PauliMatrizen

PauliMatrizen, commonly called Pauli matrices, are a set of three 2x2 complex matrices used to describe the spin of spin-1/2 particles in quantum mechanics. The matrices are σ_x = [[0, 1], [1, 0]], σ_y = [[0, -i], [i, 0]], and σ_z = [[1, 0], [0, -1]]. They form the basic algebraic toolkit for representing qubit states and spin observables.

These matrices are Hermitian and unitary, and each squares to the identity. They are traceless and satisfy the commutation and anti-commutation relations: [σ_i, σ_j] = 2i ε_ijk σ_k and {σ_i, σ_j} = 2 δ_ij I, where ε_ijk is the Levi-Civita symbol and I is the 2x2 identity. Their eigenvalues are ±1.

In quantum mechanics, PauliMatrizen describe the spin operators for spin-1/2 systems: S_i = (ħ/2) σ_i. They generate the su(2) Lie algebra and underpin rotations in quantum state space. A rotation by angle θ about a unit vector n is represented by U = exp(-i θ n · σ / 2). On the Bloch sphere, a pure qubit state is characterized by the expectation values of the Pauli matrices, providing a geometric interpretation of quantum states.

For multi-qubit systems, tensor products of Pauli matrices form a basis for 2^n by 2^n operators, enabling compact representation of quantum gates and Hamiltonians. Any 2x2 Hermitian matrix can be written as a0 I + a · σ. The matrices are named after Wolfgang Pauli and are central to quantum information, condensed matter physics, and many areas of quantum theory.

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