https://r2.erweima.ai/mathbot/image/1726662144339wdikeryy.jpg

This is a problem involving the Pauli matrices \( S_x \), \( S_y \), and \( S_z \), which are used in quantum mechanics to describe the spin operators for a spin-1/2 particle. Here's a breakdown of the steps required to solve the problem presented in the image:

### Given:
- Pauli matrices \( S_x \), \( S_y \), and \( S_z \) are defined as:
  \[
  S_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad S_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad S_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
  \]
  
### Part (1) 
To prove:
\[
S_x S_y + S_y S_x = 0
\]

This is essentially showing the anti-commutator relation between \( S_x \) and \( S_y \). To do this, calculate the matrix products and show their sum equals zero.

- **Step 1**: Compute \( S_x S_y \)
  \[
  S_x S_y = \left( \frac{\hbar}{2} \right)^2 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}
  \]
  Carry out the matrix multiplication:
  \[
  S_x S_y = \left( \frac{\hbar}{2} \right)^2 \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}
  \]

- **Step 2**: Compute \( S_y S_x \)
  \[
  S_y S_x = \left( \frac{\hbar}{2} \right)^2 \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
  \]
  Perform the multiplication:
  \[
  S_y S_x = \left( \frac{\hbar}{2} \right)^2 \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}
  \]

- **Step 3**: Add the two products
  \[
  S_x S_y + S_y S_x = \left( \frac{\hbar}{2} \right)^2 \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} + \left( \frac{\hbar}{2} \right)^2 \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} = 0
  \]
Thus, the first part is proven.

---

### Part (2)
To prove the relation:
\[
S_i S_j + S_j S_i = \frac{\hbar^2}{2} \delta_{ij} I
\]
Where \( \delta_{ij} \) is the Kronecker delta and \( I \) is the identity matrix.

- **Step 1**: When \( i = j \), we have:
  \[
  S_i^2 = \frac{\hbar^2}{4} I
  \]
  which satisfies the equation because the identity matrix \( I \) appears.

- **Step 2**: When \( i \neq j \), the anti-commutator of different Pauli matrices equals zero, as shown in part (1).

Therefore, this equation holds for both cases when \( i = j \) and \( i \neq j \).

---

### Part (3)
Prove the commutator relation:
\[
[S_x, S_y] = \frac{i\hbar}{2} S_z
\]
- **Step 1**: Compute the commutator \( [S_x, S_y] = S_x S_y - S_y S_x \).
  Using the results from part (1), we already have:
  \[
  S_x S_y = \frac{\hbar^2}{4} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \quad S_y S_x = \frac{\hbar^2}{4} \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}
  \]
  Subtract these two results:
  \[
  [S_x, S_y] = \frac{\hbar^2}{4} \begin{pmatrix} 2i & 0 \\ 0 & -2i \end{pmatrix} = \frac{i\hbar^2}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
  \]

- **Step 2**: Recognize that the result is proportional to \( S_z \), since:
  \[
  S_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
  \]
Thus, we have:
\[
[S_x, S_y] = i\hbar S_z
\]
This completes the proof for part (3).

---

Do you need further details on these steps or any specific explanations? Here are some related questions to consider:

1. Can you explain the physical significance of the commutator between spin operators?
2. How do Pauli matrices relate to angular momentum in quantum mechanics?
3. Can you show the commutators for other combinations of \( S_x \), \( S_y \), and \( S_z \)?
4. How do the Pauli matrices operate on spinors in quantum mechanics?
5. What is the role of \( \hbar \) in quantum mechanical spin operators?

**Tip:** The commutator relations between spin matrices are crucial for deriving quantum spin behavior and understanding angular momentum algebra.

证明 (1) S_x S_y + S_y S_x = 0; (2) S_i S_j + S_j S_i = \frac{\hbar^2}{2} \delta_{ij} I; (3) [S_x, S_y] = \frac{i\hbar}{2} S_z

Detailed proof of Pauli matrix relations including S_x S_y + S_y S_x = 0, S_i S_j + S_j S_i = \frac{\hbar^2}{2} \delta_{ij} I, and commutator [S_x, S_y] = \frac{i\hbar}{2} S_z. This guide covers key concepts in quantum mechanics, matrix multiplication, and spin operators.

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