The GATE Grind

GATE 2022 PH – Question 23

Quantum Mechanics · orbital and spin angular momenta · 1 mark · Multiple select

Which identities hold for Pauli spin matrices?

  1. $[\sigma_\alpha,\sigma_\beta]=i\epsilon_{\alpha\beta\gamma}\sigma_\gamma$
  2. $[\sigma_\alpha,\sigma_\beta]=2i\epsilon_{\alpha\beta\gamma}\sigma_\gamma$
  3. $\{\sigma_\alpha,\sigma_\beta\}=\epsilon_{\alpha\beta\gamma}\sigma_\gamma$
  4. $\{\sigma_\alpha,\sigma_\beta\}=2\delta_{\alpha\beta}I$

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Correct answer: (B) $[\sigma_\alpha,\sigma_\beta]=2i\epsilon_{\alpha\beta\gamma}\sigma_\gamma$; (D) $\{\sigma_\alpha,\sigma_\beta\}=2\delta_{\alpha\beta}I$

Explanation

The Pauli matrices satisfy
$$\sigma_\alpha\sigma_\beta=\delta_{\alpha\beta}I+i\epsilon_{\alpha\beta\gamma}\sigma_\gamma .$$

**Commutator:** subtract the same relation with $\alpha\leftrightarrow\beta$:
$$[\sigma_\alpha,\sigma_\beta]=\sigma_\alpha\sigma_\beta-\sigma_\beta\sigma_\alpha=2i\epsilon_{\alpha\beta\gamma}\sigma_\gamma .$$
So B is true and A (missing the factor 2) is false.

**Anticommutator:** add the two relations (the $\epsilon$ terms cancel):
$$\{\sigma_\alpha,\sigma_\beta\}=2\delta_{\alpha\beta}I .$$
So D is true and C is false.

Answer **B and D**.