Physics • Year 12 • Module 8 • Lesson 13
Quantum Mechanics and the Atom
Build HSC Band 5–6 extended-response technique on wave–particle duality, the uncertainty principle, quantum number constraints, and evaluating models of the atom.
1. Multi-step calculation — electron diffraction at the Australian Synchrotron (Band 5–6)
8 marks Band 5–6
Electrons at the Australian Synchrotron in Clayton, Victoria can be accelerated through large potential differences to study crystal structure by diffraction — analogous to X-ray diffraction. In one experiment, electrons are accelerated through a potential difference of 150 V.
Given: e = 1.60×10−19 C, me = 9.11×10−31 kg, h = 6.63×10−34 J·s.
(a) Calculate the kinetic energy (in joules) gained by an electron accelerated through 150 V. (2 marks)
(b) Hence calculate the momentum and de Broglie wavelength of these electrons. (3 marks)
(c) The spacing between crystal planes in a typical material is approximately 2.0×10−10 m. Explain, with reference to your answer in (b), why these electrons are well-suited for crystal diffraction studies. (1 mark)
(d) State one assumption made in part (a) and identify one limitation of using classical kinetic energy to describe very high-energy electrons. (2 marks)
2. Data + scenario — evaluating models of the hydrogen atom (Band 5–6)
8 marks Band 5–6
Scenario. In 1913 Niels Bohr proposed that electrons orbit the hydrogen nucleus in fixed circular paths with quantised angular momentum L = nℏ. His model correctly predicted the hydrogen emission spectrum. However, it failed for multi-electron atoms and could not explain the fine structure of spectral lines. In 1926 Erwin Schrödinger published a wave equation whose solutions gave the same hydrogen energy levels but described electrons as probability clouds (orbitals), not orbits. The table below compares predictions from the two models.
| Criterion | Bohr model | Quantum model (Schrödinger) |
|---|---|---|
| Hydrogen energy levels | En = −13.6/n² eV ✓ | En = −13.6/n² eV ✓ |
| Multi-electron atoms | Fails — no electron–electron repulsion ✗ | Accounts for electron–electron repulsion via quantum numbers ✓ |
| Spectral fine structure | Cannot explain ✗ | Explained by l and ml splitting ✓ |
| Electron position | Defined circular orbit ✗ | Probability distribution (|ψ|²) ✓ |
| Uncertainty principle | Violates ΔxΔp ≥ ℏ/2 ✗ | Consistent ✓ |
Based on Bohr (1913) and Schrödinger (1926). ✓ = model consistent with observation; ✗ = model inconsistent or silent.
Q2. Analyse and evaluate the two models of the hydrogen atom using the data above. In your response you must:
- Identify at least two specific criteria on which the Bohr model succeeds and explain why it was initially accepted.
- Identify and explain at least two specific failures of the Bohr model, with reference to the data in the table.
- Explain how de Broglie’s wave hypothesis provides the physical justification for Bohr’s quantisation postulate.
- Explain how the quantum model resolves the failures of Bohr’s model, with specific reference to quantum numbers and the uncertainty principle.
- Reach an evidence-based judgement about whether the Bohr model should still be taught in schools, acknowledging its successes and limitations.
Q1 — Multi-step calculation (8 marks)
(a) Kinetic energy [2 marks]: KE = eV = 1.60×10−19 C × 150 V = 2.40×10−17 J [1]. Units: joules (1 eV = 1.60×10−19 J; 150 eV = 2.40×10−17 J) [1].
(b) Momentum and wavelength [3 marks]: p = √(2meKE) = √(2 × 9.11×10−31 × 2.40×10−17) = √(4.37×10−47) = 6.61×10−24 kg·m/s [1]. λ = h/p = 6.63×10−34 / 6.61×10−24 = 1.00×10−10 m = 0.100 nm [2].
(c) Suitability for diffraction [1 mark]: For diffraction to occur, the wavelength of the wave must be comparable to the spacing between scattering centres. Since λ ≈ 1.00×10−10 m is of the same order of magnitude as the crystal plane spacing (2.0×10−10 m), strong diffraction effects will occur — the electrons are well-suited to crystal structure studies [1].
