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
\[
p = \frac{h}{\lambda}= \frac{6.626\times10^{-34}}{5.0\times10^{-7}}
= 1.33\times10^{-27}\ \text{kg m s}^{-1}.
\]
The experiment shows that light can eject electrons from a metal surface. The governing equation is
\[
hf = \Phi + \tfrac{1}{2}mv^{2},
\]
where
Worked example
Metal with \(\Phi = 2.1\ \text{eV}\) illuminated by light of \(\lambda = 300\ \text{nm}\):
\[
f = \frac{c}{\lambda}= \frac{3.00\times10^{8}}{3.00\times10^{-7}}
= 1.0\times10^{15}\ \text{Hz}
\]
\[
E_{\text{photon}} = hf = (6.626\times10^{-34})(1.0\times10^{15})
= 6.63\times10^{-19}\ \text{J}
= \frac{6.63\times10^{-19}}{1.60\times10^{-19}}\ \text{eV}
= 4.14\ \text{eV}
\]
\[
\text{K.E.}=E_{\text{photon}}-\Phi = 4.14-2.10 = 2.04\ \text{eV}
\]
\[
V_{s}= \frac{\text{K.E.}}{e}= \frac{2.04\ \text{eV}}{1\ \text{eV V}^{-1}}
= 2.04\ \text{V}
\]
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 \(v\):
\[
p = mv \qquad\Longrightarrow\qquad \lambda = \frac{h}{mv}.
\]
| 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 |
= 2.40\times10^{-17}\ \text{J}\).
\(v = \sqrt{\dfrac{2eV}{m}}\)
\(= \sqrt{\dfrac{2(2.40\times10^{-17})}{9.11\times10^{-31}}}
= 9.2\times10^{6}\ \text{m s}^{-1}\).
= 8.4\times10^{-24}\ \text{kg m s}^{-1}\).
\[
\lambda = \frac{h}{p}= \frac{6.626\times10^{-34}}{8.4\times10^{-24}}
= 7.9\times10^{-11}\ \text{m}=0.079\ \text{nm}.
\]
\[
n\lambda = 2d\sin\theta,
\]
where \(d\) is the lattice spacing, \(\theta\) the diffraction angle and \(n\) an integer.
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.
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.