Final Exam Review | Source: Knight, Ch. 28.4–28.6
Difficulty: Intermediate | Prerequisites: Basic wave concepts (wavelength, frequency, standing waves), photon energy (E = hf), the electron-volt unit.
This material bridges classical physics and quantum mechanics. You have already met photons (light behaving as particles); now the tables turn and particles such as electrons behave as waves. That wave nature forces energy to come in discrete packets rather than any value you like, which is the central idea behind atomic spectra, lasers, and semiconductor physics. If you are comfortable with wavelength, frequency, and the relation E = hf, you have the toolkit to follow everything here.
Every moving particle has a wavelength given by λ = h/mv (the de Broglie relation). Confine that particle and its wave nature forces only certain energies to exist, labelled by a quantum number n. When a particle jumps between energy levels, a photon whose energy equals the gap is emitted or absorbed.
De Broglie wavelength
The wavelength associated with any moving particle, defined as λ = h / p = h / (mv), where h is Planck's constant. In simple terms, this means every object has a wave side, but only tiny particles like electrons have wavelengths large enough to notice.
Matter wave
The wave behaviour exhibited by particles such as electrons, confirmed by diffraction and interference experiments. Think of it as the electron acting like a ripple passing through slits, just as light does.
Quantised energy
The restriction that a confined particle can possess only specific, discrete energy values rather than a continuous range. In simple terms, the particle's energy comes in fixed steps, like rungs on a ladder, not a smooth ramp.
Quantum number (n)
A positive integer (n = 1, 2, 3, ...) that labels each allowed energy level of a confined particle. The lowest energy state has n = 1.
Ground state
The lowest allowed energy level (n = 1). It is stable and the particle can remain there indefinitely.
Excited state
Any energy level above the ground state (n = 2, 3, ...). Particles in excited states tend to drop back down, emitting photons as they go.
Quantum jump (transition)
A change from one allowed energy level to another. The particle does not pass through energies in between; it simply appears in the new state.
Photon emission
When a particle drops from a higher energy level to a lower one, the energy difference leaves as a photon with frequency f = ΔE / h.
Photon absorption
When an incoming photon has exactly the right energy to match the gap between two levels, the particle absorbs it and jumps up.
Particle in a box
A model where a particle is confined between two rigid walls separated by length L. Standing-wave conditions force the allowed wavelengths to be λ_n = 2L / n, producing quantised energies E_n = n² E_1.
De Broglie wavelength of a moving particle
The wavelength is λ = h / p = h / (mv).
h = 6.626 × 10⁻³⁴ J·s (Planck's constant).
Wavelength is inversely proportional to speed: faster particle, shorter wavelength.
Electron diffraction
Electrons diffract and interfere exactly like x-rays when passed through a crystal or double slit.
An electron accelerated through a larger potential difference moves faster and therefore has a shorter de Broglie wavelength.
Worked example: de Broglie wavelength of a 1.0 eV electron
Convert kinetic energy: KE = 1.0 eV = 1.6 × 10⁻¹⁹ J.
Solve for speed: ½ mv² = 1.6 × 10⁻¹⁹ J, with m_e = 9.1 × 10⁻³¹ kg, giving v = 5.9 × 10⁵ m/s.
Compute wavelength: λ = h / (mv) = 6.626 × 10⁻³⁴ / (9.1 × 10⁻³¹ × 5.9 × 10⁵) ≈ 1.2 × 10⁻⁹ m = 1.2 nm.
This is comparable to x-ray wavelengths and to atomic spacings, which is why electron diffraction works.
Particle in a box model
A particle of mass m bounces between two perfectly rigid walls separated by distance L.
Collisions with the walls are perfectly elastic (no kinetic energy lost).
Matter waves travelling in both directions create standing waves, just like standing waves on a guitar string.
Allowed wavelengths
Standing-wave condition: λ_n = 2L / n, where n = 1, 2, 3, ...
Combining with de Broglie: λ_n = h / (mv), so only certain speeds (and therefore certain energies) are permitted.
Quantised energy levels
The allowed energies are E_n = (h² / 8mL²) × n².
Equivalently, E_n = n² E_1, where E_1 = h² / (8mL²) is the ground-state energy.
The particle can never have zero energy. The lowest it can sit is E_1 (the ground state).
Worked example: electron in a 0.19 nm box
E_1 = h² / (8mL²) = 1.7 × 10⁻¹⁹ J = 10 eV.
E_2 = 2² × E_1 = 4 × 10 = 40 eV.
E_3 = 3² × E_1 = 9 × 10 = 90 eV.
Notice the spacings grow: the gap between levels widens as n increases.
Energy-level diagrams
Horizontal lines represent the allowed energies, labelled by quantum number n.
The lowest line is the ground state (n = 1); lines above it are excited states.
Photon emission (jumping down)
When a particle drops from level n_high to n_low, a photon is emitted.
The photon's frequency: f_photon = ΔE_system / h, where ΔE = E_high – E_low.
The photon's wavelength: λ = c / f.
Photon absorption (jumping up)
A photon is absorbed only if its energy exactly matches the gap between two levels.
Absorption transitions must start from the ground state (n = 1) in a cold sample where atoms sit in their lowest state.
Emission transitions can start and end at any pair of levels.
Worked example: emission wavelengths for levels at 1.0, 4.0, and 6.0 eV
Transition n = 3 to n = 2: ΔE = 6.0 – 4.0 = 2.0 eV. f = 2.0 / 4.141 × 10⁻¹⁵ = 4.8 × 10¹⁴ Hz. λ = c / f = 3.0 × 10⁸ / 4.8 × 10¹⁴ ≈ 625 nm.
Transition n = 3 to n = 1: ΔE = 5.0 eV. f = 1.2 × 10¹⁵ Hz. λ ≈ 250 nm.
