Wave function of a proton and a neutron with and without isospin

Question

Suppose I have a proton and a neutron in an external harmonic oscillator potential well. Let us first neglect all interactions between the two: since they are distinguishable particles I conclude their Hamiltonian eigenfunctions to be:

$$ \Psi(r_1, r_2) = \phi_p (r_1) \phi_n (r_2),\tag{1}$$

where the $\phi_i$s are any HO Hamiltonian eigenfunction. Now if I do the same but thinking in terms of isospin I conclude that the eigenfunction should instead be

$$ \Psi(r_1, r_2) = \phi_{\tau = \frac{1}{2} } (r_1) \phi_{\tau = - \frac{1}{2} } (r_2) - \phi_{\tau = - \frac{1}{2} }(r_1) \phi_{\tau = \frac{1}{2}} (r_2)$$

because of antisymmetry. These two expressions are clearly different. Are these two formalisms equivalent when dealing with problems involving the strong interaction? If I introduce an electromagnetic interaction does the antisymmetry requirement decay, making $(1)$ best suited for the new problem at hand?

Answer 1

I think you're barking up the wrong tree here with the ideas about interactions. This isn't about interactions, it's just about symmetry.

Let A be one HO state and B the other.

If you consider the neutron and the proton to be non-identical particles, and let the first quantum number be a label for the proton, then the states |A,B> and |B,A> are orthogonal. In the first state, the proton is in state A and the neutron is in state B. In the second state, it's the other way around. Because they're orthogonal, we expect them to be distinguishable experimentally. For example, one state could have a vanishing electric quadrupole moment and the other a nonvanishing one.

If you consider the neutron and proton to be identical particles with different isospin, then these states could be notated as |A,p;B,n> and |A,n;B,p>. These are again orthogonal, and everything is as before, including the links to observable quantities. Antisymmetrization has no effect on these facts.

If you now consider |A,p;B,n> compared to |B,n;A,p>, then these are not orthogonal to each other. We want these to be two different labels for the same state. If you want to represent them in terms of explicit spatial wavefunctions, then you could ensure that by using a Slater determinant.