How Does Hund's Rule Influence the Symmetry of Wave Functions in a 2p² Shell?

[Joqe]

Hi

According top Hunds rule I have a $$^{3}P$$ term which should be the term for the ground state for a $$2p^{2}$$ shell (in this case the outer sub-shell), this means that I have a triplet state and thus a symmetric wave function for the spinn. Since the electrons are femions the total wave function should be anti-symmetric, this means that my spartial wave functions must be anti-symmetric.

My problem arises when I'm trying to construct a anti-symmetric spartial wave function and give it in Hydronic spatial wave function. I use

$$\psi_{spatial}=\frac{1}{\sqrt{2}}\left[\psi_{a}(1)\psi_{b}(2)-\psi_{b}(1)\psi_{a}(2)\right]$$​

to create the anti-symmetric wave function. But quantum number $${l} = {1}$$ for all the electron and quantum number $${m_{l}} = {-1},{0},{1}$$ for all the electorns. This will lead to that the anti-symmetric wave function allways is equal to 0.

Can someone please help me?

Regareds
/Joqe

 

[Jhero]

Dear Joqe,

Thank you for your post. It seems like you are on the right track with your understanding of Hunds rule and the requirement for an anti-symmetric wave function for fermions. However, there are a few things to consider when constructing the spatial wave function for a ^{3}P term.

Firstly, the wave function you have written is correct, but it is missing a crucial component - the spin wave function. Remember that the total wave function for an atom must also take into account the spin of the electrons. In this case, since you have a triplet state, the spin wave function must be symmetric. This means that your total wave function will be a product of the spatial wave function you have written and the spin wave function, which is given by:

\psi_{spin}=\frac{1}{\sqrt{2}}\left[\alpha(1)\beta(2)+\beta(1)\alpha(2)\right]

where \alpha and \beta represent the spin states of the two electrons.

Secondly, when constructing the spatial wave function, you must also consider the principle quantum number (n) and the azimuthal quantum number (l). In this case, since you are dealing with a 2p^{2} shell, n=2 and l=1. This means that your spatial wave function should also include the appropriate radial wave function for the 2p orbital.

Overall, your total wave function for the ^{3}P term should look something like this:

\Psi=\psi_{radial}(1)\psi_{radial}(2)\psi_{spatial}(1)\psi_{spatial}(2)\psi_{spin}

I hope this helps to clarify things for you. If you have any further questions, please don't hesitate to ask.