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## Main Question or Discussion Point

In the Anderson model, it cost an energy [tex]Un_{\Uparrow}n_{\Downarrow}[/tex] for a quantum dot level to be occupied by two electrons. Here [tex]n_{\Uparrow}[/tex] is the second quantized number operator, counting the number of particles with spin [tex]\Uparrow[/tex]. I need the term [tex]Un_{\Uparrow}n_{\Downarrow}[/tex] in first quantization. Here is what I know:

[tex]Un_{\Uparrow}n_{\Downarrow} =

Ud_{\Uparrow}^{\dagger}d_{\Uparrow}d_{\Downarrow}^{\dagger}d_{\Downarrow}

=

-Ud_{\Uparrow}^{\dagger}d_{\Downarrow}^{\dagger}d_{\Uparrow}d_{\Downarrow}

=

\frac{1}{2}\sum_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}d_{\eta_{1}}^{\dagger}d_{\eta_{2}}^{\dagger}d_{\eta_{3}}d_{\eta_{4}}

[/tex]

where

[tex]

V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}=\Big\{

\begin{array}{c}

-2U \qquad \text{for} \qquad \eta_{1}=\eta_{2}=\Uparrow,\: \eta_{2}=\eta_{4}=\Downarrow\\

0 \qquad \text{elsewhere}

\end{array}[/tex].

V is also given by

[tex]

V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}=\int dx_{j} dx_{k} \psi_{\eta_{1}}^{\ast}(x_{j})\psi_{\eta_{2}}^{\ast}(x_{k})V(x_{j}-x_{k})

\psi_{\eta_{3}}(x_{j})\psi_{\eta_{4}}(x_{k})[/tex]

Now, what is [tex]V(x_{j}-x_{k})[/tex] ?

[tex]Un_{\Uparrow}n_{\Downarrow} =

Ud_{\Uparrow}^{\dagger}d_{\Uparrow}d_{\Downarrow}^{\dagger}d_{\Downarrow}

=

-Ud_{\Uparrow}^{\dagger}d_{\Downarrow}^{\dagger}d_{\Uparrow}d_{\Downarrow}

=

\frac{1}{2}\sum_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}d_{\eta_{1}}^{\dagger}d_{\eta_{2}}^{\dagger}d_{\eta_{3}}d_{\eta_{4}}

[/tex]

where

[tex]

V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}=\Big\{

\begin{array}{c}

-2U \qquad \text{for} \qquad \eta_{1}=\eta_{2}=\Uparrow,\: \eta_{2}=\eta_{4}=\Downarrow\\

0 \qquad \text{elsewhere}

\end{array}[/tex].

V is also given by

[tex]

V_{\eta_{1}\eta_{2}\eta_{3}\eta_{4}}=\int dx_{j} dx_{k} \psi_{\eta_{1}}^{\ast}(x_{j})\psi_{\eta_{2}}^{\ast}(x_{k})V(x_{j}-x_{k})

\psi_{\eta_{3}}(x_{j})\psi_{\eta_{4}}(x_{k})[/tex]

Now, what is [tex]V(x_{j}-x_{k})[/tex] ?