Questions: Pauli Matrices

Question 1

You know that σz = [[1,0],[0,−1]] has eigenstates |↑⟩ and |↓⟩. What are the eigenstates of σₓ = [[0,1],[1,0]]?

• A. |↑⟩ and |↓⟩ — the same eigenstates, since all Pauli matrices share eigenstates
• B. (|↑⟩ ± |↓⟩)/√2 — equal real superpositions of spin-up and spin-down
• C. (|↑⟩ ± i|↓⟩)/√2 — superpositions with a relative phase of ±i
• D. Only |↑⟩ is an eigenstate of σₓ; |↓⟩ is not

Answer: (B)

Solving σₓ|ψ⟩ = ±|ψ⟩ with |ψ⟩ = (a, b)ᵀ gives b = ±a, so the normalized eigenstates are (|↑⟩ + |↓⟩)/√2 (eigenvalue +1) and (|↑⟩ − |↓⟩)/√2 (eigenvalue −1) — equal superpositions with real coefficients, corresponding to spin pointing in the ±x directions. Option C gives the eigenstates of σᵧ, not σₓ; the imaginary unit i in σᵧ is precisely what encodes the y direction. The Pauli matrices do not share eigenstates (they don't commute), reflecting that spin components along different axes are incompatible observables.

Question 2

A student computes σₓσᵧ = iσz and σᵧσₓ = −iσz. What property of the Pauli matrices does this illustrate?

• A. The Pauli matrices commute, since both products give ±iσz (the same magnitude)
• B. The Pauli matrices anticommute: σₓσᵧ + σᵧσₓ = 0, and also do not commute
• C. The Pauli matrices fail to close under multiplication and are not a group
• D. Both products being nonzero means the Pauli matrices are not Hermitian

Answer: (B)

σₓσᵧ = iσz and σᵧσₓ = −iσz, so σₓσᵧ + σᵧσₓ = 0: the anticommutator {σₓ, σᵧ} = 0. Different Pauli matrices anticommute. Also, σₓσᵧ ≠ σᵧσₓ, so they do NOT commute: [σₓ, σᵧ] = 2iσz ≠ 0. The full identity σᵢσⱼ = δᵢⱼI + iεᵢⱼₖσₖ captures both rules simultaneously. The non-commutativity encodes the geometry of 3D rotations and is the algebraic basis for the uncertainty principle between spin components.

Question 3

The three Pauli matrices commute with each other (σᵢσⱼ = σⱼσᵢ for i ≠ j).

• T. True
• F. False

Answer: False

The Pauli matrices anticommute for i ≠ j: {σᵢ, σⱼ} = σᵢσⱼ + σⱼσᵢ = 0, meaning σᵢσⱼ = −σⱼσᵢ. Equivalently, [σᵢ, σⱼ] = 2iεᵢⱼₖσₖ ≠ 0. Non-commutativity is physically significant: measuring spin along x then y is not the same as measuring y then x, and this gives rise to the Robertson-Heisenberg uncertainty relation ΔSₓΔSᵧ ≥ (ℏ/2)|⟨Sz⟩|.

Question 4

Any 2×2 Hermitian matrix can be written as a real linear combination of the identity I and the three Pauli matrices {σₓ, σᵧ, σz}.

• T. True
• F. False

Answer: True

The set {I, σₓ, σᵧ, σz} forms a basis for the four-dimensional real vector space of 2×2 Hermitian matrices. Any Hermitian M can be written as M = aI + bσz + cσₓ + dσᵧ with a, b, c, d real. This completeness means any spin-½ density matrix, Hamiltonian, or observable is fully characterized by its four components in this basis. The three Pauli components give the Bloch vector — the direction and magnitude of the spin polarization.

Question 5

What physical role does the imaginary unit i play in σᵧ = [[0, −i],[i, 0]]? Why can it not simply be replaced by a real number?

Model answer: The i in σᵧ encodes the phase relationship between z-eigenstates that corresponds to spin pointing along the y axis. The eigenstates of σᵧ are (|↑⟩ ± i|↓⟩)/√2 — the relative phase between up and down components is ±i (a 90° rotation in the complex plane). This phase is what geometrically distinguishes the y direction from x: the eigenstates of σₓ are real superpositions, while the eigenstates of σᵧ have imaginary relative phase. If σᵧ were replaced by a real matrix, it would be linearly dependent on σₓ and σz within the real numbers, and the three matrices would no longer span the space of traceless Hermitian matrices.

More formally: the commutation relations [σᵢ, σⱼ] = 2iεᵢⱼₖσₖ require the commutator to produce an imaginary coefficient times another Pauli matrix. A real σᵧ would violate this algebra. The i is the algebraic necessity that encodes the three-dimensional geometry of spin space — specifically, the fact that 3D rotations form a non-abelian group, which requires complex structure in the 2×2 matrix representation.