Question

In the quantum mechanics class we were told that the total angular momentum basis (LHS) is equal to basis in product basis (RHS) as below. However I don't have any clue how to reach the result. Professor gave a clue that it is about clebsch-gordan coefficient and use lowering operator both sides but I don't know how to apply that to get the result below. So, could you please show how to switch to product basis from the total angular momentum basis in the example below step by step? $|l, s; j \pm \frac{1}{2}, m_j\rangle = \pm \sqrt{\frac{l \pm m_j + \frac{1}{2}}{2l+1}} |l, s, m_j - \frac{1}{2}, \frac{1}{2}\rangle + \sqrt{\frac{l \mp m_j + \frac{1}{2}}{2l+1}} |l, s, m_j + \frac{1}{2}, -\frac{1}{2}\rangle$ As hint he said that spin is 1/2, and j = l+1/2 or j = l-1/2 because J = L + S and $m_l = m_j - m_s$

          In the quantum mechanics class we were told that the total angular momentum basis (LHS) is equal to basis in product basis (RHS) as below.
However I don't have any clue how to reach the result. Professor gave a clue that it is about clebsch-gordan coefficient and use lowering operator both sides but I don't know how to apply that to get the result below.
So, could you please show how to switch to product basis from the total angular momentum basis in the example below step by step?
$|l, s; j \pm \frac{1}{2}, m_j\rangle = \pm \sqrt{\frac{l \pm m_j + \frac{1}{2}}{2l+1}} |l, s, m_j - \frac{1}{2}, \frac{1}{2}\rangle + \sqrt{\frac{l \mp m_j + \frac{1}{2}}{2l+1}} |l, s, m_j + \frac{1}{2}, -\frac{1}{2}\rangle$
As hint he said that spin is 1/2, and j = l+1/2 or j = l-1/2 because J = L + S and $m_l = m_j - m_s$

In the quantum mechanics class we were told that the total angular momentum basis (LHS) is equal to basis in product basis (RHS) as below.
However I don't have any clue how to reach the result. Professor gave a clue that it is about clebsch-gordan coefficient and use lowering operator both sides but I don't know how to apply that to get the result below.
So, could you please show how to switch to product basis from the total angular momentum basis in the example below step by step?
|l, s; j ±(1)/(2), mj⟩ = ±√((l ± mj + (1)/(2))/(2l+1)) |l, s, mj - (1)/(2), (1)/(2)⟩ + √((l ∓ mj + (1)/(2))/(2l+1)) |l, s, mj + (1)/(2), -(1)/(2)⟩
As hint he said that spin is 1/2, and j = l+1/2 or j = l-1/2 because J = L + S and ml = mj - ms

Added by Albert M.

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