Spin Operator Eigenvalues Calculator

Spin Operator Eigenvalues Calculator finds S_z, S^2 eigenvalues and probabilities for quantum spin systems, aiding quantum mechanics and quantum computing studies.

Spin Quantum Number (s):

Spin-Up Coefficient (α):

Spin-Down Coefficient (β):

Formulas Used in Spin Operator Eigenvalues Calculator

The calculator uses the following formulas to compute spin operator eigenvalues:

S_z Eigenvalues:

\\[ m_s \hbar, \quad m_s = -s, -s + 1, \ldots, s – 1, s \\]

S^2 Eigenvalue:

\\[ s(s + 1) \hbar^2 \\]

Probabilities for S_z Measurement (s = 1/2):

\\[ P(\uparrow) = |\alpha|^2, \quad P(\downarrow) = |\beta|^2 \\]

Where:

\\( m_s \\): Magnetic quantum number
\\( s \\): Spin quantum number
\\( \hbar \\): Reduced Planck’s constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
\\( \alpha, \beta \\): Coefficients of spin-up and spin-down states (\\( |\alpha|^2 + |\beta|^2 = 1 \\))
\\( P(\uparrow), P(\downarrow) \\): Probabilities for spin-up and spin-down (s = 1/2)

Example Calculations

Example 1: s = 1/2, Equal Superposition

Input: Spin Quantum Number = 0.5, Spin-Up Coefficient = 0.707, Spin-Down Coefficient = 0.707

\\[ S_z \text{ eigenvalues} = \left\{ +\frac{\hbar}{2}, -\frac{\hbar}{2} \right\} = \left\{ +5.272859 \times 10^{-35}, -5.272859 \times 10^{-35} \right\} \ \text{J·s} \\] \\[ S^2 \text{ eigenvalue} = s(s + 1) \hbar^2 = 0.5 \cdot 1.5 \cdot (1.0545718 \times 10^{-34})^2 = 8.337468 \times 10^{-69} \ \text{J}^2\text{·s}^2 \\] \\[ P(\uparrow) = |\alpha|^2 = 0.707^2 = 0.5 \ (50\%) \\] \\[ P(\downarrow) = |\beta|^2 = 0.707^2 = 0.5 \ (50\%) \\]

Result: S_z Eigenvalues: {+5.27e-35, -5.27e-35} J·s, S^2 Eigenvalue: 8.34e-69 J²·s², Probabilities: 50% (up), 50% (down)

Example 2: s = 1, No Probabilities

Input: Spin Quantum Number = 1

\\[ S_z \text{ eigenvalues} = \{-1, 0, 1\} \cdot \hbar = \{-1.0545718 \times 10^{-34}, 0, 1.0545718 \times 10^{-34}\} \ \text{J·s} \\] \\[ S^2 \text{ eigenvalue} = s(s + 1) \hbar^2 = 1 \cdot 2 \cdot (1.0545718 \times 10^{-34})^2 = 2.2233248 \times 10^{-68} \ \text{J}^2\text{·s}^2 \\]

Result: S_z Eigenvalues: {-1.05e-34, 0, 1.05e-34} J·s, S^2 Eigenvalue: 2.22e-68 J²·s²

Example 3: s = 1/2, Spin-Up State

Input: Spin Quantum Number = 0.5, Spin-Up Coefficient = 1, Spin-Down Coefficient = 0

\\[ S_z \text{ eigenvalues} = \left\{ +\frac{\hbar}{2}, -\frac{\hbar}{2} \right\} = \left\{ +5.272859 \times 10^{-35}, -5.272859 \times 10^{-35} \right\} \ \text{J·s} \\] \\[ S^2 \text{ eigenvalue} = 0.5 \cdot 1.5 \cdot (1.0545718 \times 10^{-34})^2 = 8.337468 \times 10^{-69} \ \text{J}^2\text{·s}^2 \\] \\[ P(\uparrow) = |\alpha|^2 = 1^2 = 1 \ (100\%) \\] \\[ P(\downarrow) = |\beta|^2 = 0^2 = 0 \ (0\%) \\]

Result: S_z Eigenvalues: {+5.27e-35, -5.27e-35} J·s, S^2 Eigenvalue: 8.34e-69 J²·s², Probabilities: 100% (up), 0% (down)

Formulas Used in Spin Operator Eigenvalues Calculator

Example Calculations

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