QNT-04

Bohr radius

a0 = 4π ε0 ħ² /(m e²) ≈ 0.0529 nm.

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HydrogenBohr 1913

Governing equation

a_n=\dfrac{4\pi\varepsilon_0\hbar^2}{m e^2}\dfrac{n^2}{Z}=a_0\dfrac{n^2}{Z}

where

n: n (—)
Z: Atomic number (—)
a_n: Orbital radius (nm)

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-04 — Bohr radius) is the form associated with Bohr 1913. Working symbols: $n$ , $Z$ $\rightarrow$ $a_n$ . Quantising angular momentum as n ħ in a Coulomb orbit gives a discrete radius a0 n²/Z, with a0 the hydrogen ground-state radius.

Purpose

Purpose: compute

a_n

from

n

Z

in Quantum mechanics via

a_n=\dfrac{4\pi\varepsilon_0\hbar^2}{m e^2}\dfrac{n^2}{Z}=a_0\dfrac{n^2}{Z}

a0 = 4π ε0 ħ² /(m e²) ≈ 0.0529 nm. 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

n = 1.000\,\mathrm{—}

Z = 1.000\,\mathrm{—}

, the governing relation

a_n=\dfrac{4\pi\varepsilon_0\hbar^2}{m e^2}\dfrac{n^2}{Z}=a_0\dfrac{n^2}{Z}

yields

a_n = 0.05292\,\mathrm{nm}

. Hydrogen-like ion, reduced mass ≈ me. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

n n1 —Atomic number Z1 —

Outputs

Orbital radius a_n0.05292 nm

Reading speed

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Narration of this film

Hydrogen-like ion, reduced mass ≈ me.

Quantising angular momentum as n ħ in a Coulomb orbit gives a discrete radius a0 n²/Z, with a0 the hydrogen ground-state radius.

Reading speed

Watch on YouTube