Hydrogen Energy Spectrum Simulator

Hydrogen Atom Emission Spectrum Teach me i

Guided Inquiry

1 · The bound electron

Let’s meet our atom. A single electron is bound to a proton. Unlike a particle in a box, it’s held by the Coulomb attraction, so every allowed energy is negative: E_n = −13.6/n² eV for n = 1, 2, 3, …, with E = 0 the ionisation limit. When it moves between two levels, the electron emits or absorbs one photon carrying exactly the energy difference, and you’ll see that photon’s wavelength recorded on the strip.

2 · The classical atom

Let’s start with the old, pre-quantum view. In classical physics the orbiting electron radiates, spiralling inward through every orbit size on the way. What do you think — what light would a gas of such atoms give off?

3 · The three series

Now let’s sort those lines into families. A series is the set of transitions ending on one fixed lower level. Lyman ends on n = 1, and every one of its lines is ultraviolet. So which series do you think lands in the visible band?

4 · Above the seventh rung

Here’s a question about the very top of the ladder. The diagram draws levels up to n = 7. How many bound levels do you think lie between n = 7 and the ionisation limit E = 0?

5 · Light through cold gas

One last experiment. A lamp with a continuous spectrum, from the ultraviolet through the visible to the infrared, shines through cold hydrogen gas, whose electrons all sit in n = 1. What’s your guess — what does the transmitted spectrum show?

Listen

CONTROLS

Mode

Emission Absorption

Manual Transition

⬇ Emission: Electron transitions from a higher energy state to a lower energy state, releasing a photon.

From n=

To n=

Spectral Series

Visuals

Speed 1×

Last transition

Type:—

Transition:—

ΔE:—

λ:—

Frequency:—

Region:—

Ready — Emission Mode

The Physics

The hydrogen spectrum · what this sim shows, the physics behind it, and every control

At a glance

A hydrogen atom can sit only on certain energy levels, so a jump between two of them releases or absorbs a photon of one exact energy. This sim lets you make those jumps, one at a time or in groups, and records each photon's wavelength on a spectrum strip.

What you'll learn

How the energy levels E_n = −13.6/n² eV fix the size of every possible jump.
How to turn a jump's energy ΔE into a wavelength with λ = hc/ΔE.
How the level a jump ends on decides where its photon lands on the strip.
How the same ladder is read in emission and in absorption.

A sketch of three levels: a jump from n = 3 to n = 1 releases one photon.

What's on screen

Energy Levels (eV) (left): one blue line per level, n = 1 to 7. The dashed red line is E = 0, the ionisation limit. Each jump is drawn as an arrow coloured by its wavelength.
Hydrogen Atom (right): the proton p⁺ at the centre, dashed rings for the levels, and the electron e⁻ on the ring of its current level.
Spectrum strip (bottom): the wavelength of every photon so far, on a logarithmic axis from 50 to 2500 nm. Dashed marks at 380 and 780 nm bound the visible window.
Last transition (sidebar): type, ΔE, wavelength, frequency and region of the most recent jump.

See itMake the jump n = 3 → n = 1. Watch for: the electron dropping to the inner ring and one photon crossing to the strip, where it leaves a line.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Mode	Emission: the electron drops to a lower level and a photon leaves. Absorption: the electron rises to a higher level by taking in a photon.	Emission / Absorption opens in Emission
From n, To n	The two levels of one jump. In Emission, From is the higher level. In Absorption, From is the lower one. The sim corrects an impossible pair.	n = 1–7 opens at From 3, To 1
Make Transition	Fires the one jump set by From and To.	button
Spectral Series	Fires every jump between one fixed level and the levels above it, up to n = 7, one after another: Lyman (→ n = 1), Balmer (→ n = 2), Paschen (→ n = 3). The jump controls are locked until the run ends.	3 buttons
Speed	How fast the electron and photons move. The energies and wavelengths do not change.	0.25×–1× opens at 1×
Clear Spectrum Strip	Removes every recorded line from the strip.	button
Last transition	Reads out the most recent jump: type, ΔE, λ, frequency and region.	panel
Reset	Restores the opening state: Emission, From 3, To 1, speed 1×, empty strip.	top bar

Fixed: Ry = 13.6 eV, h = 6.626×10⁻³⁴ J s, c = 2.998×10⁸ m/s. Levels run from n = 1 to 7.

One jump, one photon

A jump between levels n_hi and n_lo changes the atom's energy by ΔE = Ry (1/n_lo² − 1/n_hi²). That energy goes into one photon, or comes from one, and the photon's wavelength follows from it:

Energy of the jump, and the photon that carries it.

The sim's own constants. In nm and eV, hc ≈ 1240 eV nm.

Example: n = 3 → 1 gives ΔE = 13.6 (1 − 1/9) = 12.09 eV, so λ = 1240/12.09 = 102.6 nm.

From n To n

These sliders move the sim's From n and To n.

Only the two level numbers matter. Doubling ΔE halves λ.
The largest jump the sim can make is 7 → 1: ΔE = 13.32 eV, λ = 93.1 nm.

See itFire the jump set in the panel above. Watch for: a line landing on the strip at the λ the panel shows, matching the Last transition readout.

Only certain wavelengths appear

The allowed energies of the bound electron are

The sim's energy levels.

