Relativistic Electrons: When Classical Momentum Fails
1 · Predict
A 400 keV electron moves fast enough that relativistic effects matter (β ≈ 0.83). Does using the simple non-relativistic de Broglie formula λ = h/√(2mK) still give an accurate wavelength at this energy?
- No — at this energy the electron's momentum must be computed relativistically; the non-relativistic formula would be noticeably wrong.
- Yes — the non-relativistic formula still works fine at any electron energy.
- Wavelength calculations don't depend on relativistic effects at all.
2 · Set Up
- Open the electron-ke-comparison preset and press Reset. This preset's default kinetic energy is 400 keV, giving β ≈ 0.83.
- Enable the de Broglie wavelength readout.
- Set the electron's kinetic energy for each trial (all in the relativistic regime) and record its de Broglie wavelength using the relativistic momentum formula.
3 · Collect Data
| Kinetic energy K (eV) | De Broglie wavelength λ (pm) |
|---|---|
| 200000 | |
| 400000 | |
| 600000 |
Plot wavelength λ (y-axis) against kinetic energy K (x-axis) for your three trials.
4 · Analyze
- For one trial, compute the relativistic momentum via pc = √(K(K + 2m_ec²)) (with m_ec² ≈ 511 keV, the electron's rest energy), then λ = hc/(pc) — converting units carefully. Compare to the table.
- Explain why the relativistic momentum formula pc = √(K(K + 2mc²)) reduces to the familiar non-relativistic p = √(2mK) when K is much smaller than the rest energy mc² — and why that approximation breaks down once K becomes comparable to or larger than mc².
5 · Extend
- An electron's rest energy m_ec² ≈ 511 keV. At 400 keV kinetic energy, the electron's total energy is already more than 80% additional to its rest energy — solidly in the relativistic regime. Explain why physicists have a rule of thumb that relativistic corrections become important once kinetic energy approaches roughly 10% of rest energy.
- Particle accelerators routinely reach kinetic energies enormously larger than a particle's rest energy (ultra-relativistic). Explain why, in that extreme limit, pc ≈ K (momentum-energy and kinetic energy become nearly equal) even though that's clearly not true at everyday, non-relativistic speeds.
The Physics Behind This Experiment
Relativistic Momentum-Energy Relation
For a relativistic particle, momentum and kinetic energy relate through (pc)² = K(K + 2mc²), derived from the full relativistic energy-momentum relation E² = (pc)² + (mc²)². This reduces to the familiar non-relativistic p = √(2mK) only when K ≪ mc².