Spin-1 Operators

1. Spin Operators for Spin-1

The spin operators for spin-1 can be expressed in the basis states \( | 1, 1 \rangle \), \( | 1, 0 \rangle \), and \( | 1, -1 \rangle \). Here are the matrices:

1.1. \( S_z \) Matrix

\[ S_z = \hbar \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \]

1.2. Raising Operator \( S_+ \) Matrix

\[ S_+ = \hbar \begin{pmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \]

1.3. Lowering Operator \( S_- \) Matrix

\[ S_- = \hbar \begin{pmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \]

2. Clebsch-Gordon Relations

The Clebsch-Gordon coefficients for the raising and lowering operators are:

\[ S_+ |S, m \rangle = \hbar \sqrt{(S - m)(S + m + 1)} |S, m + 1 \rangle \] \[ S_- |S, m \rangle = \hbar \sqrt{(S + m)(S - m + 1)} |S, m - 1 \rangle \]

3. Equations for Spin-1 States

For a spin-1 system, the actions of \( S_+ \) and \( S_- \) on the states are:

3.1. Raising Operator \( S_+ \)

1. For \( |1, 1\rangle \): \[ S_+ |1, 1\rangle = 0 \]

2. For \( |1, 0\rangle \): \[ S_+ |1, 0\rangle = \sqrt{2} \hbar |1, 1\rangle \]

3. For \( |1, -1\rangle \): \[ S_+ |1, -1\rangle = \hbar |1, 0\rangle \]

3.2. Lowering Operator \( S_- \)

1. For \( |1, 1\rangle \): \[ S_- |1, 1\rangle = \hbar \sqrt{2} |1, 0\rangle \]

2. For \( |1, 0\rangle \): \[ S_- |1, 0\rangle = \hbar |1, -1\rangle \]

3. For \( |1, -1\rangle \): \[ S_- |1, -1\rangle = 0 \]

4. Generating \( S_x \) and \( S_y \)

To generate \( S_x \) and \( S_y \) from \( S_+ \) and \( S_- \), we use the following relationships:

4.1. Formulas

1. For \( S_x \): \[ S_x = \frac{1}{2} (S_+ + S_-) \]

2. For \( S_y \): \[ S_y = \frac{1}{2i} (S_+ - S_-) \]

5. Calculating \( S_x \) and \( S_y \)

5.1. Calculating \( S_x \)

\[ S_x = \frac{1}{2} (S_+ + S_-) = \frac{\hbar}{2} \begin{pmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} + \frac{\hbar}{2} \begin{pmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \]

\[ = \frac{\hbar}{2} \begin{pmatrix} 0 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \]

5.2. Calculating \( S_y \)

\[ S_y = \frac{1}{2i} (S_+ - S_-) = \frac{\hbar}{2i} \begin{pmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} - \frac{\hbar}{2i} \begin{pmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \]

\[ = \frac{\hbar}{2i} \begin{pmatrix} 0 & -\sqrt{2} & 0 \\ \sqrt{2} & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} \]

6. Final Matrices

The final matrices for \( S_x \) and \( S_y \) for spin-1 are:

Matrix for \( S_x \)

\[ S_x = \frac{\hbar}{2} \begin{pmatrix} 0 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \]

Matrix for \( S_y \)

\[ S_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -\sqrt{2}i & 0 \\ \sqrt{2}i & 0 & -1i \\ 0 & 1i & 0 \end{pmatrix} \]