Addition of Angular Momenta, problem with notation

[joelcponte]

Homework Statement

The electron in a hydrogen atom is described by the following superposition of two states:

$|\psi> = \frac{1}{\sqrt{2}}(|n=2,l=1,m=0,s_z= +1/2> + |n=2,l=0,m=0,s_z= +1/2>)$

(b) Let J = L + S be the total angular momentum. Express state $\psi$ in basis $|n, l, J, J_z>$
Hint: use the Clebsch -Gordan coffecients from the table on pg. 188 of Griffiths.

Homework Equations

$|s_1 m_1>|s_2 m_2> = \sum C^{s_1s_2s}_{m_1m_2m}|s m>$

The Attempt at a Solution

For the first state:

$|n=2,l=1,m=0,s_z= +1/2> ====> l=1, s = 1/2$
$|1,0>|1/2,1/2> = \sqrt{\frac{2}{3}}|3/2,1/2> - \sqrt{\frac{1}{3}}|1/2,1/2>$

For the second state:

$|n=2,l=0,m=0,s_z= +1/2>$ ====> there's no orbital angular momentum

My question is: I don't know if what I've done is correct and, if it is, I don't know how to transform to the notation he asks. For me, the "l" in the new notation doesn't make sense or have to be 0.

 

[vela]

Homework Statement

The electron in a hydrogen atom is described by the following superposition of two states:

$|\psi> = \frac{1}{\sqrt{2}}(|n=2,l=1,m=0,s_z= +1/2> + |n=2,l=0,m=0,s_z= +1/2>)$

(b) Let J = L + S be the total angular momentum. Express state $\psi$ in basis $|n, l, J, J_z>$
Hint: use the Clebsch -Gordan coffecients from the table on pg. 188 of Griffiths.

Homework Equations

$|s_1 m_1>|s_2 m_2> = \sum C^{s_1s_2s}_{m_1m_2m}|s m>$

The Attempt at a Solution

For the first state:

$|n=2,l=1,m=0,s_z= +1/2> ====> l=1, s = 1/2$
$|1,0>|1/2,1/2> = \sqrt{\frac{2}{3}}|3/2,1/2> - \sqrt{\frac{1}{3}}|1/2,1/2>$

That's right.

For the second state:

$|n=2,l=0,m=0,s_z= +1/2>$ ====> there's no orbital angular momentum

So what are j and j_z equal to in this case?

My question is: I don't know if what I've done is correct and, if it is, I don't know how to transform to the notation he asks. For me, the "l" in the new notation doesn't make sense or have to be 0.

You have one basis with states of the form $\lvert \ l\ m_l \rangle \lvert\ s\ m_s \rangle$. These states are eigenstates of L², L_z, S², and S_z. The other basis has states of the form $\lvert \ l\ s\ j\ j_z\rangle$, which are eigenstates of L², S², J², and J_z. These two bases are related through the Clebsch-Gordan coefficients.

When you wrote
$$\lvert 1,0\rangle \lvert 1/2,1/2\rangle = \sqrt{\frac{2}{3}} \lvert 3/2,1/2\rangle - \sqrt{\frac{1}{3}} \lvert1/2,1/2\rangle,$$ what it really means is
$$\lvert 1,0\rangle \lvert 1/2,1/2\rangle = \sqrt{\frac{2}{3}} \lvert l=1, s=1/2, j=3/2, j_z=1/2\rangle - \sqrt{\frac{1}{3}} \lvert l=1, s=1/2, j=1/2, j_z=1/2\rangle,$$