understand that there are discrete electron energy levels in isolated atoms (e.g. atomic hydrogen)

Energy Levels in Atoms and Line Spectra

Learning Objective

Explain that an isolated atom (e.g. hydrogen) can occupy only certain, discrete electron energy levels and that transitions between these levels produce photons of characteristic wavelengths, giving rise to line spectra.

Syllabus Alignment (Cambridge International AS & A Level Physics 9702)

Syllabus Requirement	Coverage in These Notes	Action Required
1 – 11 (AS core topics)	Only sub‑topic 22.4 (Energy levels & line spectra) is covered.	Create a separate set of lecture notes for the remaining AS topics (e.g. kinematics, dynamics, waves, electricity). Include a mapping table linking each syllabus number to its dedicated notes.
12 – 25 (A‑level extensions)	No A‑level material is present.	Add a “Further Reading / Next Steps” section (see below) that points to the next units: 12 Motion in a circle, 13 Gravitational fields, 22 Quantum physics, 23 Nuclear physics, etc.
Depth & Accuracy (AO1)	Correct Bohr model, energy formulae, Rydberg equation and a worked example are provided, but a few technical slips exist.	Correct the Bohr‑radius expression, state explicitly that the 13.6 eV constant applies to hydrogen‑like (single‑electron) atoms, and add a footnote for the Rydberg constant.

Key Concepts

• Quantisation of electron energy in atoms.
• Bohr model of the hydrogen atom (historical model).
• Energy of a photon: hf = E_photon = hc/λ.
• Energy change for an electronic transition: hf = E_i – E_f.
• Rydberg formula for hydrogen spectral lines.
• Hydrogen spectral series (Lyman, Balmer, Paschen, Brackett, Pfund).
• Quantum numbers and selection rules (brief overview).
• Limitations of the Bohr model and the need for full quantum‑mechanical treatment.

Bohr Model of the Hydrogen Atom

The Bohr model was the first successful attempt to explain why atoms emit discrete spectra. It treats the electron as moving in a circular orbit around the nucleus, but only certain orbits are allowed.

Quantised Angular Momentum

\[ L = m_e v r = n\hbar ,\qquad n = 1,2,3,\dots \]

Force Balance (Centripetal = Coulomb)

\[ \frac{m_e v^{2}}{r}= \frac{1}{4\pi\varepsilon_0}\frac{e^{2}}{r^{2}} \]

Allowed Radii

Eliminating the speed \(v\) using the angular‑momentum condition gives the Bohr radius and its higher‑order equivalents:

\[ r_n = \frac{n^{2}\hbar^{2}}{k m_e e^{2}} = n^{2}a_0, \qquad a_0 = \frac{4\pi\varepsilon_0\hbar^{2}}{m_e e^{2}} \approx 5.29\times10^{-11}\,\text{m}, \] where \(k = \dfrac{1}{4\pi\varepsilon_0}\).

Energy of the n‑th Orbit

The total energy (kinetic + potential) of an electron in the \(n\)th orbit is

\[ E_n = -\frac{k e^{2}}{2r_n} = -\frac{k^{2} m_e e^{4}}{2\hbar^{2}}\frac{1}{n^{2}} = -\frac{13.6\ \text{eV}}{n^{2}} . \]

**Note:** The 13.6 eV value is valid for hydrogen‑like (single‑electron) atoms only.

Limitations of the Bohr Model (Box)

Photon Emission and Absorption

When an electron moves between two allowed levels, a photon is emitted (if the electron drops) or absorbed (if it is raised). The photon energy is given exactly by

\[ hf = E_i - E_f , \]

where \(E_i\) (higher) and \(E_f\) (lower) are the energies of the initial and final states. Substituting the Bohr‑hydrogen expression for \(E_n\) yields

\[ hf = 13.6\ \text{eV}\left(\frac{1}{n_f^{2}}-\frac{1}{n_i^{2}}\right). \]

Rydberg Formula

In terms of wavelength the relationship becomes

\[ \frac{1}{\lambda}=R_{\infty}\!\left(\frac{1}{n_f^{2}}-\frac{1}{n_i^{2}}\right), \qquad R_{\infty}=1.097\,\times10^{7}\ \text{m}^{-1}\;^†. \]

