understand the de Broglie wavelength as the wavelength associated with a moving particle

Wave‑Particle Duality and the de Broglie Wavelength (Cambridge 9702 – 22.1 to 22.5)

22.1 Energy and Momentum of a Photon

• Energy: \(E = h\,f = \dfrac{h\,c}{\lambda}\)
• Momentum: \(p = \dfrac{E}{c} = \dfrac{h}{\lambda}\)

Key constants

Planck’s constant, \(h\)	\(6.626\times10^{-34}\ \text{J·s}\)
Speed of light, \(c\)	\(3.00\times10^{8}\ \text{m s}^{-1}\)
Elementary charge, \(e\)	\(1.60\times10^{-19}\ \text{C}\)
1 eV (electron‑volt)	\(1.60\times10^{-19}\ \text{J}\)

Worked example – photon momentum

A photon of visible light with \(\lambda = 500\ \text{nm}\) has

\[ p = \frac{h}{\lambda}= \frac{6.626\times10^{-34}}{5.0\times10^{-7}} = 1.33\times10^{-27}\ \text{kg m s}^{-1}. \]

22.2 Photoelectric Effect

The experiment shows that light can eject electrons from a metal surface. The governing equation is

\[ hf = \Phi + \tfrac{1}{2}mv^{2}, \] where

• \(f\) – frequency of the incident light
• \(\Phi\) – work function (minimum energy to free an electron)
• \(\tfrac{1}{2}mv^{2}\) – kinetic energy of the emitted electron
• Stopping potential \(V_{s}\) satisfies \(eV_{s}= \tfrac{1}{2}mv^{2}\)

Worked example

Metal with \(\Phi = 2.1\ \text{eV}\) illuminated by light of \(\lambda = 300\ \text{nm}\):

\[ f = \frac{c}{\lambda}= \frac{3.00\times10^{8}}{3.00\times10^{-7}} = 1.0\times10^{15}\ \text{Hz} \] \[ E_{\text{photon}} = hf = (6.626\times10^{-34})(1.0\times10^{15}) = 6.63\times10^{-19}\ \text{J} = \frac{6.63\times10^{-19}}{1.60\times10^{-19}}\ \text{eV} = 4.14\ \text{eV} \] \[ \text{K.E.}=E_{\text{photon}}-\Phi = 4.14-2.10 = 2.04\ \text{eV} \] \[ V_{s}= \frac{\text{K.E.}}{e}= \frac{2.04\ \text{eV}}{1\ \text{eV V}^{-1}} = 2.04\ \text{V} \]

22.3 Wave‑Particle Duality

• All quantum entities (photons, electrons, neutrons, atoms, molecules) display both wave‑like and particle‑like behaviour.
• The observed aspect depends on the experimental arrangement:
  • Diffraction or interference → wave nature.
  • Photoelectric effect, Compton scattering → particle nature.
• The dual nature is expressed mathematically by the de Broglie relation \(\displaystyle \lambda = \frac{h}{p}\).

22.4 de Broglie Hypothesis

Louis de Broglie (1924) proposed that any moving particle of momentum \(p\) is associated with a wave of wavelength

\[ \lambda = \frac{h}{p}. \]

For a non‑relativistic particle of mass \(m\) travelling at speed \(v\):

\[ p = mv \qquad\Longrightarrow\qquad \lambda = \frac{h}{mv}. \]

Typical de Broglie Wavelengths

Particle	Speed / Kinetic Energy	De Broglie wavelength \(\lambda\)	Observable wave effect?
Electron (thermal, \(T\approx300\ \text{K}\))	\(v\approx1.0\times10^{5}\ \text{m s}^{-1}\)	\(1.2\times10^{-10}\ \text{m}\) (0.12 nm)	Yes – electron diffraction
Neutron (cold, \(v\approx2.0\times10^{3}\ \text{m s}^{-1}\))	–	\(1.8\times10^{-10}\ \text{m}\) (0.18 nm)	Yes – neutron scattering
Proton (accelerated, 1 MeV)	\(v\approx1.4\times10^{7}\ \text{m s}^{-1}\)	\(9.0\times10^{-13}\ \text{m}\) (0.009 nm)	Yes – high‑energy diffraction
Macroscopic object (baseball, \(m=0.145\ \text{kg}\), \(v=30\ \text{m s}^{-1}\))	–	\(1.5\times10^{-34}\ \text{m}\)	No – far below any measurable length scale

Worked calculation – electron accelerated through 150 V

1. Energy gained: \(eV = (1.60\times10^{-19}\ \text{C})(150\ \text{V}) = 2.40\times10^{-17}\ \text{J}\).
2. Assuming non‑relativistic motion, \(\tfrac{1}{2}mv^{2}=eV\) ⇒ \(v = \sqrt{\dfrac{2eV}{m}}\) \(= \sqrt{\dfrac{2(2.40\times10^{-17})}{9.11\times10^{-31}}} = 9.2\times10^{6}\ \text{m s}^{-1}\).
3. Momentum: \(p = mv = (9.11\times10^{-31})(9.2\times10^{6}) = 8.4\times10^{-24}\ \text{kg m s}^{-1}\).
4. De Broglie wavelength: \[ \lambda = \frac{h}{p}= \frac{6.626\times10^{-34}}{8.4\times10^{-24}} = 7.9\times10^{-11}\ \text{m}=0.079\ \text{nm}. \]

22.5 Matter Waves and Diffraction

• Davisson–Germer experiment (1927) – Electrons reflected from a nickel crystal produced a diffraction pattern that obeyed Bragg’s law when the electron wavelength was taken as \(\lambda = h/p\). This was the first direct confirmation of de Broglie’s hypothesis for matter.
• Bragg’s law for matter waves: \[ n\lambda = 2d\sin\theta, \] where \(d\) is the lattice spacing, \(\theta\) the diffraction angle and \(n\) an integer.
• Neutron diffraction – Cold neutrons (λ ≈ 0.1 nm) are scattered by crystal lattices, revealing atomic positions because neutrons interact with nuclei rather than electron clouds.
• Electron diffraction in thin films – A beam of electrons of known accelerating voltage forms concentric rings on a phosphor screen; the ring radii give \(\lambda\) and hence the crystal spacing.

Example – Using Bragg’s law with electrons

Electrons accelerated through 54 V have \(\lambda = 0.167\ \text{nm}\) (from the calculation in 22.4). For a crystal with spacing \(d = 0.215\ \text{nm}\) and first‑order diffraction (\(n=1\)):

\[ \sin\theta = \frac{n\lambda}{2d}= \frac{0.167}{2\times0.215}=0.388 \quad\Rightarrow\quad \theta = 22.8^{\circ}. \]

The measured diffraction angle matches the prediction, confirming the wave nature of electrons.

Applications in Technology

• Electron microscopes – Use electrons with \(\lambda\) ≈ 0.01 nm to resolve structures far smaller than possible with visible light.
• Neutron scattering – Probes magnetic ordering and light atoms (e.g., hydrogen) because neutrons are electrically neutral.
• Scanning tunnelling microscopy (STM) – Relies on the wave nature of electrons tunnelling between a conductive tip and a surface to image atomic‑scale topography.

Summary

De Broglie’s relation \(\lambda = h/p\) extends wave‑particle duality from photons to all matter. It explains why sub‑atomic particles exhibit diffraction and interference, while macroscopic objects do not. Mastery of the associated equations (photon energy/momentum, photoelectric effect, Bragg’s law) equips students to answer the full range of Cambridge 9702 questions on wave‑particle duality, matter waves, and their experimental verification.