If a neutron in a nucleus is in a 1p state, before splitting this up in to seperate j-state (due to spin-orbit effect) this neutron has 6 possible states.(adsbygoogle = window.adsbygoogle || []).push({});

l = 1

m_l = 1, 0 or -1

m_s = 1/2 or -1/2

Splitting this in to j-states corresponding to l+s and l-s, as expected there are 6 states.

j = 3/2 (l+s)

m_j = 3/2, 1/2, -1/2, -3/2

j = 1/2 (l-s)

m_j = 1/2, -1/2.

I've tried to think of this result classically. How exactly the j-vector is made of the l and s vectors. For a particular j and mj, we have on a vector diagram that the j-vector can revolve in a circle around that particular m_j value. For any particular direction of this j-vector, the l and s vectors add vectorially and can lie anywhere on a circle that precesses around the j-vector.

But exactly what m_l, m_s state corresponds to which j/m_j state?

For j = 3/2, m_j = 3/2, the only state that could correspond to this is m_l = 1, m_s = 1/2. Similarly for j = 3/2, m_j = -3/2, we have m_l = -1, m_s = -1/2.

But what about the other four states? I've tried for a few hours now to figure out how you can deduce this. For j =1/2, m_j = 1/2, there are two possible states which give the correct z-component and, I believe, the correct magnitude (l = 1, m_s = -1/2, or l = 0, m_s = 1/2). So how is it possible to know which of these two it is? Similarly for j = 3/2, m_j = 1/2, there are two possible states which give the correct z-component and, I believe again, the correct magnitude (l = 1, m_s = -1/2, or l = 0, m_s = 1/2). The same two possible states for both j = 3/2, j = 1/2. So I need to know which of l = 1, m_s = -1/2, or l = 0, m_s = 1/2 corresponds to j = 3/2, or j = 1/2. How is it possible to know?

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# Having a hard time thinking about quantum mechanical vector addition.

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