Hydrogen Spectral Series

Series	Final level \(n_f\)	Typical region of the spectrum
Lyman	1	Ultraviolet
Balmer	2	Visible
Paschen	3	Infrared
Brackett	4	Infrared
Pfund	5	Infrared

Worked Example – Balmer α Line

1. Transition: \(n_i = 3 \rightarrow n_f = 2\).
2. Apply the Rydberg formula: \[ \frac{1}{\lambda}=R_{\infty}\!\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right). \]
3. Calculate: \[ \frac{1}{\lambda}=1.097\times10^{7}\left(\frac{1}{4}-\frac{1}{9}\right) =1.097\times10^{7}\left(\frac{5}{36}\right) =1.523\times10^{6}\ \text{m}^{-1}. \] \[ \lambda = \frac{1}{1.523\times10^{6}} \approx 6.57\times10^{-7}\,\text{m}=657\ \text{nm}. \]
4. Result: a visible red line (Balmer α) at 657 nm.

Quantum Numbers & Selection Rules (A‑level Extension)

• Principal quantum number \(n\): determines the energy level (as above).
• Azimuthal quantum number \(\ell\): \(0 \le \ell \le n-1\); defines the orbital shape.
• Magnetic quantum number \(m_\ell\): \(-\ell \le m_\ell \le +\ell\); defines orientation.
• Spin quantum number \(m_s\): \(\pm \tfrac{1}{2}\).
• Selection rules for electric‑dipole transitions:
  • \(\Delta n\) unrestricted (any change).
  • \(\Delta \ell = \pm 1\).
  • \(\Delta m_\ell = 0, \pm 1\).

Multi‑Electron Atoms (A‑level Extension)

In atoms with more than one electron:

• Electron‑electron repulsion splits the energy levels into sub‑levels (fine structure).
• Screening/shielding reduces the effective nuclear charge, so the simple \(-13.6\,\text{eV}/n^{2}\) formula no longer holds.
• Spectra become more complex; line positions are described by quantum‑mechanical calculations or empirical term‑value formulas.

Further Reading / Next Steps

These notes form a self‑contained module on discrete energy levels. To complete the Cambridge syllabus, plan the following follow‑up sessions:

1. Motion in a circle, gravitation, and orbital dynamics (Units 12 & 13).
2. Thermodynamics and kinetic theory (Units 14 & 15).
3. Full quantum physics: Schrödinger equation, quantum numbers, and probability densities (Unit 22).
4. Nuclear physics, medical applications, and modern instrumentation (Units 23 & 24).
5. Practical spectroscopy: using diffraction gratings, calibration, and data analysis.

Common Misconceptions

• Electrons spiral into the nucleus. – Incorrect. Quantised orbits prevent continuous energy loss.
• All atoms emit the same spectral lines. – Incorrect. Each element has a unique set of energy levels.
• Energy levels are continuous. – Incorrect. Observed line spectra prove that the levels are discrete.
• The Bohr model is exact for all atoms. – Incorrect. It works only for hydrogen‑like atoms; multi‑electron systems require quantum‑mechanical treatment.

Summary

1. Isolated atoms possess quantised electron energy levels.
2. For a transition \(n_i \rightarrow n_f\), the photon energy is \(hf = E_{n_i} - E_{n_f}\).
3. The Rydberg formula accurately predicts the wavelengths of hydrogen’s line spectra.
4. Observation of line spectra provides direct experimental evidence for the quantum nature of atomic structure.
5. Beyond hydrogen, quantum numbers, selection rules, and electron‑electron interactions must be considered.

References

• Cambridge International AS & A Level Physics (9702) – Specification, 2023.
• J. J. Thompson, *Modern Physics for Scientists and Engineers*, 5th ed., 2020.
• R. A. Fleming, *Spectroscopy: The Key to the Stars*, 2018.

^†Rydberg constant \(R_{\infty}\) is defined for an infinitely massive nucleus; for real hydrogen the value is slightly lower (\(R_H = 1.09678\times10^{7}\ \text{m}^{-1}\)).