![The Pauli spin matrices are defined as Which of these matrices has an eigenvector [1 -1]T? σx σy σz None have real eigenvectors.](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTkyMCAxMDgwIiB3aWR0aD0iMTkyMCIgaGVpZ2h0PSIxMDgwIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==)
The Pauli spin matrices are defined as
Which of these matrices has an eigenvector [1 -1]T?
σx
σy
σz
None have real eigenvectors.
Quantum Mechanics MCQ – Pauli Matrices & Eigenvectors
The Pauli spin matrices are defined as:
σx = ( 0 1
1 0 )
1 0 )
σy = ( 0 −i
i 0 )
i 0 )
σz = ( 1 0
0 −1 )
0 −1 )
Which of these matrices has [1 −1]ᵀ as an eigenvector?
Options:
- A) σx
- B) σy
- C) σz
- D) None have real eigenvectors
Step-by-Step Explanation
✔ For σx
σₓ [1 -1]ᵀ = [-1 1]ᵀ = -[1 -1]ᵀ → Eigenvector with eigenvalue -1 ✔
❌ For σy
σᵧ [1 -1]ᵀ = [i i]ᵀ ≠ λ[1 -1]ᵀ → Not proportional → Not an eigenvector ❌
✔ For σz
σ𝓏 [1 -1]ᵀ = [1 1]ᵀ = +1 [1 -1]ᵀ → Eigenvalue +1 ✔
Summary Table
| Matrix | Test Result | Eigenvalue | Correct? |
|---|---|---|---|
| σx | Eigenvector | -1 | ✔ |
| σy | Not an eigenvector | — | ❌ |
| σz | Eigenvector | +1 | ✔ Correct Choice |
🎯 Final Answer: C) σz
