HSC Exam Practice

Sample response. The de Broglie wavelength is the wavelength associated with a moving particle, arising from its wave-like nature. The formula is λ = h/p = h/(mv), where λ is the de Broglie wavelength (m), h is Planck’s constant (6.63×10⁻³⁴ J·s), p is the particle’s momentum (kg·m/s), m is its mass (kg), and v is its speed (m/s).

Marking notes. 1 mark for a correct description of the de Broglie wavelength as the wavelength associated with a moving particle. 1 mark for stating the formula λ = h/p (or equivalent). 1 mark for correctly identifying all three symbols (h, p or mv, and λ).

Sample response. In 1927, Clinton Davisson and Lester Germer fired electrons at a nickel crystal and observed a diffraction pattern — the same pattern produced by X-rays (waves) diffracting from a crystal lattice. Diffraction is a wave phenomenon: it only occurs when the wavelength of the incident wave is comparable to the spacing between scatterers. The observation that electrons produced a diffraction pattern therefore demonstrated that moving electrons have an associated wavelength, confirming that matter (electrons) has wave-like properties as de Broglie predicted.

Marking notes. 1 mark for identifying the evidence (diffraction pattern produced by electrons). 1 mark for explaining that diffraction is a wave phenomenon that requires the wavelength to be comparable to the obstacle spacing. 1 mark for concluding that this demonstrates the wave nature of electrons (matter waves).

Sample response. The wave function ψ is a mathematical function that completely describes the quantum state of a particle. It has no direct physical interpretation itself — it can be complex-valued. Its square, |ψ|², is the probability density: it gives the probability per unit volume of finding the electron at a specific point in space. Regions where |ψ|² is large are where the electron is most likely to be found; regions where |ψ|² = 0 are nodes where the electron cannot be found. The total probability over all space equals 1 (normalisation condition).

Marking notes. 1 mark for describing ψ as a mathematical description of the quantum state (not a physical wave). 1 mark for stating |ψ|² gives the probability density (probability per unit volume of finding the electron). 1 mark for a correct additional detail such as nodes, normalisation, or the relationship to orbital shape.

Sample response. (1) Trajectory: A Bohr orbit is a defined circular path with a precise radius; a quantum orbital is a probability cloud — the electron has no defined trajectory. (2) Certainty of position: In the Bohr model, the electron’s position is known at every instant; in the quantum model, only the probability of finding the electron at any point is defined (|ψ|²). (3) Physical basis: Bohr orbits were postulated without physical justification; quantum orbitals emerge as solutions to the Schrödinger equation and are consistent with the Heisenberg uncertainty principle, which the Bohr model violates.

Marking notes. 1 mark per clearly stated and correct distinction. Accept any three from the following valid criteria: trajectory (precise vs no trajectory), position certainty (exact vs probabilistic), basis for quantisation (postulated vs derived from Schrödinger), consistency with uncertainty principle, shape (circle vs 3D probability cloud), applicability to multi-electron atoms.

Sample response. (1) Principal quantum number n: n = 1, 2, 3, … — determines the energy level and average distance from the nucleus. (2) Angular momentum quantum number l: l = 0, 1, …, n−1 — determines the shape of the orbital (s, p, d, f). (3) Magnetic quantum number m_l: m_l = −l, …, 0, …, +l — determines the orientation of the orbital in space. (4) Spin quantum number m_s: m_s = ± ½ — describes the intrinsic spin of the electron.

Marking notes. 1 mark per quantum number correctly identified with its symbol, a valid range of allowed values, and what it specifies. Allow ½ mark per quantum number if only 2 of the 3 elements (symbol, values, meaning) are correct — but sum to whole marks only.

Sample response. The Bohr model treats the electron as experiencing only the Coulomb attraction from the nucleus, with no other interactions. For hydrogen (one electron), this is an excellent approximation and gives E_n = −13.6/n² eV, which matches observation. For multi-electron atoms, each electron also experiences repulsion from all other electrons. The Bohr model has no mechanism to account for electron–electron repulsion; it cannot calculate how each electron’s energy is modified by the presence of the others, so its predictions fail for helium and heavier atoms.

Marking notes. 1 mark for stating the Bohr model works for hydrogen because it has only one electron (no electron–electron repulsion). 1 mark for identifying that the Bohr model neglects electron–electron repulsion in multi-electron atoms. 1 mark for explaining that electron–electron repulsion modifies each electron’s energy in a way the Bohr model cannot account for.

Sample response (a). λ = h/(m_ev₁) = 6.63×10⁻³⁴ / (9.11×10⁻³¹ × 2.19×10⁶) = 6.63×10⁻³⁴ / 1.994×10⁻²⁴ = 3.32×10⁻¹⁰ m. [1 substitution correct, 1 answer correct]

Sample response (b). Circumference = 2πr₁ = 2π × 5.29×10⁻¹¹ = 3.32×10⁻¹⁰ m [1]. This equals the de Broglie wavelength calculated in (a) [1]. For n = 1, the standing wave condition requires 2πr = 1 × λ, i.e. exactly one wavelength fits the circumference, confirming the condition is satisfied [1].

Sample response (c). The standing wave condition 2πr = nλ = nh/(mv) rearranges to mvr = nh/(2π) = nℏ, giving L = nℏ [1]. This shows Bohr’s angular momentum quantisation is not an arbitrary postulate but a consequence of requiring the electron wave to form a standing wave — only integer-wavelength orbits can persist without destructive interference [1].

