QNT-12

Electron magnetic moment

μ = −g μB S/ħ with g ≈ 2, μB = e ħ /(2 me).

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SpinUhlenbeck–GoudsmitDirac

Governing equation

\mu=-g\mu_B m_s,\quad \mu_B=\dfrac{e\hbar}{2m_e}

$B \in \left[0, 20\right]$

where

g: g-factor (—)
m_s: Spin projection (—)
B: Field (for ΔE) (T)
\mu: Moment (μB) (μB)
\Delta E: Zeeman energy (µeV)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\mu=-g\mu_B m_s,\quad \mu_B=\dfrac{e\hbar}{2m_e}

$B \in \left[0, 20\right]$

$B = 0$ $\mu = 0$

Along the curve

②

Domain of the variable — blue

B_{\min} = 0,\quad B_{\max} = 20

$B \in \left[0, 20\right]$

$B = 0$ $B = 0$

Along the curve

③

Response curve — green

\mu=-g\mu_B m_s,\quad \mu_B=\dfrac{e\hbar}{2m_e}

$B \in \left[0, 20\right]$

$B = 0$ $\mu = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\mu}{\mathrm{d}B}

$B \in \left[0, 20\right]$

$B = 0.126$ $\mu^{\prime} = 57.94$

Along the curve

Lecture brief

Historical brief

Planck (1900), Einstein’s photoelectric law, Bohr, de Broglie, Heisenberg and Schrödinger’s 1926 equation rebuilt matter as amplitude. These sheets are the first solvable models: wells, spin, tunneling, uncertainty. This sheet (QNT-12 — Electron magnetic moment) is the form associated with Uhlenbeck–Goudsmit · Dirac. Working symbols: $g$ , $m_s$ , $B$ $\rightarrow$ $\mu$ , $\Delta E$ . Dirac's equation predicts g = 2 for the electron; the moment μB is the natural magneton. Stern–Gerlach splits beams by 2 μB B.

Purpose

Purpose: compute

\mu

\Delta E

from

g

m_s

B

in Quantum mechanics via

\mu=-g\mu_B m_s,\quad \mu_B=\dfrac{e\hbar}{2m_e}

μ = −g μB S/ħ with g ≈ 2, μB = e ħ /(2 me). Use it when a real quantum mechanics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

g = 2.002\,\mathrm{—}

m_s = 0.500\,\mathrm{—}

B = 1.000\,\mathrm{T}

, the governing relation

\mu=-g\mu_B m_s,\quad \mu_B=\dfrac{e\hbar}{2m_e}

yields

\mu = -1.0010\,\mathrm{\mu B}

\Delta E = 57.942\,\mathrm{\mu eV}

. g-factor input, spin ½ projection ms = ±½. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

g-factor g2.002 —Spin projection m_s0.5 —Field (for ΔE) B1 T

Outputs

Moment (μB) \mu-1.0010 μB
Zeeman energy \Delta E57.942 µeV

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Narration of this film

g-factor input, spin ½ projection ms = ±½.

Dirac's equation predicts g = 2 for the electron; the moment μB is the natural magneton. Stern–Gerlach splits beams by 2 μB B.

Reading speed

Watch on YouTube