No energy between two levels is allowed. A photon can therefore carry away only one of the gaps ΔE, and each gap gives exactly one wavelength.
A classical electron circling a proton would radiate continuously and spiral inward, passing through every orbit size and emitting every wavelength on the way. The quantum levels rule that out: no in-between energies, no in-between wavelengths.
The sim records one line per pair of levels. It draws every line equally bright and does not model how likely each jump is.

See itFire the five jumps that end on n = 2. Watch for: five separate narrow lines on the strip with empty gaps between them.

Lines fall into families

All jumps that end on the same lower level form a series. Written for 1/λ, the energy formula becomes the Rydberg formula:

Wavelength of any jump, and the limit each series approaches as n_hi grows.

series	energy gap	wavelength	strip region
Lyman, n = 1	10.20–13.32 eV	121.6–93.1 nm	ultraviolet
Balmer, n = 2	1.889–3.122 eV	656.5–397.1 nm	visible
Paschen, n = 3	0.661–1.234 eV	1875.6–1005.2 nm	infrared

Each range runs over the jumps from the levels above, up to n = 7. The visible window is the strip's 380–780 nm. Lines crowd together towards each series limit.

The three series on the sim's strip.

See itFire Lyman, then Balmer, then Paschen on a cleared strip. Watch for: the Lyman lines first, all to the left of the 380 nm mark; then Balmer inside the visible window, then Paschen beyond 780 nm.

Emission and absorption share one ladder

One ladder, read in either direction.

Going down from n_hi to n_lo releases ΔE. Going up from n_lo to n_hi needs a photon of that same ΔE, so both directions use the same wavelength.
In Absorption mode the strip shows a continuous spectrum, and each jump removes a dark bar at its wavelength. An ideal atom takes in only those wavelengths and lets the rest pass.
The sim's Lyman button in Absorption mode fires jumps up from n = 1, as if every atom started in the ground state. It does not model temperature or how many atoms sit in each level.

See itSwitch to Absorption and fire the Lyman series. Watch for: dark bars cut out of the continuous strip, at the same wavelengths as the bright Lyman lines in section 05.

The levels crowd towards E = 0

Ionisation energy, and the spacing of neighbouring levels.

The gaps shrink fast: 1 → 2 is 10.20 eV, 2 → 3 is 1.889 eV, 3 → 4 is 0.661 eV, and 6 → 7 only 0.100 eV, about 100 times smaller than the first.
The formula has no largest n, so infinitely many bound levels lie between n = 7 and E = 0. The sim draws the first seven.
A small gap means a long wavelength. The jump 7 → 6 gives 12.4 µm, beyond the strip's 2500 nm edge, so the sim pins its line to the right-hand edge.
A photon with 13.6 eV or more (λ ≤ 91.2 nm) can ionise an atom in n = 1. The sim only makes jumps between its bound levels.

See itFire the jump n = 7 → n = 6. Watch for: the Last transition ΔE of 0.1002 eV, and the line pinned at the strip's right-hand edge.

Where it matters

Stars. Hydrogen's Balmer lines appear as dark lines in the light of hot stars, because atoms in n = 2 absorb them. How strong they are depends on the star's temperature.
Glowing gas clouds. Hα, the 3 → 2 line near 656 nm, is one of the main red emission lines of ionised-hydrogen nebulae.
Identifying elements. The pattern of wavelengths is a fingerprint of the element. The Rydberg formula fitted hydrogen's lines before anyone could explain them; Bohr's 1913 model gave the first explanation.

The five Balmer wavelengths the sim fires, removed from a continuum.

Go deeper: what the sim leaves out

The sim's rings are evenly spaced, so a ring's radius grows as n. In the Bohr picture it grows as n²: r_n = n² a₀ with a₀ = 0.0529 nm, so r₇ = 49 a₀ = 2.59 nm. The ring pictures are schematic. The energies are the part that carries over to the full theory.
Real spectra have fine structure, selection rules for which jumps are allowed, and intensities. The sim treats every pair n_hi → n_lo as available and draws all lines equally bright.

Summary

In one line. Hydrogen's levels E_n = −13.6/n² eV are fixed, so every jump releases or absorbs one photon of wavelength λ = hc/ΔE, and only these wavelengths appear.

Key points

ΔE = Ry (1/n_lo² − 1/n_hi²) and λ = 1240 eV nm / ΔE: a bigger gap gives a shorter wavelength.
Jumps sharing a lower level form a series: Lyman (n = 1) is ultraviolet, Balmer (n = 2) is visible, Paschen (n = 3) is infrared.
Emission and absorption use the same gaps and the same wavelengths.
The levels crowd together towards E = 0, so each series piles up at its limit.

Common mix-ups

"A bigger jump gives a longer wavelength." The opposite: λ = hc/ΔE, so a bigger ΔE gives a shorter λ.
"The electron travels along the rings." The rings are a picture of the levels. The energies are what the sim computes; the circular path is not an observation.
"Brighter line, more likely jump." In this sim every line is drawn equally bright, so brightness says nothing about probability.

Things to try on your own

Calculate ΔE and λ for 5 → 2 by hand, then fire it and compare with the Last transition panel.
Fire 2 → 1 and 7 → 6 and compare their ΔE. Note which one lands on the strip.
Run Balmer in Emission, then in Absorption, and compare the positions of the lines and the dark bars.