I'm trying to understand the wave function of two identical particles with spin but unfortunately my textbook does a poor job explaining any of it.

For two identical bosons, for instance, I know that ignoring spin $$\psi_+(x_1,x_2)=\frac{1}{\sqrt{2}}[\psi_a(x_1)\psi_b(x_2)+\psi_a(x_2)\psi_b(x_1)]$$ I also know that for the ground state and first excited state, the energies are non-degenerate and doubly degenerate respectively.

But what happens to the wave function when I include spin? Also what happens to the degeneracy of energies when including spin?

If someone can give a thorough explanation of identical particles with spin that would be awesome!


1 Answer 1


The total wave function for a particle is made by a combination of a spin part (say $\chi$) and of a position part (lets say $\psi$), so that we have : $$ \Psi_t(x_i,s_i)=\chi (s_i) \, \psi (x_i) $$

When you consider bosons, you want to ensure that the total wave function is symmetric. As such, you would have either two cases: a symmetric spin wave function and a symmetric spatial wave function or you would have that the spin and spatial wave functions are both anti-symmetric.

So, for bosons (+ sign for symmetric and - for anti-symmetric):

$$ \Psi_t = \chi_+ \ \psi_+ $$ Or $$ \Psi_t = \chi_- \ \psi_- $$

Your final wave function is then a combination or those two cases.

For fermions, same idea but remember that fermions have anti-symmetric total wave function. So that for fermions, both accepted wave functions are: $$ \Psi_t = \chi_- \ \psi_+ $$ Or $$ \Psi_t = \chi_+ \ \psi_- $$

Now say you consider a spin $\frac{1}{2}$ particle, there are four spin states, which we separate into two categories, the singlet state and the triplet states.

For the singlet state (anti-symmetric state), you have: $$ |\chi\rangle = \frac{1}{\sqrt{2}}[|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle] $$

As for the triplet states (all symmetric states), they are : $$ |\chi\rangle = \frac{1}{\sqrt{2}}[|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle] \\ |\chi\rangle= |\uparrow\uparrow\rangle \\|\chi\rangle= |\downarrow\downarrow\rangle $$

You can then see that it is possible to get extra degeneracy via the spin parts (you have more than one state that gives a symmetric spin part).

For extra reference, you can check out Sakurai's or Zettili's textbook.


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