ATM-02

Bohr level energy

E_n = −13.6 Z² / n² eV. Bound hydrogenic levels.

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SpectraBohr

Governing equation

E_n=-13.6\,Z^2/n^2\,\mathrm{eV}

$n \in \left[1, 12\right]$

where

Z: Atomic number (—)
n: Principal n (—)
E_n: Level energy (eV)

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

E_n=-13.6\,Z^2/n^2\,\mathrm{eV}

$n \in \left[1, 12\right]$

$n = 1$ $E_n = -13.61$

Along the curve

②

Domain of the variable — blue

n_{\min} = 1,\quad n_{\max} = 12

$n \in \left[1, 12\right]$

$n = 1$ $n = 1$

Along the curve

③

Response curve — green

E_n=-13.6\,Z^2/n^2\,\mathrm{eV}

$n \in \left[1, 12\right]$

$n = 1$ $E_n = -13.61$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E_n}{\mathrm{d}n}

$n \in \left[1, 12\right]$

$n = 1.069$ $E_n^{\prime} = 22.45$

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-02 — Bohr level energy) is the form associated with Bohr. Working symbols: $Z$ , $n$ $\rightarrow$ $E_n$ . Matches the Rydberg formula. Finite-mass and fine-structure corrections sit on top.

Purpose

Purpose: compute

E_n

from

Z

n

in Atomic physics via

E_n=-13.6\,Z^2/n^2\,\mathrm{eV}

E_n = −13.6 Z² / n² eV. Bound hydrogenic levels. 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 = 1.000\,\mathrm{—}

n = 2.000\,\mathrm{—}

, the governing relation

E_n=-13.6\,Z^2/n^2\,\mathrm{eV}

yields

E_n = -3.4014\,\mathrm{eV}

. A ladder of n rungs, an electron dropping. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Atomic number Z1 —Principal n n2 —

Outputs

Level energy E_n-3.4014 eV

Reading speed

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Narration of this film

A ladder of n rungs, an electron dropping.

Matches the Rydberg formula. Finite-mass and fine-structure corrections sit on top.

Reading speed

Watch on YouTube