Physics • Year 12 • Module 8 • Lesson 13
Quantum Mechanics and the Atom
Lock in the core vocabulary, the de Broglie wavelength formula, quantum numbers, and the key principles of the quantum mechanical model before tackling harder questions.
1. Term–definition match
The definitions below are shuffled. In the right-hand column write the matching term from this list: de Broglie wavelength, wave function (ψ), probability density, Heisenberg uncertainty principle, quantum number, Pauli exclusion principle, orbital, standing wave, quantisation, Schrödinger equation. 10 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The wavelength associated with a moving particle, given by λ = h/p, where h is Planck’s constant and p is momentum. | |
| 1.2 | A mathematical description of the quantum state of a particle; its symbol is ψ. | |
| 1.3 | The quantity |ψ|², which gives the probability per unit volume of finding a particle at a given location. | |
| 1.4 | The fundamental limit Δx Δp ≥ ℏ/2; position and momentum cannot both be known with arbitrary precision simultaneously. | |
| 1.5 | A discrete integer or half-integer value (n, l, ml, or ms) that specifies a particular allowed state of an electron in an atom. | |
| 1.6 | The rule stating that no two electrons in an atom can share the same set of four quantum numbers. | |
| 1.7 | A three-dimensional probability cloud that describes where an electron is likely to be found in an atom; replaces the Bohr orbit concept. | |
| 1.8 | A wave pattern that fits exactly into a confined region, with nodes at fixed positions; explains why only certain electron orbits are allowed. | |
| 1.9 | The restriction of a physical quantity (such as energy or angular momentum) to discrete, allowed values rather than a continuous range. | |
| 1.10 | The wave equation &Ĥψ = Eψ whose solutions give the allowed energy levels and probability distributions for electrons in atoms. |
2. True or false — with correction
Circle T or F for each statement. If the statement is false, write the corrected version on the line below it. 12 marks (1 T/F + 1 correction each)
2.1 The de Broglie wavelength of a particle increases when its momentum increases. T / F
2.2 In the quantum mechanical model, electrons travel in well-defined circular orbits around the nucleus. T / F
2.3 The Davisson–Germer experiment confirmed matter waves by demonstrating electron diffraction. T / F
2.4 The angular momentum quantum number l can take any integer value from 0 to n. T / F
2.5 The Heisenberg uncertainty principle is a limitation of our current measurement technology, not a fundamental law of nature. T / F
2.6 The n = 2 shell of hydrogen can hold a maximum of 8 electrons. T / F
3. Fill-in-the-blank paragraph
Use the word bank to complete the passage. Each word is used once. 8 marks (1 per blank)
Word bank:
circumference · integer · momentum · orbitals · Pauli · probability · quantisation · standing wave
Louis de Broglie proposed that the wavelength of a particle is inversely proportional to its ___________, according to the relation λ = h/p. For an electron in a Bohr orbit, the wave must form a ___________ around the nucleus, meaning the ___________ of the orbit must contain an ___________ number of wavelengths. This condition naturally produces the ___________ of angular momentum that Bohr had to postulate. In the quantum mechanical model, electrons do not orbit at fixed radii; instead, they occupy ___________ described by probability clouds. The square of the wave function gives the ___________ density of finding the electron at any point. The ___________ exclusion principle limits the number of electrons in each subshell by requiring unique sets of quantum numbers.
4. Function recall
Answer each question in 1–2 sentences using precise terms from the lesson. 8 marks (2 each)
4.1 What is the physical significance of the quantity |ψ|² in quantum mechanics?
4.2 Explain how the de Broglie standing wave condition accounts for Bohr’s angular momentum quantisation rule (L = nℏ).
4.3 State what each of the four quantum numbers (n, l, ml, ms) specifies about an electron’s state.
4.4 Why does the Heisenberg uncertainty principle represent a fundamental limit rather than a technical measurement limitation?
