Energy Levels & Wave-Particle Duality

Collisions of electrons with atoms, discrete energy levels, photon emission and absorption, line spectra, and de Broglie wavelength.

Spec Points Covered
  • Describe ionisation and excitation of atoms by electron collision.
  • Explain discrete energyThe capacity to do work. Measured in joules (J). levels in atoms and the significance of the ground stateThe lowest energyThe capacity to do work. Measured in joules (J). level of an atom. The state in which all electrons are in their lowest possible energyThe capacity to do work. Measured in joules (J). levels..
  • Calculate the frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz). and wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m). of photons emitted or absorbed during electron transitions.
  • Explain the origin of emission and absorption line spectra.
  • State what is meant by wave-particle dualityThe concept that all matter and radiation exhibit both wave-like and particle-like properties. Particles have a de Broglie wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m).; photons exhibit particle behaviour in the photoelectric effectThe emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz). is incident on it...
  • Apply the de Broglie equation $\lambda = h/mv$ to calculate the wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m). of a moving particle.
  • Describe evidence for wave-particle dualityThe concept that all matter and radiation exhibit both wave-like and particle-like properties. Particles have a de Broglie wavelength; photons exhibit particle behaviour in the photoelectric effectThe emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz). is incident on it..: photoelectric effectThe emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency is incident on it. (particle), electron diffraction (wave).
Notes
01 Electrons can excite or ionise atoms through collisions Ionisation 3.2.2.2 → 02 Atoms have discrete energy levels 3.2.2.3 → 03 Photons are emitted when electrons drop to lower energy levels $\Delta E = hf = \frac{hc}{\lambda}$ 3.2.2.3 → 04 Line spectra are fingerprints of discrete energy levels 3.2.2.3 → 05 Wave-particle duality: everything has both wave and particle properties Wave-particle duality 3.2.2.4 → 06 The de Broglie wavelength: λ = h/mv $\lambda = \frac{h}{mv} = \frac{h}{p}$ 3.2.2.4 → 07 Electron diffraction is evidence for the wave nature of particles 3.2.2.4 →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Energy of an emitted/absorbed photon
$$\begin{aligned} \Delta E &= hf \\ &= \frac{hc}{\lambda} \end{aligned}$$
  • Where:
    • $ΔE$ = energy difference between levels (J)
    • $h$ = Planck's constant (J s)
    • $f$ = frequency of photon (Hz)
    • $λ$ = wavelength of photon (m)
ΔE is always positive. Use |E₂ − E₁| if working with negative energy levels.
De Broglie wavelength of an accelerated electron
$$\lambda = \frac{h}{\sqrt{2meV}}$$
  • Where:
    • $λ$ = de Broglie wavelength (m)
    • $h$ = Planck's constant (J s)
    • $m$ = electron mass (kg)
    • $e$ = electron charge (C)
    • $V$ = accelerating potential difference (V)
Derived by substituting v = √(2eV/m) from eV = ½mv² into λ = h/mv.
De Broglie wavelength
$$\begin{aligned} \lambda &= \frac{h}{mv} \\ &= \frac{h}{p} \end{aligned}$$
  • Where:
    • $λ$ = de Broglie wavelength (m)
    • $h$ = Planck's constant (6.63 × 10⁻³⁴ J s)
    • $m$ = mass (kg)
    • $v$ = speed (m s⁻¹)
    • $p$ = momentum (kg m s⁻¹)
Applies to all particles. Wavelength decreases with increasing speed or mass.
Q Retrieval Practice All 13 Questions →
Q1. Define excitation.
When an electron in an atom absorbs energy and moves to a higher energy levelA discrete amount of energy that an electron in an atom can have. Electrons can only exist at specific energy levels, not between them., without leaving the atom.
Q2. Define ionisation.
The removal of an electron from an atom, leaving a positive ion.
Q3. What is the ionisation energyThe minimum energy required to remove an electron completely from an atom in its ground stateThe lowest energy level of an atom. The state in which all electrons are in their lowest possible energy levels. to infinity. of hydrogen?
13.6 eV — the energy needed to remove an electron from the ground stateThe lowest energy level of an atom. The state in which all electrons are in their lowest possible energy levels. (n = 1) to n = ∞.
Q4. Why must the energy transferred in excitation exactly match an energy gap?
  • Energy levels are discrete.
  • An electron can only exist at specific levels, so it can only absorb the exact amount of energy needed to jump between two levels.
Q5. Write the equation linking photonA quantum (discrete packet) of electromagnetic radiation. Its energy is proportional to its frequency. energy to an electron transition.
ΔE = hf = hc/λ, where ΔE is the energy difference between the two levels.