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What is the hamiltonian for an atom with two electrons and a nuclear charge $Z$?

The Hamiltonian always takes the general form: $\hat H=\hat T + \hat V $

The kinetic energy of each of the electrons needs to be taken into account so: $\hat T=-\frac{\hbar^2}{2\mu}(\nabla_1^2+\nabla_2^2)$

The potential operator is where I am finding trouble. There are two terms for the attraction towards the nucleus - one for each electron: $-\frac{Ze^2}{4\pi\epsilon_0r_1}-\frac{Ze^2}{4\pi\epsilon_0r_2}$. However, the potential energy involved in the electron electron repulsion needs to be taken into account. My question is: are there two terms for this? To clarify do I need to calculate the increase in potential of electron 1 caused by electron 2 AND the increase in potential energy of electron 2 caused by the presence of electron 1? Or would adding: $\frac{e^2}{4\pi\epsilon_0r_{12}}$ take into account all of the potential energy involved in the electron electron interaction?

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    $\begingroup$ Potential is a characteristic of the system as a whole, and not of individual electrons. There are three charges, hence three pairwise interactions. Two of them, namely nucleus-electron1 and nucleus-electron2, you took into account once. Why in the world would you want to add third such value twice? $\endgroup$ – Ivan Neretin Oct 8 '15 at 15:16
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The electron-electron repulsion potential is an interaction potential, therefore it has one term per interaction. If you have two electrons, you will have only one term, representing the interaction between them. The number of potential terms will be, in your case:

  • Attraction between electron 1 and the nucleus (one interaction)
  • Attraction between electron 2 and the nucleus (one interaction)
  • Repulsion between electron 1 and electron 2 (one interaction)

In total, you will have three interactions and therefore only three potential terms (two for electron-nucleus and one for electron-electon interactions).

In general, an interaction potential that only depends on the distance of two particles can be written as $$ U = \sum_{i<j} u(|\vec{r}_i - \vec{r}_j|), $$ where $u$ represents the interaction between only two particles. You see that for two electrons you will have only one term, since $i<j$. For three electrons you will have three terms (interaction between 1 and 2, between 1 and 3 and between 2 and 3) and so on.

Maybe your confusion comes from the fact that, sometimes, this interaction potential is written as $$ U = \frac{1}{2}\sum_{i\neq j} u(|\vec{r}_i - \vec{r}_j|) $$ In this case, for two electrons, you will have two terms. However there is the factor $1/2$ which cancels out the double counting, since $$u(|\vec{r}_i - \vec{r}_j|)=u(|\vec{r}_j - \vec{r}_i|).$$

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