# In ground state, are spins aligned or anti-aligned?

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1. Jun 26, 2014

### Aziza

In a past physics gre question (https://www.physicsforums.com/showthread.php?t=192334), we make use of the idea that the ground state energy of two ions that have spin is when the spins are aligned.

However, the ground state of the helium atom is a spin singlet state, which is a linear combo of two spins ANTI-aligned.

Can anyone please explain this discrepancy to me? Thank you!

2. Jun 26, 2014

### Einj

Which state is the ground state depends on the expression for the Hamiltonian. Take for example the easiest spin-spin interaction for two particles:
$$H=\lambda \vec{S}_1\cdot \vec{S}_2.$$
Suppose $\lambda>0$, then, in order to minimize the energy you want $\vec{S}_1\cdot \vec{S}_2<0$, i.e. anti-alligned spins. On the other hand if $\lambda<0$ you want the spins to be alligned.

So, there is no definitive way of determining if the ground state is same- or opposite-spins. It depends on the interaction.

3. Jun 26, 2014

### dipstik

well, you can't have two electrons in the same spin state if they have the same quantum numbers for everything else, since they are fermions.