QM: Changing basis

  • Thread starter Niles
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  • #1
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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 [itex]\left| \sigma \right\rangle [/itex]-representation. My book says it is given by

[tex]
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|},
[/tex]

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

[tex]
\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|} ,
[/tex]

when in fact it is given by

[tex]
\hat 1 = \sum\limits_\sigma {\left| {\psi _\sigma \left( {r_i } \right)} \right\rangle \left\langle {\psi _\sigma \left( {r_i } \right)} \right|}
[/tex]

I hope you will shed some light on this.

Best,
Niles.
 

Answers and Replies

  • #2
fzero
Science Advisor
Homework Helper
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He is, but part of the expression has been contracted to form the matrix elements of T. Specifically, start with

[tex] 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) [/tex]

[tex] = \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|}[/tex]
 
  • #3
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You are right, thanks!
 

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