Question

Which of the following is not correct about Pauli spin matrices?

• A. \( [\sigma_x, \sigma_y] = 0 \)
• B. \( \sigma_x \sigma_y = -i\sigma_z \)
• C. \( Tr(\sigma) = 0 \)
• D. \( \sigma^2 = I \)

Answer 1

The Correct Option is A

Let's analyze each statement about the Pauli spin matrices:

1. \( [\sigma_x, \sigma_y] = 0 \)

The commutator of \( \sigma_x \) and \( \sigma_y \) is given by:

\[ [\sigma_x, \sigma_y] = \sigma_x \sigma_y - \sigma_y \sigma_x \]

Using the Pauli matrix multiplication rules, we get: \[ [\sigma_x, \sigma_y] = 2i \sigma_z \] Therefore, this statement is incorrect. The correct commutation relation is: \[ [\sigma_x, \sigma_y] = 2i \sigma_z \] 

2. \( \sigma_x \sigma_y = -i\sigma_z \)

Multiplying \( \sigma_x \) and \( \sigma_y \), we get:

\[ \sigma_x \sigma_y = -i \sigma_z \]

This is correct according to the Pauli matrices multiplication rules. 

3. \( \text{Tr}(\sigma) = 0 \)

The trace of any Pauli matrix \( \sigma_x \), \( \sigma_y \), or \( \sigma_z \) is zero. That is:

\[ \text{Tr}(\sigma_x) = \text{Tr}(\sigma_y) = \text{Tr}(\sigma_z) = 0 \]

This is also correct. 

4. \( \sigma^2 = I \)

For any Pauli matrix \( \sigma_x \), \( \sigma_y \), or \( \sigma_z \), we have:

\[ \sigma_x^2 = \sigma_y^2 = \sigma_z^2 = I \]

This is true for all the Pauli matrices, so this statement is correct. 

Conclusion: The incorrect statement is:

\( [\sigma_x, \sigma_y] = 0 \)