Physics HL · Chapter 22: Quantum Physics
22.4 de Broglie Hypothesis and Matter-Wave Scaling
Assign wavelength to particles through momentum, then connect accelerating voltage to measurable electron wavelengths.
Estimated time: 36 minutes
From Photon Duality to Matter Duality
If light, previously understood as a wave, can show particle-like transfer events, de Broglie's proposal asks the reciprocal question: can matter, previously understood as particles, show wave behavior? The answer is yes, with wavelength determined by momentum. This is not a metaphorical wave and not literal mechanical wobbling of a particle path. It is a predictive wavelength that controls interference and diffraction outcomes.
Bigger momentum means shorter matter wavelength and less visible diffraction in everyday scales.
This equation explains why quantum wave behavior is dramatic for electrons but invisible for a moving brick. Large mass and ordinary speeds give huge momentum, so wavelength becomes extraordinarily tiny compared with slit sizes or obstacle scales in the macroscopic world.
Accelerating Voltage to Electron Wavelength
For non-relativistic electrons accelerated from rest through V.
The inverse-square-root dependence on voltage is a key exam trend: raising V increases momentum, so de Broglie wavelength shrinks as 1/sqrt(V). That means diffraction angles from fixed crystal spacing get smaller as accelerating voltage rises. If you remember only one scaling for this section, remember lambda proportional to 1/sqrt(V).
What the Wave Represents
In modern interpretation, matter-wave amplitude is linked to probability amplitude, and squared magnitude gives detection probability density. For IB-level problem solving, you usually do not need full wavefunction mathematics, but this interpretation explains why interference patterns emerge statistically after many particle detections rather than as a single continuous material ripple.
Simulation: Matter-Wave Scaling and Bragg Feasibility
Compare electron, proton, alpha, and neutron de Broglie wavelengths while changing kinetic energy and crystal spacing to test when Bragg peaks are possible.
Diffraction Intensity vs Angle
A clear diffraction peak requires wavelength comparable to lattice spacing. Heavy particles at the same kinetic energy have shorter de Broglie wavelengths and weaker angular spread.
De Broglie Lambda
0.167 nm
Momentum
3.97e-24 kg m/s
Particle Charge
-e
Bragg Angle
51.250 deg
Electron accelerator view: kinetic energy of 54.000 eV corresponds to acceleration through approximately the same number of volts.
Test Yourself
An electron is accelerated from rest through 54 V. Enter its de Broglie wavelength.
Hint: Use lambda = h / sqrt(2meV).