# Matrix elements of a one-fermion operator (first and second quantizations)

I'm currently struggling with the expression of operators in second quantization. I did an exercise in which I had to consider a fermion in a central potential $V(\vec{r})$ and show that the matrix elements of $V(\vec{r})$ and $V_{ij} a_i a^{\dagger}_j$ were identical. I think I get this right. Yet, when I try to do the same with two fermions in a central potential (not considering any interaction between them), I end up completely lost.

On the one hand (first quantization), I have

$$<a b ∣ V(r) ∣ cd > = \frac{1}{2} \int \int d^3 r_1 d^3 r_2 \Big(\phi_a^{\ast}(r_1)\phi_b^{\ast}(r_2) - \phi_a^{\ast}(r_2) \phi_b^{\ast}(r_1) \Big) [V(r_1) + V(r_2)] \Big(\phi_c^{\ast}(r_1) \phi_d^{\ast}(r_2) - \phi_c^{\ast}(r_2) \phi_d^{\ast}(r_1)\Big)$$

(I should be able to further develop and simplify this expression, but I'm stuck with eight terms with no simplification on the horizon) and on the other hand (second quantization), I have

$$<a b ∣ \sum_{ij} V_{ij} a_i a_j^{\dagger} ∣ cd > = < ab ∣ V_{ac} a_a a_c^{\dagger} + V_{ad} a_a a_d^{\dagger} + V_{bc} a_b a_c^{\dagger} + V_{bd} a_b a_d^{\dagger} ∣ cd>$$ I tried applying the anti-commutation relationship to this expression so as to simplify it, but unsuccessfully so far.

I am more or less sure I'm missing something terribly obvious and I should feel ashamed about it. I would gladly appreciate some fresh insight on this problem.

$$<a b ∣ V(r) ∣ cd > = \frac{1}{2} \Bigg[ \int d^3 r_1 \phi_a^{\ast}(r_1) V(r_1) \phi_c (r_1) \int d^3 r_2 \phi_b^{\ast}(r_2) \phi_d (r_2) + \dots \Bigg] = \frac{1}{2} \Bigg[ V_{ac} \delta_{bd} + \dots \Bigg] = V_{ac} \delta_{bd} - V_{ad} \delta_{bc} + V_{bd} \delta_{ac} - V_{bc} \delta_{ad}$$
$$<a b ∣ \sum_{ij} V_{ij} a_i a_j^{\dagger} ∣ cd > = < ab ∣ V_{ac} a_a a_c^{\dagger} + V_{ad} a_a a_d^{\dagger} + V_{bc} a_b a_c^{\dagger} + V_{bd} a_b a_d^{\dagger} ∣ cd> \\ = V_{ac} <ab ∣ a_a a_c^{\dagger} ∣ cd > + V_{ad} <ab ∣ a_a a_d^{\dagger} ∣ cd > + V_{bd} <ab ∣ a_b a_c^{\dagger} ∣ cd > + V_{bd} <ab ∣ a_b a_d^{\dagger} ∣ cd > \\ = V_{ac} <b ∣ d > - V_{ad} <b ∣ c > + V_{bd} <a ∣ c > - V_{bc} <a ∣ d > \\ = V_{ac} \delta_{bd} - V_{ad} \delta_{bc} + V_{bd} \delta_{ac} - V_{bc} \delta_{ad}$$ where I used $∣ ab > = - ∣ ba >$.