How Do You Convert Spin Basis Using Clebsch-Gordan Coefficients?

[Domnu]

Problem

Express $$\bold{S}^2$$ in the $$|j_1 j_2 ; m_1 m_2\rangle$$ and $$|j_1 j_2; s m\rangle$$ bases and find the unitary matrix, $$U$$, which takes the $$|j_1 j_2 ; m_1 m_2\rangle$$ basis to the $$|j_1 j_2; s m\rangle$$ basis, for spin $$1/2$$ (s corresponds to the total angular momentum while m corresponds to the total z angular momentum). This is basically "adding" together two angular momenta of spin 1/2.

My attempt at a Solution

Well, we know that $$\bold{S}^2 = \bold{S_1}^2+\bold{S_2}^2 + 2 S_{1z} S_{2z} + S_{1+}S_{2-} + S_{1-}S_{2+}$$. Now, using this and the fact that $$j_1 = j_2 = 1/2$$, I got that

$$\bold{S^2} = \hbar^2\begin{bmatrix} 2 & 0 & 0 & 0\\ 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 1\\ 0 & 0 & 2 & 0\end{bmatrix}$$​

in the $$|j_1 j_2 ; m_1 m_2\rangle$$ basis. The above columns are, from left to right, |+ +>, |+ ->, |- +>, and |- ->, respectively. Now, for the other basis, I got

$$\bold{S^2} = \hbar^2\begin{bmatrix} 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0\end{bmatrix}$$​

to be the representation; the columns above represent |1 1>, |1 0>, |1 -1>, and |0 0>, respectively. First of all, are these correct? If they are, how can there be a unitary transformation from one matrix to another? Two of the first matrices' columns are equal, but the corresponding columns in the second matrix are not equal. How can we resolve this?

Thanks very much for the help.

 

[Jhero]

Hi there,

Your calculations seem to be correct. The two matrices are indeed different, as they represent different bases. In the first matrix, the basis states are |j_1 j_2 ; m_1 m_2\rangle, while in the second matrix, the basis states are |j_1 j_2; s m\rangle. These two bases are related by a unitary transformation, which we can find by diagonalizing the first matrix and then transforming it to the second basis.

To find the unitary matrix, U, we first need to find the eigenvectors of the first matrix. These eigenvectors will be the columns of U. Since the first matrix is already in diagonal form, the eigenvectors are simply the columns of the matrix. We can write them as:

|+ +>, |+ ->, |- +>, and |- ->.

Now, we can transform these vectors to the second basis by multiplying them with the transformation matrix, which can be written as:

U = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 1 & 0 & -1 & 0\\ 0 & 1 & 0 & -1\end{bmatrix}

So, the unitary transformation is given by:

|+ +> = \frac{1}{\sqrt{2}}(|1 1> + |1 -1>)

|+ -> = \frac{1}{\sqrt{2}}(|1 1> - |1 -1>)

|- +> = \frac{1}{\sqrt{2}}(|1 0> + |1 0>)

|- -> = \frac{1}{\sqrt{2}}(|1 0> - |1 0>)

I hope this helps. Let me know if you have any further questions.