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?

• Yes — the non-relativistic formula still works fine at any electron energy.
• No — at this energy the electron's momentum must be computed relativistically; the non-relativistic formula would be noticeably wrong.
• Wavelength calculations don't depend on relativistic effects at all.

2 · Set Up

1. Open the electron-ke-comparison preset and press Reset. This preset's default kinetic energy is 400 keV, giving β ≈ 0.83.
2. Enable the de Broglie wavelength readout.
3. 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

Plot wavelength λ (y-axis) against kinetic energy K (x-axis) for your three trials.

4 · Analyze

1. 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.
2. 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

1. An electron's rest energy m_ec² ≈ 511 keV. At 400 keV kinetic energy, the electron's total energy is already about 78% more than 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.
2. 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.

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