# Motivation for symmetric and antisymmetric atoms in QM

For a two particle system in QM, if we have identical particles, which are fermions then we require that the overall wave function (position and spin) is antisymmetric and if the the particles are identical bosons then we require that overall wave function is symmetric, i.e.

$$\psi(r_{1},r_{2}) = A[\psi_{a}(r_{1})\psi_{b}(r_{2})\pm\psi_{b}(r_{1})\psi_{a}(r_{2}) ].$$

Question: But what are the rules when considering atoms. For example, helium or hydrogen seem to follow the fermion rules regarding when a wave function is symmetric or antisymmetric, that is the overall wave function should be antisymmetric. Do we follow the rules related to fermions for atoms as well since we have electrons involved? What is the basic motivation for these rules? I am just starting to learn about spin in QM, hence am using an introductory text.

Thanks.

• Not a direct answer, but I strongly recommend that you read this related post. Jan 24, 2017 at 21:58
• You mixed up fermions and bosons. Fermions are antisymmetric and bosons are symmetric. Jan 24, 2017 at 21:58

Indeed certain atoms are bosons and other atoms are fermions. Just to provide you with an example, Bose-Einstein condensates are often realized with ultracold bosonic atoms, such as $^4\mathrm{He}$ or Rubidium. Since these kinds of atoms are bosons, if you cool them down below a critical temperature, the wavefunctions of each single atom overlap and you can start talking about a macroscopic wavefunction: that's condensation!

The basic motivation is that every object is either a boson or a fermion, no matter if we are talking about a single electron, an hydrogen atom, a photon or an atom. Once you know what kind of particle you are dealing with, you automatically know which rule you have to follow.

• Okay but how do you know which atoms are considered fermions and which are considered bosons? And what is it based on?
– Alex
Jan 25, 2017 at 18:03
• Your question is very similar to this one: physics.stackexchange.com/questions/59753/… To make thing simple, you have to count the number of constituents of your atom, i.e. sum up the number of protons, neutrons and electrons. If it's even, then you have a bosonic atom, if it's odd then you have a fermionic atom. A classical example are $^3\mathrm{He}$ and $^4\mathrm{He}$. The first one has 2 electrons, 2 protons and a neutron: 2+2+1 = 5, so it's a fermion. The second has one neutron more so it's a boson. Jan 25, 2017 at 18:15
• Okay I see, thanks. Just to confirm then, a diatomic hydrogen atom $H_2$ has two electrons, two protons and no neutrons and hence is a boson with spin 2. Therefore the overall wave function should be symmetric, so if the spatial part of the wave function is symmetric then the spin part should also be symmetric and if the spatial part is antisymmetric then the spin part should also be antsymmetric. Is this correct?
– Alex
Jan 25, 2017 at 20:19
• Correct! Except that you said "diatonic hydrogen atom". It's a molecule, but I'm sure it was a typo! Jan 25, 2017 at 21:25
• Okay great thanks. One more thing...Could we instead of considering the quantum state of the diatomic hydrogen molecule consider just the two particle quantum state of the two electrons in the $H_2$ molecule. These electrons would be two fermions, so the total wave function of this two particle system should be antisymmetric and hence if the spatial wave function is symmetric then the spin state should be antisymmetric and if the spatial part is antisymmetric then the spin part should be symmetric. Is this fine?
– Alex
Jan 25, 2017 at 21:32