Why Do Helium Atom Energy Levels Split in Different Interaction Scenarios?

[captainjack2000]

Homework Statement

I have having difficulty explaining the splitting of helium atom energy levels for three different cases:
1)Coulomb interaction is ignored
2)Coulomb interaction is present but exchange interaction is neglected
3)Coulomb and exchange interactions are present

The Attempt at a Solution

For 1) The helium atom has two outer electron. These electrons must have opposite spins due to pauli exclusion principle (-1/2 and +1/2). So we will have the case {n1=1 l=0} and {n2=1 l=0} for the first energy level. and {n1=1 l=0} {n2=2 l=0 or 1} for the second?

For 2) Electron electron interaction is repulsive so it raises all energy levels wrt levels in 1). Coulomb potential is inversely proportional to separation of electrons.
?

3)I know that triplet states are lower in energy than singlet states since in the triplet state electrons want to keep away from each other.

I don't really understand these effects and where degeneracy in l is lifted and when it isn't. If someone could clarify this I would really appreciate it!

 

[mark726]

Thank you for your question. The splitting of energy levels in the helium atom is a complex phenomenon that involves both the Coulomb and exchange interactions between the two electrons. In order to understand this, we must first understand the concept of degeneracy.

In the absence of any interactions (case 1), the energy levels of the helium atom would be degenerate, meaning that the energy of the electrons would not depend on their angular momentum quantum numbers (l). This is because the electrons would not interact with each other and their energy would only depend on their principal quantum number (n).

However, in reality, the Coulomb interaction between the two electrons causes a splitting of the energy levels. This is because the electrons repel each other, and the energy of the system is lower when the electrons are further apart. This splitting is more pronounced for higher values of l, as the electrons with higher l values are closer to each other and experience a stronger repulsion.

In the case where the Coulomb interaction is present but the exchange interaction is neglected (case 2), the energy levels will still be split, but not as much as in case 1. This is because the exchange interaction, which arises from the quantum mechanical exchange of the electrons, also contributes to the energy of the system and helps to lower the energy of the system.

Finally, in the case where both Coulomb and exchange interactions are present (case 3), the energy levels will be split even more, as both interactions are working together to lower the energy of the system. In this case, the triplet states (where the spins of the electrons are parallel) will have lower energy than the singlet states (where the spins are anti-parallel), as you have correctly noted.

In summary, the splitting of energy levels in the helium atom is a result of the interplay between the Coulomb and exchange interactions between the two electrons. I hope this explanation helps to clarify your understanding. If you have any further questions, please don't hesitate to ask.