ATM-09

Term degeneracy

A term ²ˢ⁺¹L_J has multiplicity 2S+1 and (2J+1) magnetic sublevels.

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SpectraTerm symbols

Governing equation

g=2J+1,\quad M=2S+1

$J \in \left[0, 8\right]$

where

S: Spin S (—)
J: J (—)
2S+1: Multiplicity (—)
2J+1: Degeneracy (—)

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

g=2J+1,\quad M=2S+1

$J \in \left[0, 8\right]$

$J = 0$ $2S+1 = 1$

Along the curve

②

Domain of the variable — blue

J_{\min} = 0,\quad J_{\max} = 8

$J \in \left[0, 8\right]$

$J = 0$ $J = 0$

Along the curve

③

Response curve — green

g=2J+1,\quad M=2S+1

$J \in \left[0, 8\right]$

$J = 0$ $2S+1 = 1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}2S+1}{\mathrm{d}J}

$J \in \left[0, 8\right]$

$J = 0.05$ $2S+1^{\prime} = 2$

Along the curve

Lecture brief

Historical brief

Balmer, Rydberg, Bohr (1913), fine structure, Zeeman and Stern–Gerlach built the spectrum of one atom. The lab computes levels, selection and magnetic splitting. This sheet (ATM-09 — Term degeneracy) is the form associated with Term symbols. Working symbols: $S$ , $J$ $\rightarrow$ $2S+1$ , $2J+1$ . L = 0,1,2,… → S,P,D. J runs from |L−S| to L+S. Hund fills the ground term.

Purpose

Purpose: compute

2S+1

2J+1

from

S

J

in Atomic physics via

g=2J+1,\quad M=2S+1

A term ²ˢ⁺¹L_J has multiplicity 2S+1 and (2J+1) magnetic sublevels. Use it when a real atomic physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

S = 0.500\,\mathrm{—}

J = 1.500\,\mathrm{—}

, the governing relation

g=2J+1,\quad M=2S+1

yields

2S+1 = 2\,\mathrm{—}

2J+1 = 4\,\mathrm{—}

. A term label, a degeneracy bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Spin S S0.5 —J J1.5 —

Outputs

Multiplicity 2S+12 —
Degeneracy 2J+14 —

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Narration of this film

A term label, a degeneracy bar.

L = 0,1,2,… → S,P,D. J runs from |L−S| to L+S. Hund fills the ground term.

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