Question

I have a couple of questions regarding the following problem:

Let the interaction between two nucleons given by: ˆV(1,2) = σ^(1) σ^(2).
What I need help with is the following: Evaluate the following two-body matrix element in the coupled base:
being the angular momentum couplings: J = L+S with L = l1 + l2 and S = s1 + s2. Now, what I have so far is:
ˆV(1,2) = ˆσ(1)x ˆσ(2)x + ˆσ(1)y ˆσ(2)y + ˆσ(1)z ˆσ(2)z which I think only affects the coordinates ms1 and ms2, since the σ's are the Pauli matrices and the states are = (1 0)and < s1 = 1/2, ms1 = −1/2 > = (0 1), but these coordinates (msi) don't appear explicitly on the matrix element, so I don't know how the operator should act. On the other hand, I know the only bound state is the proton-neutron pair, which has J^π = 1^+, so the only possibilities are S=1, L=0(⟹J=1) and S=1, L=2(⟹J=1,2,3). This I guess implies the state | l1 s1 l2 s2; LS ; JMj > should be decomposed into two possible states, but I don't know how to do that, or how the operator ˆV(1,2) will act on the state in the end. My first guess is that ˆV(1,2) doesn't affect the coordinates l1, s1 ,l2 ,s2 ,L, S, J, Mj so that would mean the matrix element is 1, but I suppose that is not correct. Could anyone help me on getting the concepts clear?

Answer 1

The interaction between two nucleons can be represented by the operator ˆV(1,2) = σ^(1) σ^(2). To evaluate the two-body matrix element in the coupled basis, we need to decompose the state | l1 s1 l2 s2; LS ; JMj > into the possible states allowed by the interaction.

Step-by-step explanation:

The interaction between two nucleons can be represented by the operator ˆV(1,2) = σ^(1) σ^(2), where σ^(i) are the Pauli matrices.

To evaluate the two-body matrix element in the coupled basis, we need to decompose the state | l1 s1 l2 s2; LS ; JMj > into the possible states allowed by the interaction.

As the bound state has J^π = 1^+, the only possibilities are S=1, L=0(L=2 will not yield J=1).

The operator ˆV(1,2) acts on the spin space of the nucleons, not on the coordinates l1, s1, l2, s2, L, S, J, Mj, so we can't say that the matrix element is 1 directly.