[물리] Deuteron(1) [MIT]

The deuteron is a bound state of a proton and a neutron. The Hamiltonian in the center of mass system has the form

.

Here, x = x_n - x_p, r = |x|, σ_p and σ_n are the Pauli matrices for the spins of the proton and neutron, μ is the reduced mass, and p is conjugate to x.

(a) The total angular momentum J² = J(J+1) and parity are good quantum numbers. show that if V₃ = 0, the total orbital angular momentum L² = L(L+1), total spin S² = S(S+1) and S = (1/2)σ_p + (1/2)σ_n are good quantum numbers. Show that if V₃ ≠ 0, S is still a good quantum number.

sol)

Use units s.t. hbar = 1.

Consider V₃ = 0 case. As

we have

and so

Thus L² is a good quantum number.

Now consider the total spin S². As

We have

Thus [S², H] = 0 and S² is a good quantum number.

Consider V₃ ≠ 0 case.

As σ_p + σ_n = 2(s_p +s_n) = 2S and

using the formula

the above becomes

So we have

Thus S is still a good quantum number if V₃ ≠ 0

(b) The deuteron has J = 1 and positive parity. What are the possible value of L? What is the value of S?

sol)

The parity of the deuteron nucleus is

Since the deuteron has positive parity, L = even.

Then S can only be 0 or 1 and J = 1.

Thus, we must have S = 1 and L = 0 or 2.