• Support PF! Buy your school textbooks, materials and every day products Here!

CG coefficient in Wigner Eckhart Theorem-how to find

  • #1
186
2

Homework Statement


The Clebsh Gordon coefficient is < 11;00 |11> = < l_1,m_1;l_2m_2 | kq >

Homework Equations


My professor determined the CG to be 1

The Attempt at a Solution


How do we look up these numbers in the table? What is the direct product basis? 1x1? What is JM and what is m_1 and m_2... very confused. Any help greatly appreciated
 

Answers and Replies

  • #2
blue_leaf77
Science Advisor
Homework Helper
2,629
784
Suppose you have two angular momenta vector operator ##\mathbf{L}_1## and ##\mathbf{L}_2## acting on different vector spaces. The eigenket of ##L_{1z}## in the first vector space is denoted by ##|l_1,m_1\rangle##, while the eigenket of ##L_{2z}## in the second space is denoted by ##|l_2,m_2\rangle##. These kets can also serve as the bases in their respective vector spaces.

But now you want to combine the two spaces together through the tensor product. In this new space, the two angular momenta add together ##\mathbf{J} = \mathbf{L}_1 + \mathbf{L}_2##. One of many possible bases in this composite space is formed by directly pairing the basis kets from its two composition spaces: ##|l_1,m_1\rangle |l_2,m_2\rangle##. This form of basis kets is also an eigenket of both ##L_{1z}## and ##L_{2z}## (more precisely ##L_{1z} \otimes \mathbf{1}## and ##\mathbf{1} \otimes L_{2z}##). But you can also find a different basis by forming certain linear combination of the old basis ##|l_1,m_1\rangle |l_2,m_2\rangle##, namely
$$
|J,m_j\rangle = \sum_{m_1} \sum_{m_2} C_{m_1m_2} |l_1,m_1\rangle |l_2,m_2\rangle
$$
such that ##|J,m_j\rangle## is an eigenket of both ##J^2## and ##J_z## operators. The coefficient ##C_{m_1m_2}## is called the Clebsch-Gordan coefficient.
 

Related Threads on CG coefficient in Wigner Eckhart Theorem-how to find

  • Last Post
Replies
2
Views
611
Replies
0
Views
2K
Replies
0
Views
5K
Replies
2
Views
7K
Replies
4
Views
658
Replies
0
Views
3K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
3
Views
903
  • Last Post
Replies
5
Views
2K
Top