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Formal Dirac spin coefficient in a uniform field

Facts & Assumptions

Given: Use the formal spin term and magnetic-moment convention of Minimal Dirac magnetic square and formal Pauli benchmark. Let $m > 0$ be its mass in kilograms, $q = - e$ with $e > 0$ in coulombs, and define $A : R^{3} \to R^{3}$ by $A (x) = \frac{1}{2} B_{0} e_{3} \times x$ , where $B_{0} > 0$ is in tesla; this is smooth with tesla-metre components and its curl is the constant field $B = \nabla \times A : R^{3} \to R^{3}$ , $B (x) = B_{0} e_{3}$ , where $e_{3} = (0, 0, 1)$ . Use $χ_{+} = (1, 0)^{T}$ and $χ_{-} = (0, 1)^{T}$ in $C^{2}$ , which satisfy $σ_{3} χ_{\pm} = \pm χ_{\pm}$ for the matrix in the supplier.

[F1]

Minimal Dirac magnetic square and formal Pauli benchmark: the formal magnetic term is $- (q / m) S \cdot B$ with $S = ℏ Σ / 2$ and the minimal coefficient convention has $g_{s} = 2$ .

Example

The two spin eigenvalues give opposite formal energy coefficients in the same uniform field. For comparison only, define the formal frequency coefficient $ν_{c o m p} : = e B_{0} / (2 π m)$ , whose SI unit is $C T {k g}^{- 1} = s^{- 1}$ , equivalently the formal angular-frequency coefficient $ω_{c o m p} : = 2 π ν_{c o m p} = e B_{0} / m$ . This is a formal comparison only, not a trajectory law or empirical prediction.

Verification

1.1givenF1algebra

With $q = - e$ , the supplier's term is $(e / m) S_{3} B_{0}$ . Since $σ_{3}$ has eigenvalues $\pm 1$ , $S_{3}$ has eigenvalues $\pm ℏ / 2$ , so the two coefficients are $\pm e ℏ B_{0} / (2 m) = \pm μ_{B} B_{0}$ , where $μ_{B} = e ℏ / (2 m)$ . Their gap is $e ℏ B_{0} / m$ .

2.1step 1.1F1algebra∎

Dividing that formal energy gap by $ℏ$ gives the angular-frequency coefficient $ω_{s} = e B_{0} / m$ in $r a d s^{- 1}$ . The cyclic-frequency coefficient is $ν_{s} : = ω_{s} / (2 π) = e B_{0} / (2 π m)$ in hertz, so $ν_{s} = ν_{c o m p}$ and $ω_{s} = ω_{c o m p}$ . This compares formal coefficients only; it is not a full Landau-spectrum result or an empirical prediction.

Depends on

Minimal Dirac magnetic square and formal Pauli benchmark

Formal Dirac spin coefficient in a uniform field

Facts & Assumptions

Example

Verification

Depends on

Dependency tree · 2 levels

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