# The Quantum Atom

> Bohr's energy levels, the hydrogen spectrum and Rydberg's equation, de Broglie's matter waves and Schrödinger's electron clouds.

From HSC Physics on WhizzNotes, free study notes for NSW students: https://whizznotes.com/physics/quantum-atom/. Updated 2 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The barcode of hydrogen
Pass electricity through a tube of hydrogen gas and it glows pink. Look at that glow through a prism and you do not see a rainbow. You see just four sharp coloured lines: red, blue-green, blue and violet, always at exactly the same wavelengths.

In 1885, a Swiss school teacher called Johann Balmer found a simple formula that gave all four wavelengths. He had no idea why it worked. It took almost 30 years before Niels Bohr explained it: electrons in atoms can only have certain energies, like standing only on the rungs of a ladder, never in between.

**In short:** In the quantum model of the atom, electrons can only have certain energy levels. An electron gives out or takes in a photon only when it jumps between levels, with $hf = E_i - E_f$, which is why each element has its own line spectrum.

## Bohr's model of the hydrogen atom

In 1913, Niels Bohr kept Rutherford's nucleus but added new rules, called **postulates**:

1. Electrons move in certain allowed orbits, called **stationary states**, without giving off radiation. Each has a fixed energy.
2. An electron gives off or absorbs a **photon** only when it jumps between allowed orbits. The photon's energy equals the difference in energy:

$$E = hf = E_i - E_f$$

3. Allowed orbits have angular momentum that is a whole number multiple of $\dfrac{h}{2\pi}$:

$$mvr = \frac{nh}{2\pi}$$

The whole number $n$ is the **principal quantum number**. $n = 1$ is the **ground state**, the lowest energy.

## Energy levels of hydrogen

For hydrogen, the allowed energies are:

$$E_n = -\frac{13.6}{n^2} \text{ eV}$$

| $n$ | Energy |
|---|---|
| 1 | −13.6 eV |
| 2 | −3.40 eV |
| 3 | −1.51 eV |
| 4 | −0.85 eV |
| ∞ | 0 (the electron is free) |

The energies are **negative** because the electron is bound to the nucleus. To free an electron from the ground state (**ionise** the atom) takes 13.6 eV.

### Worked example: Red light from hydrogen
An electron drops from $n = 3$ to $n = 2$.

$E = -1.51 - (-3.40) = 1.89$ eV $= 3.03 \times 10^{-19}$ J

$\lambda = \dfrac{hc}{E} = \dfrac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{3.03 \times 10^{-19}} = 656$ nm, which is red.

## Emission and absorption

- **Emission**: an electron drops to a **lower** level and gives out a photon. A bigger drop means a higher-energy photon with a shorter wavelength.
- **Absorption**: an atom absorbs a photon only if its energy **exactly matches** a jump up. This makes the dark lines in absorption spectra.

## The Rydberg equation

The wavelengths of hydrogen's lines are given by:

$$\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

- $R = 1.097 \times 10^7$ m⁻¹ (the Rydberg constant)
- $n_f$ = final level, $n_i$ = initial level

Lines are grouped into **series** by where the electron lands:

| Series | Ends on | Type of light |
|---|---|---|
| Lyman | $n_f = 1$ | ultraviolet |
| **Balmer** | $n_f = 2$ | **visible** |
| Paschen | $n_f = 3$ | infrared |

## Limits of Bohr's model

Bohr's model worked brilliantly for hydrogen, but:

- It only worked for atoms with **one electron**.
- It could not explain why some lines are **brighter** than others.
- It could not explain the fine splitting of lines into close pairs, or their splitting in a magnetic field.
- It mixed old physics (orbits) with new rules, without explaining **why** the orbits are allowed.

## de Broglie: electrons are waves too

In 1924, Louis de Broglie suggested that if light waves can act like particles, then particles might act like waves. Any moving particle has a wavelength:

$$\lambda = \frac{h}{mv}$$

where $mv$ is the particle's momentum.

- For everyday objects, $\lambda$ is far too small to notice.
- For electrons, $\lambda$ is about the size of the gaps between atoms in a crystal.

In 1927, Davisson and Germer fired electrons at a nickel crystal and saw a **diffraction pattern**, just like X-rays. Electrons really do behave as waves.

### Why Bohr's orbits are allowed

de Broglie explained Bohr's rule. An electron's wave must fit around its orbit as a **standing wave**, a whole number of wavelengths:

$$2\pi r = n\lambda$$

Any other orbit would make the wave cancel itself out. Substituting $\lambda = \dfrac{h}{mv}$ gives exactly Bohr's rule $mvr = \dfrac{nh}{2\pi}$.

