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Hint(n)=ℏΩ J⋅σ(n)2j,Un=exp⁡ ⁣(−iκJ⋅σ(n)2j)H_{\mathrm{int}}^{(n)} =\hbar\Omega\, \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}, \qquad U_n=\exp\!\left(-i\kappa \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}\right)

Why this formula appears here

Let J\mathbf J be the spin- j angular-momentum operator and σ(n)\boldsymbol\sigma^{(n)} the Pauli vector for probe n . A candidate rotationally invariant interaction is Hint(n)=ℏΩ J⋅σ(n)2j,Un=exp⁡ ⁣(−iκJ⋅σ(n)2j)H_{\mathrm{int}}^{(n)} =\hbar\Omega\, \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}, \qquad U_n=\exp\!\left(-i\kappa \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}\right). where Ω\Omega has units of inverse seconds, interaction time τ\tau has seconds, and κ\kappa=Ω\Omegaτ\tau is dimensionless. The factor 2j is a declared normalization choice, not an inherited law. Alternative normalizations must be compared because they change the large- j limit if coupling strength is held fixed under different conventions.

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σ(n)\sigma^{(n)}

Symbol σ^(n)

σ^(n) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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jj

Symbol j

j occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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J⋅σ(n)\mathbf J\cdot\boldsymbol\sigma^{(n)}

Numerator: mathbf J × boldsymbolσ^(n)

The complete quantity above the fraction bar.

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J⋅σ(n)\mathbf J\cdot\boldsymbol\sigma^{(n)}

Numerator: mathbf J × boldsymbolσ^(n)

The complete quantity above the fraction bar.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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Hint(n)=ℏΩ J⋅σ(n)2j,Un=exp⁡ ⁣(−iκJ⋅σ(n)2j),H_{\mathrm{int}}^{(n)} =\hbar\Omega\, \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}, \qquad U_n=\exp\!\left(-i\kappa \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}\right),

Equation 80 · Evolutionary Physics

A Reference Frame Becomes Classical by Publishing Its Orientation

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Let J\mathbf J be the spin- j angular-momentum operator and σ(n)\boldsymbol\sigma^{(n)} the Pauli vector for probe n . A candidate rotationally invariant interaction is Hint(n)=ℏΩ J⋅σ(n)2j,Un=exp⁡ ⁣(−iκJ⋅σ(n)2j)H_{\mathrm{int}}^{(n)} =\hbar\Omega\, \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}, \qquad U_n=\exp\!\left(-i\kappa \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}\right). where Ω\Omega has units of inverse seconds, interaction time τ\tau has seconds, and κ\kappa=Ω\Omegaτ\tau is dimensionless. The factor 2j is a declared normalization choice, not an inherited law. Alternative normalizations must be compared because they change the large- j limit if coupling strength is held fixed under different conventions.

Meanings in this article

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