ATM-19

Thomas–Reiche–Kuhn sum

Σ f = Z. The remaining oscillator strength after two known lines.

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SpectraThomas–Reiche–Kuhn

Governing equation

\sum_n f_{gn}=Z

$f_1 \in \left[0, 2\right]$

where

Z: Electrons Z (—)
f_1: Line 1 (—)
f_2: Line 2 (—)
f_{rest}: Remaining f (—)

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

\sum_n f_{gn}=Z

$f_1 \in \left[0, 2\right]$

$f_1 = 0$ $f_{rest} = 1.8$

Along the curve

②

Domain of the variable — blue

f_1_{\min} = 0,\quad f_1_{\max} = 2

$f_1 \in \left[0, 2\right]$

$f_1 = 0$ $f_1 = 0$

Along the curve

③

Response curve — green

\sum_n f_{gn}=Z

$f_1 \in \left[0, 2\right]$

$f_1 = 0$ $f_{rest} = 1.8$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f_{rest}}{\mathrm{d}f_1}

$f_1 \in \left[0, 2\right]$

$f_1 = 0.013$ $f_{rest}^{\prime} = -1$

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-19 — Thomas–Reiche–Kuhn sum) is the form associated with Thomas–Reiche–Kuhn. Working symbols: $Z$ , $f_1$ , $f_2$ $\rightarrow$ $f_{rest}$ . A direct consequence of the double commutator [x,[H,x]]. Bound plus continuum.

Purpose

Purpose: compute

f_{rest}

from

Z

f_1

f_2

in Atomic physics via

\sum_n f_{gn}=Z

Σ f = Z. The remaining oscillator strength after two known lines. 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

Z = 2.000\,\mathrm{—}

f_1 = 0.400\,\mathrm{—}

f_2 = 0.200\,\mathrm{—}

, the governing relation

\sum_n f_{gn}=Z

yields

f_{rest} = 1.400\,\mathrm{—}

. A stack of f bars that must fill Z. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Electrons Z Z2 —Line 1 f_10.4 —Line 2 f_20.2 —

Outputs

Remaining f f_{rest}1.400 —

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Narration of this film

A stack of f bars that must fill Z.

A direct consequence of the double commutator [x,[H,x]]. Bound plus continuum.

Reading speed

Watch on YouTube