# How can I apply this central force projector operator?

I'm working with two neutrons bound in a 3D harmonic trap (so, living in a harmonic oscillator potential) which also interact with each other with a given potential. The states of the individual neutrons can be written as $$|n_1l_1 s_1 \rangle$$, $$|n_2 l_2 s_2 \rangle$$ (I have ommited the $$m$$ quantum number), and the product state as $$|n_1 l_1 s_1, n_2 l_2 s_2\rangle$$, with the convenient antisymmetrization.

One of the terms of the interaction potential between the neutrons is of the form $$V(r)\frac{1}{2}(1 + P_r)$$, where $$P_r$$ is an operator that exchanges the spatial degrees of freedom of the two particles ($$P_r|n_1 l_1 s_1, n_2 l_2 s_2\rangle =|n_2 l_2 s_1, n_1 l_1 s_2\rangle$$). So, $$\frac{1}{2}(1 + P_r)$$ projects onto the "even" part of the nuclear two body wavefunction (part that does not change sign on the exchange of the spatial degrees of freedom).

My question is: How do I apply this projector in the calculation of a matrix element of the form $$\langle n_1l_1s_1, n_2l_2s_2 |V(r)P_r| n_3l_3s_3, n_4l_4s_4\rangle$$. In other words, how is the state $$|n_2 l_2 s_1, n_1 l_1 s_2\rangle$$ related to $$|n_1 l_1 s_1, n_2 l_2 s_2\rangle$$ in general?

• Aren't those states the same? $s_1 = s_2 = \frac{1}{2}$ for neutrons. Mar 8 at 4:06