Sample response (d). The calculation assumes the electron behaves as a non-relativistic particle, so the classical relation p = mv is valid [1]. At speeds approaching a significant fraction of c (e.g. >0.1c), relativistic momentum p = γmv would need to be used, and the classical wavelength formula would underestimate the true de Broglie wavelength.

Marking notes. Part (a): 1 = correct substitution; 1 = correct answer 3.32×10⁻¹⁰ m. Part (b): 1 = correct circumference value; 1 = comparison shows circumference = λ; 1 = explicit reference to n=1 standing wave condition. Part (c): 1 = algebra showing mvr = nℏ from 2πr = nλ; 1 = physical explanation (standing wave requires integer wavelengths). Part (d): 1 = valid assumption (non-relativistic / classical mechanics / electron starts from rest / v << c).

Sample response (a). The graph shows an inverse (hyperbolic) relationship between Δp_min and Δx on a log-log scale, appearing as a straight line with slope −1 [1]. This is consistent with Δp_min = ℏ/(2Δx): doubling the confinement width halves the minimum momentum uncertainty, and vice versa — they are inversely proportional [1].

Sample response (b). From the graph at Δx = 1.0×10⁻¹⁰ m, Δp_min ≈ 5.3×10⁻²⁵ kg·m/s [1]. Using the formula: Δp_min = 1.055×10⁻³⁴/(2 × 1.0×10⁻¹⁰) = 5.28×10⁻²⁵ kg·m/s [1]. Minimum speed implied: v_min = Δp_min/m_e = 5.28×10⁻²⁵/9.11×10⁻³¹ = 5.8×10⁵ m/s [1].

Sample response (c). For an electron to be stationary, its momentum would be exactly zero, giving Δp = 0 [1]. The graph shows that as Δx decreases (as the electron is confined to smaller regions of the atom), Δp_min increases without bound — a finite confinement width always imposes a non-zero minimum momentum uncertainty, meaning the electron must always be moving. An electron cannot be stationary inside an atom of finite size [1].

Marking notes. Part (a): 1 = identifies inverse/hyperbolic relationship; 1 = justifies using ΔpΔx = constant (product is constant = ℏ/2). Part (b): 1 = correct graph read (~5.3×10⁻²⁵ kg·m/s); 1 = verified by formula calculation; 1 = correct minimum speed. Part (c): 1 = stationary electron would require Δp = 0; 1 = graph shows Δp can never be zero for finite Δx, so the electron must always be moving.

Sample response. The Bohr model (1913) was a landmark advance: it introduced quantised energy levels E_n = −13.6/n² eV and correctly predicted the hydrogen emission spectrum. However, it postulated angular momentum quantisation L = nℏ without any physical justification, and it assigned electrons to precise circular orbits with well-defined positions and momenta simultaneously — a direct violation of what would later become the Heisenberg uncertainty principle. De Broglie’s 1924 hypothesis that matter has wave-like properties resolved the first deficiency: if the electron has a wavelength λ = h/(mv), then for a stable orbit the electron wave must form a standing wave around the nucleus, requiring the circumference 2πr to be an integer multiple of λ. Substituting λ = h/(mv) and rearranging immediately gives mvr = nℏ = L — Bohr’s postulate emerges as a natural consequence of wave mechanics rather than an arbitrary assumption. This is a qualitative advance: quantisation is no longer imposed but explained. The Heisenberg uncertainty principle (Δx Δp ≥ ℏ/2) resolved the second fundamental problem. Because the Bohr model assigned exact positions and momenta to electrons, it violated this principle, which forbids such simultaneous precision. The quantum model’s wave function and probability density |ψ|² replace exact trajectories with probability distributions, inherently satisfying the uncertainty principle. The electron’s position is defined only as a probability cloud (orbital), not a specific location. This represents a fundamentally different description of physical reality: in classical mechanics, a particle has definite position and momentum at all times, and its future trajectory is precisely predictable given initial conditions (determinism). In quantum mechanics, the best possible description is probabilistic — even in principle, the exact position and momentum of an electron cannot both be known. The physical universe is intrinsically probabilistic at the quantum scale, not merely uncertain due to limited measurement. Overall, de Broglie’s hypothesis provided the physical justification for quantisation, while the Heisenberg uncertainty principle forced the abandonment of classical trajectories and the adoption of probability densities. Together, they enabled Schrödinger’s wave equation to produce a self-consistent model that correctly describes all atoms — not just hydrogen — and that is consistent with the fundamental nature of quantum reality.

Marking criteria (7 marks). 1 = identifies a specific limitation of the Bohr model addressed by de Broglie (postulated quantisation without justification); 1 = correctly explains how de Broglie’s standing wave condition (λ = h/(mv), 2πr = nλ) derives Bohr’s quantisation condition L = nℏ; 1 = identifies a specific limitation of the Bohr model addressed by the uncertainty principle (exact orbits violate ΔxΔp ≥ ℏ/2); 1 = correctly explains how the uncertainty principle necessitates probability distributions (orbitals) rather than trajectories; 1 = analyses how quantum mechanics differs fundamentally from classical mechanics (determinism vs inherent probability); 1 = evidence-based assessment of the extent to which quantum mechanics represents a fundamentally different view of reality (not just more precise, but different in kind); 1 = logical, cohesive response with precise terminology (wave function, probability density, orbital, uncertainty principle, standing wave, quantisation, determinism).