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Consider $|^3\text{He}\rangle$ in the ground state (2 protons and 1 neutron). Assume the spatial part of the wave function is symmetric. I have to construct the spin part of the wave function. This is what I have done so far:

It is clear that the spin part of the wave function must be anti-symmetric. So denote the protons by $p_1$ and $p_2$ and the neutron by $n$. Denote the spin part of the wave function by $\psi(p_1,p_2,n)$, then we must have that $\psi(p_2,p_1,n)=-\psi(p_1,p_2,n)$. We see that the spin wave function of the protons must be $$ \frac{|p_1\uparrow\rangle|p_2\downarrow\rangle-|p_1\downarrow\rangle|p_2\uparrow\rangle}{\sqrt{2}}. $$ Now I don't understand what the spin of the neutron must be, up or down? Because in both cases the spin part of the wave function will be anti-symmetric. I need help. Thanks.

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If the protons are in a spin singlet, the neutron spin determines the spin of the $^3\text{He}$ nucleus. Without loss of generality you can define the direction of the $^3\text{He}$ polarization as $\uparrow$.

Under isospin symmetry, the proton and the neutron are two states of the same particle. If your problem includes this symmetry (and if you can assume that the isospin part of the wavefunction is symmetric), your total state will also have to be antisymmetric under the exchanges $p_1\leftrightarrow n$ and $p_2 \leftrightarrow n$. The method for finding a completely antisymmetric state of many particles is to construct a Slater determinant.

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  • $\begingroup$ so the atom being in the ground state does not have anything to do with the spin of the atom? $\endgroup$
    – Badshah
    Oct 4 '14 at 20:33
  • $\begingroup$ Is it possible for the neutron to be in a superposition of the up and down state? This doesnt affect the anti-symmetry, so I would think this is valid. $\endgroup$
    – Badshah
    Oct 4 '14 at 20:41
  • $\begingroup$ The nucleus in its ground state has spin half; I suppose there are excited states that have spin three-half. Your singlet in the question already has $p_1$ in a superposition of spin-up and -down; the neutron is no different. $\endgroup$
    – rob
    Oct 4 '14 at 22:24
  • $\begingroup$ I dont understand what $p_1$ has to do with wether the neutron is in the up or down state. $p_1$ and $p_2$ together have spin 0, so the spin of the neutron will determine if the atom has spin up or spin down. So why can the atom not be in in a superposition of spin up and spin down? Which is the same question why the neutron can not be in a superposition of spin up and spin down. $\endgroup$
    – Badshah
    Oct 5 '14 at 9:01

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