5. Build a concept map
Draw labelled arrows between the six terms below to show how they connect. Each arrow must carry a linking phrase (e.g. “gives rise to”, “limits knowledge of”, “derived from”). Aim for at least 6 labelled arrows. 6 marks (1 per valid labelled arrow)
Supplied terms: de Broglie wavelength · standing wave · quantisation · wave function · Pauli exclusion principle · periodic table.
6. Complete the quantum number table
Fill in the missing values and descriptions for the four quantum numbers of the hydrogen atom. 10 marks (1 per cell marked *)
| Symbol | Name | Allowed values | What it specifies |
|---|---|---|---|
| n | * | 1, 2, 3, … | * |
| l | Angular momentum quantum number | * | * |
| ml | * | −l, −l+1, …, +l | * |
| ms | * | ± ½ | * |
Q1 — Term–definition match
1.1 de Broglie wavelength • 1.2 wave function (ψ) • 1.3 probability density • 1.4 Heisenberg uncertainty principle • 1.5 quantum number • 1.6 Pauli exclusion principle • 1.7 orbital • 1.8 standing wave • 1.9 quantisation • 1.10 Schrödinger equation.
Q2 — True / false with correction
2.1 False. The de Broglie wavelength decreases when momentum increases, because λ = h/p — λ is inversely proportional to p.
2.2 False. In the quantum mechanical model, electrons occupy probability clouds called orbitals, not well-defined circular orbits. There is no definite trajectory.
2.3 True.
2.4 False. The angular momentum quantum number l ranges from 0 to n − 1, not n. For n = 3, l can be 0, 1, or 2.
2.5 False. The Heisenberg uncertainty principle (Δx Δp ≥ ℏ/2) is a fundamental law of nature arising from the wave nature of matter — it is not a limitation of instruments and cannot be overcome with better technology.
2.6 True. The n = 2 shell contains the 2s subshell (l = 0, 2 electrons) and the 2p subshell (l = 1, 6 electrons) giving a maximum of 8 electrons.
Q3 — Cloze paragraph
In order: momentum / standing wave / circumference / integer / quantisation / orbitals / probability / Pauli.
Q4.1 — Physical significance of |ψ|²
|ψ|² is the probability density — it gives the probability per unit volume of finding the electron at a given position in space. It is always a positive real number. Integrating it over all space gives a total probability of 1.
Q4.2 — Standing wave and Bohr quantisation
For a stable electron wave, the circumference 2πr must equal an integer multiple of the de Broglie wavelength: 2πr = nλ = nh/(mv). Rearranging: mvr = nh/(2π) = nℏ, which is exactly Bohr’s angular momentum quantisation condition L = nℏ. Quantisation emerges naturally from the wave condition rather than being imposed.
Q4.3 — What each quantum number specifies
n (principal): energy level and average distance from nucleus. l (angular momentum): shape of the orbital (s, p, d, f). ml (magnetic): orientation of the orbital in space. ms (spin): intrinsic spin of the electron (± ½).
Q4.4 — Why the uncertainty principle is fundamental
The uncertainty principle arises from the wave nature of matter: measuring position precisely requires a short wavelength (high momentum uncertainty), and measuring momentum precisely requires a well-defined wavelength (spread-out position). This trade-off is built into the mathematics of waves — it is not a deficiency of instruments but an intrinsic property of quantum objects.
Q5 — Sample concept map
Correct maps should include arrows such as:
- de Broglie wavelength — forms → standing wave
- standing wave — gives rise to → quantisation
- wave function — squared gives → probability density (orbital shape)
- wave function — solutions give → quantisation
- Pauli exclusion principle — explains structure of → periodic table
- quantisation — limits electron states in → periodic table
Award 1 mark per valid labelled arrow (minimum 6, maximum 6 marked).
Q6 — Quantum number table
n: Name — Principal quantum number; Specifies — energy level and average distance from nucleus. l: Allowed values — 0, 1, 2, …, n−1; Specifies — shape (orbital type: s, p, d, f). ml: Name — Magnetic quantum number; Specifies — orientation of orbital in space. ms: Name — Spin quantum number; Specifies — intrinsic spin of the electron (± ½).