Noob ques: probability of each J state

[empirekhoo]

Homework Statement

An electron with spin down is in a state of (n=5,l=1,ml=0) of hydrogen atom. If you could measure the total angular momentum squared of the electron, what values might you get, and what is the probability of each state.

Homework Equations

J^2 = j(j+1) * hbar^2 (i mean the eigenvalue of operator J^2)

The Attempt at a Solution

the first part is easy.
J^2 = .75 hbar^2, J^2 = 3.75 hbar^2

but I have no idea how to solve the second part (perhaps i forgot). Any idea/input is welcomed! was it insufficient information?

 

[fzero]

You have a state that is

$$| \ell = 1, m_\ell = 0\rangle\otimes |s=\tfrac{1}{2}, m_s = - \tfrac{1}{2}\rangle.$$

This state can be written in terms of the states $$|1,\tfrac{1}{2}, j,m_j\rangle$$ by using the Clebsch-Gordan coefficients.

 

[empirekhoo]

You have a state that is

$$| \ell = 1, m_\ell = 0\rangle\otimes |s=\tfrac{1}{2}, m_s = - \tfrac{1}{2}\rangle.$$

This state can be written in terms of the states $$|1,\tfrac{1}{2}, j,m_j\rangle$$ by using the Clebsch-Gordan coefficients.

Ah of course.. It wasn't taught in lecturer.. (disappointing)

Anyway I've looked into the CG table (for the first time) and I wonder was my method correct:

1. my table is alike last page of http://www3.tsl.uu.se/thep/courses/QM/061027-exam.pdf"

2. looking into 1 x 1/2, then on the 0, -1/2 row. I get:

3. sqrt(2/3) | 3/2, -1/2 > + sqrt(1/3) | 1/2, -1/2 >

4. am I right? or am I abusing the table? (I'll reply by saturday.. busy week)

 

[fzero]

Looks good to me.

 

[empirekhoo]

Looks good to me.

ah okay! thanks a bunch! I'll read on with direct product later when I'm free =)