Hydrogen energy levels Eₙ = −13.6 - n² eV: Foundations

This page assumes you know nothing. We collect every symbol, term, and idea the parent note Hydrogen energy levels leans on, and build each one from a picture. Read top to bottom — each block uses only things defined above it. We will not write the famous energy formula until we have earned every letter in it (that happens in §7).

0 · The cast of characters (the atom itself)

Figure 1 — The hydrogen atom. The electron circles the proton; the pull between them is the whole story.

Figure — Hydrogen energy levels Eₙ = −13.6 - n² eV

Look at Figure 1: the coral dot is the proton, the lavender dot is the electron on a circular path. Everything in this topic is a story about how tightly that electron is held and which circles it is allowed to travel on.

1 · Charge $e$ , distance $r$ , and the electric pull $F$

2 · Speed $v$

3 · Mass $m$ and the "turning" force

Recall A small honesty note: reduced mass

We treat the proton as perfectly fixed and use the electron mass $m$ . In reality the proton is heavy but not infinitely heavy, so both particles wobble about their shared centre. The exact fix is to replace $m$ with the reduced mass $μ = \frac{m M}{m + M}$ , where $M$ is the proton mass. Since $M \approx 1836 m$ , $μ$ is only about $0.05%$ smaller than $m$ — enough to shift $13.6$ eV in the fourth digit, but invisible in everything we do here. Keep $m$ for all our work; just know $μ$ is the precise version.

Figure 2 — Two forces, one direction. The electric pull and the centripetal need both point inward, so we set them equal.

4 · Kinetic and potential energy

Figure 3 — The energy well. Zero energy is the rim (electron escaped); any bound electron sits below it, so its total energy is negative.

5 · The whole numbers $n$ (the shelves)

6 · Putting two rules together — the Bohr radius $a_{0}$

We now have two ready-made facts. Rule A (force balance from §3): the Coulomb pull equals the centripetal need, $\frac{1}{4 π ε _{0}} \frac{e ^{2}}{r ^{2}} = \frac{m v ^{2}}{r} .$ Rule B (quantization from §5): $m v r = n ℏ$ .

Formula Deriving

$r_{n} = a_{0} n^{2}$ step by step What we do first: tidy Rule A by multiplying both sides by $r$ , so the messy $r^{2}$ becomes a single $r$ : $m v^{2} = \frac{1}{4 π ε _{0}} \frac{e ^{2}}{r} . (⋆)$ Why: this isolates $m v^{2}$ , which we can attack using Rule B. Next, solve Rule B for the speed $v$ (so we can eliminate it): $v = \frac{n ℏ}{m r} .$ Why: we have two unknowns ( $v$ and $r$ ); getting rid of $v$ leaves an equation in $r$ alone. Now substitute this $v$ into $(⋆)$ :

\;\Longrightarrow\; \frac{n^2\hbar^2}{m r^2} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}.$$ **Finally**, multiply both sides by $r^2$ and divide by the right-hand coefficient to solve for $r$: $$\boxed{\,r_n = \frac{4\pi\varepsilon_0\,\hbar^2}{m e^2}\;n^2 = a_0\,n^2\,},\qquad a_0 = \frac{4\pi\varepsilon_0\,\hbar^2}{m e^2}=0.529\ \text{Å}.$$ **What it means:** the whole clump of constants in front of $n^2$ *is* the Bohr radius $a_0$; the orbits grow as $n^2$, so the electron sits ever farther out on higher shelves.

A fixed positive charge

+ e

at the centre that pulls the electron in.

What is

e

The elementary charge,

1.6 \times 1 0^{- 19}

C, carried by proton (

+

) and electron (

-

What is the factor

\frac{1}{4 π ε _{0}}

for?

The constant in Coulomb's law that converts charges and distance into force.

What does

r

mean here?

The radius of the electron's circular orbit (proton-to-electron distance).

What is

F

Force in newtons — here the electric pull of proton on electron, given by Coulomb's law.

What is centripetal force and its formula?

The inward pull needed to keep a mass on a circle,

m v^{2} / r

Why do we set the Coulomb pull equal to the centripetal force?

The electric attraction is exactly what supplies the inward pull the circle requires.

What is kinetic energy?

Energy of motion,

\frac{1}{2} m v^{2}

; always positive.

What is potential energy here and its sign?

Stored energy of position,

- \frac{1}{4 π ε _{0}} \frac{e ^{2}}{r}

; negative, with zero chosen at infinity.

What is the total energy

E

(and

E_{n}

E = KE + PE

; when restricted to the orbit numbered

n

it becomes

E_{n}

Why is total energy negative?

Because the deep negative PE outweighs the positive KE — the electron is trapped in a well.

What is the reduced mass and why mention it?

μ = m M / (m + M)

; it replaces

m

for exact accuracy because the proton is heavy but not infinitely heavy.

What is the principal quantum number

n

A whole number

1, 2, 3, \dots

labelling the allowed orbits/energies.

What is the quantization rule?

Angular momentum

m v r = n ℏ

, forcing

n

to be an integer.

What is

ℏ

Planck's constant divided by

2 π

ℏ = h /2 π

What is the Bohr radius

a_{0}

The smallest allowed orbit radius,

a_{0} = 0.529

Å, so

r_{n} = a_{0} n^{2}

How do KE and PE combine to give

E_{n}

E = \frac{1}{2} k e^{2} / r - k e^{2} / r = - \frac{1}{2} k e^{2} / r = - KE

; substitute

r = a_{0} n^{2}

to get

- 13.6/ n^{2}

eV.

What is the speed of light

c

How fast light travels in vacuum,

3.0 \times 1 0^{8}

m/s, same for every colour.

What is

Δ E

The energy difference between two levels,

E_{n_{i}} - E_{n_{f}}

, which becomes a photon.

What is one electron-volt?

1.6 \times 1 0^{- 19}

J — the atom-sized energy unit.

What is a photon's energy in terms of wavelength?

E_{photon} = h c / λ

, and

h c = 1240

eV·nm for quick conversion.

Connections

Bohr model of the atom — supplies the quantization postulate $m v r = n ℏ$ .
Coulomb's law, Centripetal force — the two forces set equal in Step 1.
Ionization energy, Quantum numbers — meaning of the well depth and of $n$ .
Photon energy E = hf = hc/λ, Hydrogen spectral series, Rydberg constant and formula — where the levels lead next.

Foundations — Hydrogen energy levels Eₙ = −13.6 - n² eV

0 · The cast of characters (the atom itself)

1 · Charge $e$ , distance $r$ , and the electric pull $F$

2 · Speed $v$

3 · Mass $m$ and the "turning" force

4 · Kinetic and potential energy

5 · The whole numbers $n$ (the shelves)

6 · Putting two rules together — the Bohr radius $a_{0}$

7 · From radius to the energy formula $E_{n}$

8 · Light: the speed $c$ , the photon, and $λ$

How these foundations feed the topic

Equipment checklist

Connections