Transition n = 2 to n = 1: ΔE = 3.0 eV. f = 7.2 × 10¹⁴ Hz. λ ≈ 415 nm.
Each transition produces a distinct spectral line. Three levels give three possible emission wavelengths.
Formula | Meaning |
|---|---|
λ = h / p = h / (mv) | De Broglie wavelength of a particle with momentum p |
λ_n = 2L / n | Allowed wavelengths for a particle in a box of length L |
E_n = (h² / 8mL²) × n² | Allowed energies for a particle in a box |
E_n = n² E_1 | Shortcut once you know E_1 |
f_photon = ΔE / h | Frequency of the photon emitted or absorbed in a transition |
λ = c / f | Wavelength from frequency (c = 3.0 × 10⁸ m/s) |
1 eV = 1.6 × 10⁻¹⁹ J | Energy unit conversion |
h = 6.626 × 10⁻³⁴ J·s | Planck's constant |
h = 4.141 × 10⁻¹⁵ eV·s | Planck's constant in eV (handy when energies are given in eV) |
Electron diffraction is the basis of the transmission electron microscope (TEM), which images structures far smaller than visible light can resolve, down to individual atoms. Engineers tune the accelerating voltage to control the electron wavelength and therefore the resolution.
Quantised energy levels explain why elements produce characteristic spectral lines. This is how astronomers identify the chemical composition of distant stars and how neon signs produce their distinctive colours.
Students often think a particle in a box can have zero energy. It cannot. The lowest allowed state is E_1, not zero, because a standing wave needs at least one half-wavelength to fit inside the box.
Students sometimes confuse the quantum number n with the number of photons. n labels an energy level; one photon is emitted or absorbed per transition.
A common error is assuming any photon can be absorbed. Only a photon whose energy exactly matches the gap between two allowed levels will be absorbed; all others pass through.
Students sometimes forget to convert eV to joules (or vice versa) midway through a calculation. Pick one unit system at the start and stay consistent; use h in eV·s when your energies are in eV.
⚠️ Computing λ from kinetic energy is a two-step process: first solve for v from KE = ½ mv², then use λ = h/(mv). Questions love to test whether you can chain these together.
⚠️ The relationship E_n = n² E_1 is heavily tested. Be ready to find E_1 from given data, then scale up.
⚠️ Emission-spectrum questions ask for all possible wavelengths. For N energy levels, the number of distinct transitions is N(N–1)/2. Three levels give three lines, not two.
⚠️ Know the difference between absorption and emission spectra: absorption lines start from the ground state; emission lines can connect any two levels.
True or false: A particle confined in a box can have zero kinetic energy.
Answer: False. The minimum energy is E_1, which is always greater than zero.
Fill in the blank: The de Broglie wavelength is ______ proportional to a particle's speed.
Answer: inversely.
True or false: An electron can absorb a photon of any energy.
Answer: False. The photon's energy must exactly match the gap between two allowed energy levels.
Fill in the blank: For a particle in a box, E_3 = ______ times E_1.
Answer: 9 (because 3² = 9).
True or false: Emission transitions can only start from the first excited state.
Answer: False. Emission can occur between any pair of levels where the particle drops from a higher to a lower state.
Q: An electron has a kinetic energy of 2.0 eV. What is its de Broglie wavelength?
A: Convert KE to joules: 2.0 × 1.6 × 10⁻¹⁹ = 3.2 × 10⁻¹⁹ J. Solve ½ mv² = 3.2 × 10⁻¹⁹ for v = 8.4 × 10⁵ m/s. Then λ = h/(mv) = 6.626 × 10⁻³⁴ / (9.1 × 10⁻³¹ × 8.4 × 10⁵) ≈ 0.87 nm.
Q: An electron is confined in a box of length 0.10 nm. What is the energy of the ground state, and what is E_3?
A: E_1 = h² / (8mL²) = (6.626 × 10⁻³⁴)² / (8 × 9.1 × 10⁻³¹ × (0.10 × 10⁻⁹)²) ≈ 6.0 × 10⁻¹⁸ J ≈ 38 eV. E_3 = 9 × 38 = 342 eV.
Q: A system has energy levels at 0 eV, 3.0 eV, and 5.0 eV. List all possible emission wavelengths.
A: Three transitions: 5.0 to 3.0 eV (ΔE = 2.0 eV, λ ≈ 620 nm), 5.0 to 0 eV (ΔE = 5.0 eV, λ ≈ 250 nm), 3.0 to 0 eV (ΔE = 3.0 eV, λ ≈ 415 nm).
Q: Why can a particle in a box never be at rest?
A: A particle at rest has v = 0, so its de Broglie wavelength would be infinite. An infinite wavelength cannot satisfy the standing-wave boundary conditions inside a finite box. The smallest allowed wavelength is λ_1 = 2L, which corresponds to a non-zero speed and therefore a non-zero minimum energy E_1.
This material connects directly to Ch. 29 (Bohr's model and the hydrogen atom), where the same quantisation ideas are applied to a real atom with Coulomb attraction rather than rigid walls. The emission and absorption framework here is exactly what produces the hydrogen spectral series (Balmer, Lyman, Paschen).
It also links back to Ch. 28.1–28.3 on photons and the photoelectric effect: there, light behaves as particles; here, particles behave as waves. Together they form the wave-particle duality that underpins all of quantum mechanics.
de Broglie wavelength, matter wave, wave-particle duality, particle in a box, infinite square well, quantised energy, energy quantisation, quantum number, ground state, excited state, quantum jump, energy transition, photon emission, photon absorption, standing wave, spectral lines, emission spectrum, absorption spectrum, electron diffraction, Planck's constant, PHYS 116, general physics, University of Washington