**Mind map**

```text
The Quantum Atom
  Bohr's postulates
    Stationary states
    Photon: E = Ei − Ef
    mvr = nh / 2π
  Hydrogen levels
    En = −13.6 / n² eV
    Ground state n = 1
    Ionisation 13.6 eV
  Spectra
    Emission: drop down
    Absorption: exact jump up
    Rydberg equation
    Balmer series visible
  de Broglie
    λ = h / mv
    Electron diffraction
    Standing waves explain orbits
  Schrödinger
    Electron clouds
    Probability
```

### Year 11: Particles that are too big to wave
A 0.15 kg cricket ball at 30 m/s has $\lambda = \dfrac{6.626 \times 10^{-34}}{0.15 \times 30} = 1.5 \times 10^{-34}$ m. That is far smaller than an atom's nucleus, so we never see cricket balls diffract.

An electron at $2.0 \times 10^6$ m/s has $\lambda = \dfrac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 2.0 \times 10^6} = 3.6 \times 10^{-10}$ m, about the size of an atom. That is why crystals can diffract electrons.

### Year 12: Schrödinger's model
Erwin Schrödinger (1926) wrote an equation for the electron's wave in three dimensions. Its solutions do not give neat orbits. Instead, they give regions called **orbitals**, where the electron is **likely** to be found. The electron is described by a **probability cloud**, not a path.

This model works for atoms with **many electrons**, explains the brightness of spectral lines and underpins all of chemistry. It removed the certainty about where an electron is: we can only say where it probably is.

**Series limit.** The shortest wavelength in a series comes from an electron falling from $n_i = \infty$. For the Balmer series: $\dfrac{1}{\lambda} = R\left(\dfrac{1}{4} - 0\right)$, so $\lambda = 365$ nm. Lines crowd together near this limit because the energy levels get closer together as $n$ grows.

**Which transition?** The bigger the energy gap, the shorter the wavelength. In hydrogen, jumps to $n = 1$ have the biggest gaps. The **longest** Lyman wavelength is the smallest jump, $n = 2 \to 1$: $\frac{1}{\lambda} = R\left(1 - \frac{1}{4}\right)$, so $\lambda = 122$ nm.

**Matching an absorbed wavelength.** Work out the photon energy, then find the pair of levels with exactly that gap. Light at 103 nm carries 12.1 eV, the gap from $n = 1$ ($-13.6$ eV) to $n = 3$ ($-1.51$ eV). A 12.5 eV photon matches no gap from the ground state (12.09 eV or 12.75 eV are the nearest), so it is not absorbed and passes straight through.

**Ionising from a higher level** takes only the energy to reach zero: from $n = 2$, 3.40 eV.

**How many lines?** Electrons falling from level $n$ to the ground state by every route give $\dfrac{n(n - 1)}{2}$ different lines: 6 from $n = 4$.

**de Broglie wavelength after acceleration.** $qV = \frac{1}{2}mv^2 = \dfrac{p^2}{2m}$, so $\lambda = \dfrac{h}{\sqrt{2mqV}}$. An electron through 150 V has $\lambda \approx 1.0 \times 10^{-10}$ m. Going the other way, the kinetic energy for a wavelength $\lambda$ is $K = \dfrac{h^2}{2m\lambda^2}$.

**Equal wavelengths** mean equal momentum, $mv$. An alpha particle (4 times a proton's mass) with the same wavelength moves at a quarter of the proton's speed. On a graph of speed against $\frac{1}{\lambda}$, the gradient is $\dfrac{h}{m}$, which identifies the particle.

**Orbits and standing waves.** The $n$th orbit holds exactly $n$ wavelengths. Since $2\pi r = n\lambda$ and $\lambda = \dfrac{h}{mv}$, the ground-state radius is $r = \dfrac{h}{2\pi mv}$, about $5.3 \times 10^{-11}$ m.

## Quick summary

- Bohr: electrons sit in fixed energy levels and only emit or absorb photons when they jump: $hf = E_i - E_f$.
- Hydrogen levels: $E_n = -\dfrac{13.6}{n^2}$ eV.
- Rydberg: $\dfrac{1}{\lambda} = R\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$. The Balmer series ($n_f = 2$) is visible.
- Bohr's model only worked for hydrogen-like atoms.
- de Broglie: particles have a wavelength $\lambda = \dfrac{h}{mv}$. Electron diffraction confirmed it.
- Allowed orbits fit a whole number of electron wavelengths. Schrödinger replaced orbits with probability clouds.

## Quiz

Try these questions from the chapter quiz.

1. In Bohr's model, an electron gives out light when it:
   - A. Stays in the same orbit
   - B. Drops to a lower energy level
   - C. Moves to a higher energy level
   - D. Leaves the nucleus
   - Answer: B. Drops to a lower energy level. Dropping down releases the energy difference as a photon.

2. The lowest energy level of an atom is called the:
   - A. Ground state
   - B. Excited state
   - C. Ionisation level
   - D. Nucleus
   - Answer: A. Ground state. n = 1 is the ground state.

3. The energy of the photon given out when an electron drops between levels equals:
   - A. The energy of the higher level
   - B. The difference in energy between the two levels
   - C. 13.6 eV every time
   - D. The sum of the two levels
   - Answer: B. The difference in energy between the two levels. hf = E initial − E final.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/quantum-atom/#quiz