.


Niles.

 

[fzero]

He is, but part of the expression has been contracted to form the matrix elements of T. Specifically, start with

$$T = \hat{1} T \hat{1} = \left( \sum\limits_{\sigma_a} {\left| {\psi _{\sigma_a} \left( {r_i } \right)} \right\rangle \left\langle {\psi_{\sigma_a} \left( {r_i } \right)} \right|}\right) T \left( \sum\limits_{\sigma_b} {\left| {\psi _{\sigma_b} \left( {r_i } \right)} \right\rangle \left\langle {\psi_{\sigma_b} \left( {r_i } \right)} \right|} \right)$$

$$= \sum\limits_{\sigma_a,\sigma_b } {\left| {\psi _{\sigma_a} \left( {r_i } \right)} \right\rangle \left\langle {\psi_{\sigma_a} \left( {r_i } \right)} \right|} T {\left| {\psi _{\sigma_b} \left( {r_i } \right)} \right\rangle \left\langle {\psi_{\sigma_b} \left( {r_i } \right)} \right|} = \sum\limits_{\sigma_a,\sigma_b } {\left| {\psi _{\sigma_a} \left( {r_i } \right)} \right\rangle T_{\sigma_a\sigma_b \right\rangle \left\langle {\psi_{\sigma_b} \left( {r_i } \right)} \right|}$$

 

[Niles]

You are right, thanks!