# QM: Changing basis

## Homework Statement

Hi

Say I have the kinetic energy operator denoted by T(ri) 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.

Best,
Niles.

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

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

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

You are right, thanks!