(d) Assumption and limitation [2 marks]: Assumption: The electron starts from rest, so all the electrical potential energy eV is converted to kinetic energy only (no energy lost to other processes) [1]. Limitation: The classical expression KE = p²/(2m) is valid only for speeds well below c. At very high potential differences, relativistic effects become significant: the rest mass effectively increases and classical momentum underestimates the true relativistic momentum [1].
Marking criteria (8 marks): 1 = KE = eV with correct substitution; 1 = correct numerical answer (2.40×10−17 J); 1 = correct momentum formula and substitution; 1 = correct momentum (6.61×10−24 kg·m/s); 1 = correct wavelength (1.00×10−10 m); 1 = correct comparison of λ to crystal spacing with explanation; 1 = valid assumption (electron starts from rest / all energy to KE); 1 = valid relativistic limitation.
Q2 — Sample Band 6 response (8 marks), annotated
Bohr successes: The Bohr model correctly predicts the hydrogen emission spectrum, with En = −13.6/n² eV matching observed spectral lines to high precision [1 — criterion 1 from table]. Its simplicity made it immediately useful: it introduced the concept of quantised energy levels, which became a foundation for modern quantum physics. The Bohr model was accepted because it successfully explained why hydrogen only emits light at specific wavelengths — a mystery that classical physics could not resolve [1 — criterion 2 and historical reasoning].
Bohr failures: The Bohr model fails for multi-electron atoms because it has no mechanism to account for electron–electron repulsion; it treats each electron as though it experiences only the nuclear charge [1 — failure 1 from table]. It also predicts definite electron trajectories, directly violating the Heisenberg uncertainty principle (Δx Δp ≥ ℏ/2): if the electron travels in a precise orbit, both its position and momentum are simultaneously well-defined, which is fundamentally impossible [1 — failure 2 from table, with principle cited].
De Broglie link: De Broglie proposed that the electron has a wavelength λ = h/(mv). For a stable orbit, this wave must close on itself as a standing wave: 2πr = nλ = nh/(mv). Rearranging: mvr = nℏ = L — exactly Bohr’s quantisation postulate. This gives physical justification for the postulate: quantised orbits exist because only integer-wavelength standing waves are stable; non-integer configurations destructively interfere and cannot persist [1 — de Broglie derivation of Bohr postulate].
Quantum model resolution: The Schrödinger equation introduces four quantum numbers (n, l, ml, ms) that fully characterise each electron state. In multi-electron atoms, the Pauli exclusion principle (no two electrons share the same four quantum numbers) combined with electron–electron repulsion terms in the Hamiltonian account for atomic structure correctly. The orbital picture (|ψ|² as probability density) resolves the trajectory paradox: the electron has no defined path, only a probability distribution consistent with Δx Δp ≥ ℏ/2 [1 — quantum numbers; 1 — uncertainty principle resolution].
Evaluative judgement: The Bohr model should still be taught in schools as a conceptual stepping stone: it introduces quantisation and energy levels in a physically intuitive way, and it gives correct results for hydrogen. However, students must understand it is a simplified model that violates the uncertainty principle and fails for multi-electron atoms. The quantum model is superior on every criterion except conceptual simplicity. Teaching both models, in sequence, with explicit discussion of their limitations, provides the most accurate view of how scientific models develop and improve [1 — explicit evidence-based judgement with nuance].
Marking criteria (8 marks): 1 = two Bohr successes with explanation (H energy levels; initial acceptance); 1 = two Bohr failures with table reference (multi-electron; violates HUP); 1 = de Broglie standing wave derivation of Bohr’s L = nℏ quantisation; 1 = quantum model uses quantum numbers to fix multi-electron problem; 1 = quantum model uses |ψ|² and HUP to fix trajectory problem; 1 = explicit evidence-based judgement integrating both models; 1 = precise terminology (orbital, wave function, uncertainty principle, probability density, quantum numbers); 1 = logical, cohesive argument structure throughout.