QNT-09

Hydrogen energy levels

En = −13.6 Z² / n² eV.

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HydrogenBohr / Schrödinger

Governing equation

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

$n \in \left[1, 10\right]$

where

n: n (—)
Z: Z (—)
E_n: 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\,\mathrm{eV}\,\dfrac{Z^2}{n^2}

$n \in \left[1, 10\right]$

$n = 1$ $E_n = -13.6$

Along the curve

②

Domain of the variable — blue

n_{\min} = 1,\quad n_{\max} = 10

$n \in \left[1, 10\right]$

$n = 1$ $n = 1$

Along the curve

③

Response curve — green

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

$n \in \left[1, 10\right]$

$n = 1$ $E_n = -13.6$

Along the curve

④

Rate of change — amber

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

$n \in \left[1, 10\right]$

$n = 1.057$ $E_n^{\prime} = 23.19$

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-09 — Hydrogen energy levels) is the form associated with Bohr / Schrödinger. Working symbols: $n$ , $Z$ $\rightarrow$ $E_n$ . The Coulomb problem in QM reproduces Bohr's energies −13.6 Z²/n² eV, with degeneracy n² in the non-relativistic theory.

Purpose

Purpose: compute

E_n

from

n

Z

in Quantum mechanics via

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

En = −13.6 Z² / n² eV. 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

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

yields

E_n = -13.6000\,\mathrm{eV}

. Hydrogen-like, infinite nuclear mass. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

n n1 —Z Z1 —

Outputs

Energy E_n-13.6000 eV

Reading speed

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Narration of this film

Hydrogen-like, infinite nuclear mass.

The Coulomb problem in QM reproduces Bohr's energies −13.6 Z²/n² eV, with degeneracy n² in the non-relativistic theory.

Reading speed

Watch on YouTube