Pauli Matrices Calculator
Analyze spin-1/2 states and Pauli matrix operations
Pauli Matrices
σx
[0 1]
[1 0]
σy
[0 -i]
[i 0]
σz
[1 0]
[0 -1]
State Analysis
Normalized Input State
|ψ⟩ = 1.0000|↑⟩ + 0.0000|↓⟩
Result After Pauli Operation
σ|ψ⟩ = 1.0000|↑⟩ + 0.0000|↓⟩
Probability |↑⟩
100.00%
Probability |↓⟩
0.00%
Expectation Values
⟨σx⟩
0.0000
⟨σy⟩
0.0000
⟨σz⟩
1.0000
Bloch Sphere
Bloch Angle θ
0.00°
Bloch Angle φ
0.00°
Normalized?
Yes
About Pauli Matrices
The Pauli matrices are 2x2 complex matrices that are fundamental to describing spin-1/2 particles and qubits. They satisfy σi² = I and σiσj = iεijkσk. Any 2x2 Hermitian matrix can be written as a linear combination of Pauli matrices and the identity.