Niles
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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
<br /> 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|},<br />
where the first part of the sum denotes the matrix elements of T. My question is why the author is using
<br /> \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|} ,<br />
when in fact it is given by
<br /> \hat 1 = \sum\limits_\sigma {\left| {\psi _\sigma \left( {r_i } \right)} \right\rangle \left\langle {\psi _\sigma \left( {r_i } \right)} \right|} <br />
I hope you will shed some light on this.
Niles.