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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

\begin{equation} <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) \end{equation}

(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

\begin{equation}<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>\end{equation} 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.

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It turned out that some time away from my desk was what I needed to escape from this "Panic! I can no longer do any physics!" maze.

Well, then for those who might be interested and for those who might tracked down some errors, here is my development. For the first quantization :

\begin{equation} <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} \end{equation}

As for the second quantization development, I completely missed the more-than-obvious:

\begin{equation}<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} \end{equation} where I used $∣ ab > = - ∣ ba >$.

If you have comments about the developments / justifications / second quantization in general, feel free to share them - that would be appreciated.

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