Pauli Matrix Operator Calculator
Pauli Matrix Operator Calculator computes expectation values and applies Pauli operators to a qubit state, aiding quantum mechanics and quantum computing studies.
Formulas Used in Pauli Matrix Operator Calculator
The calculator computes expectation values and applies Pauli operators for a qubit state:
Qubit State:
\\[ |\psi\rangle = a |0\rangle + b |1\rangle \\]Pauli Matrices:
\\[ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\]Expectation Values:
\\[ \langle \sigma_x \rangle = a^* b + b^* a, \quad \langle \sigma_y \rangle = -i (a^* b – b^* a), \quad \langle \sigma_z \rangle = |a|^2 – |b|^2 \\]Operator Application:
\\[ \sigma_x |\psi\rangle = \begin{pmatrix} b \\ a \end{pmatrix}, \quad \sigma_y |\psi\rangle = \begin{pmatrix} -i b \\ i a \end{pmatrix}, \quad \sigma_z |\psi\rangle = \begin{pmatrix} a \\ -b \end{pmatrix} \\]Where:
- \\( a, b \\): Complex coefficients (\\( |a|^2 + |b|^2 = 1 \\))
- \\( \sigma_x, \sigma_y, \sigma_z \\): Pauli matrices
- \\( \langle \sigma_i \rangle \\): Expectation value of operator \\( \sigma_i \\)
Example Calculations
Example 1: Eigenstate of \\( \sigma_z \\)
Input: a_r = 1, a_i = 0, b_r = 0, b_i = 0, Operator = \\( \sigma_z \\)
Result: \\( \langle \sigma_x \rangle = 0 \\), \\( \langle \sigma_y \rangle = 0 \\), \\( \langle \sigma_z \rangle = 1 \\), New State: [1, 0]
Example 2: Eigenstate of \\( \sigma_x \\)
Input: a_r = 0.707, a_i = 0, b_r = 0.707, b_i = 0, Operator = \\( \sigma_x \\)
Result: \\( \langle \sigma_x \rangle = 1 \\), \\( \langle \sigma_y \rangle = 0 \\), \\( \langle \sigma_z \rangle = 0 \\), New State: [0.707, 0.707]
Example 3: General State
Input: a_r = 0.8, a_i = 0, b_r = 0.6, b_i = 0, Operator = \\( \sigma_y \\)
Result: \\( \langle \sigma_x \rangle = 0.96 \\), \\( \langle \sigma_y \rangle = 0 \\), \\( \langle \sigma_z \rangle = 0.28 \\), New State: [0 – 0.6i, 0.8i]