Rydberg formula 1 - λ = R(1 - n₁² − 1 - n₂²): Foundations

This page assumes nothing. Before you can read the parent Rydberg note, you must own every symbol it throws at you. We will meet each one, give it a plain-words meaning, draw the picture it lives in, and say why the topic needs it. Each block builds on the one before.

1. The atom as a staircase of energy levels

Picture the hydrogen atom as a single electron circling a single proton. The electron is not free to sit anywhere — it can only occupy certain fixed orbits, like being allowed to stand only on specific steps of a staircase, never between them.

Figure — Rydberg formula 1 - λ = R(1 - n₁² − 1 - n₂²)

The picture: look at the figure — the steps are drawn crowding together as you go up. This crowding is the whole personality of hydrogen, and it comes from a $1/ n^{2}$ pattern we meet below.

Why the topic needs it: the Rydberg formula's inputs $n_{1}$ and $n_{2}$ are just "which step you land on" and "which step you started on." Without the staircase idea, those numbers mean nothing. (The full derivation of the steps lives in the Bohr Model of the Atom.)

2. The symbol $E_{n}$ — energy of a step, and why it is negative

Two strange things need explaining before you can trust this.

Why the minus sign? The electron is bound — trapped by the proton's pull. We agree that a free electron infinitely far away has energy $0$ . Anything more trapped than free must have less than zero, i.e. negative energy. The deeper the trap, the more negative.

Why $1/ n^{2}$ and not just $1/ n$ ? Look at the figure again: the number $13.6/ n^{2}$ gives $- 13.6, - 3.40, - 1.51, - 0.85, \dots$ for $n = 1, 2, 3, 4$ . The gaps between neighbours shrink fast. That squeezing is exactly the "steps crowd together" picture — and it is why the coloured lines of hydrogen crowd together too.

Why the topic needs it: the colour of emitted light comes from the difference between two of these $E_{n}$ values. No energy levels, no formula.

3. The jump — $Δ E$ and the arrow between steps

The picture: the red arrow in the figure shows the electron falling from step $n_{2}$ (high) to step $n_{1}$ (low). It sheds exactly the height of that arrow's worth of energy.

Why $n_{2} > n_{1}$ always? By agreement, $n_{1}$ is the lower step (where you land), $n_{2}$ the higher step (where you start). Keeping $n_{2} > n_{1}$ is what makes the released energy come out positive — a real, outgoing flash of light. Swap them and you'd get a negative "wavelength," which is nonsense. (This same fall/rise picture powers the whole Hydrogen Spectrum and Spectral Series.)

4. The packet of light — photon, $ν$ , $c$ , and $h$

The released energy does not dribble out. It leaves as one indivisible packet of light called a photon. To describe that packet we need four symbols.

The picture: a wave. Its wavelength $λ$ is the crest-to-crest distance (marked in the figure). Short $λ$ = tightly packed wobbles = high frequency = high energy (bluer/UV). Long $λ$ = stretched wobbles = low energy (redder/IR).

Why the topic needs it: the Rydberg formula is written in $λ$ , not energy. This equation is the bridge that swaps "energy released" for "wavelength of light." (Its own page: Energy of a Photon E = hc over lambda.)

5. Wavenumber $\frac{1}{λ}$ — why flip the wavelength?

Why flip it? From $E_{photon} = h c / λ$ we can write $\frac{1}{λ} = \frac{E _{photon}}{h c}$ . So $1/ λ$ is directly proportional to energy — double the energy gap, double the wavenumber. This makes $1/ λ$ the natural quantity for the formula: the energy-difference bracket maps straight onto it, no reciprocals hiding.

6. The Rydberg constant $R_{H}$ — bundling the physics into one number

When Bohr's derivation is finished, an ugly clump of constants appears in front of the $n$ -terms. We name that whole clump $R_{H}$ .

Why the topic needs it: without bundling, the formula would be unreadable. $R_{H}$ hides all the machinery so the equation reads cleanly as "constant × (step term − step term)." (More on how $Z$ changes it: Hydrogen-like Ions and Z-scaling.)

7. Reduced mass $μ$ — why not just the electron's mass?

You'd expect the electron's mass $m_{e}$ to appear in $R_{H}$ . It nearly does — but there's a subtle correction.

Since the proton is about 1836 times heavier than the electron, $μ \approx 0.99946 m_{e}$ — almost, but not quite, $m_{e}$ . This tiny nudge makes the measured $R_{H}$ slightly smaller than the idealised $R_{\infty}$ (the value you'd get pretending the proton were infinitely heavy).

Why the topic needs it: it's the difference between $R_{H}$ (real hydrogen) and $R_{\infty}$ (the textbook idealisation). Getting this right is Bohr's precision triumph. (Full treatment: Reduced Mass in Two-Body Systems.)

The prerequisite map

Here is how every foundation above feeds into the Rydberg formula.

The two threads — energy of levels (left) and light of photons (right) — meet at the wavenumber, and the whole thing is scaled by $R_{H}$ . That meeting point is the Rydberg formula. It links onward to Quantisation of Angular Momentum, Ionisation Energy (the $n \to \infty$ limit), and Hydrogen-like Ions and Z-scaling.

Equipment checklist

Test yourself — reveal only after you've answered aloud.

What does the quantum number

n

count, and which value is the lowest step?

n

labels which allowed energy step the electron sits on;

n = 1

is the lowest (ground state).

Why is every energy level

E_{n}

negative?

The electron is bound; a free electron at infinity is defined as

0

, so anything trapped is below zero.

Why does

E_{n}

go as

1/ n^{2}

and not

1/ n

It makes the steps crowd together as

n

grows, matching the crowding of hydrogen's spectral lines.

What does

Δ E

mean, and how is it built from

E_{n_{1}}

and

E_{n_{2}}

Δ

means "change in";

Δ E = E_{n_{2}} - E_{n_{1}}

is the energy gap between the two steps.

Why must

n_{2} > n_{1}

So the released energy (and the wavenumber and wavelength) stays positive;

n_{1}

is the lower step by convention.

What is a photon, and what four symbols describe it?

An indivisible packet of light; described by

c

(speed),

ν

(frequency),

λ

(wavelength),

h

(Planck's constant).

Write the bridge between photon energy and wavelength.

E_{photon} = h ν = h c / λ

, using

c = ν λ

What is the wavenumber, and why is it the natural variable?

1/ λ

, the wobbles per metre; it is directly proportional to energy, so it maps cleanly onto the energy-gap bracket.

What does

R_{H}

bundle together?

The clump

μ e^{4} / (8 ε_{0}^{2} h^{3} c)

— all the physical constants, so the formula reads simply.

Why use reduced mass

μ

instead of

m_{e}

The proton is not infinitely heavy; both particles orbit a shared centre, so

μ = m_{e} m_{p} / (m_{e} + m_{p})

is the effective mass.