Questions: Compton Scattering and Wavelength Shift

Question 1

Classical wave theory (Thomson scattering) predicted that X-rays scattered off electrons would have the same wavelength as the incident X-rays. What did Compton actually observe, and why was it decisive evidence for the photon model?

• A. Scattered X-rays were shorter in wavelength, proving electrons absorb photon energy and re-emit it at higher frequencies
• B. Scattered X-rays had the same wavelength as predicted, confirming the classical wave model
• C. Scattered X-rays were longer in wavelength, and the shift depended on scattering angle — consistent with photons losing momentum and energy in a particle collision
• D. The wavelength shift depended on the incident wavelength, consistent with wave interference patterns

Answer: (C)

Compton observed wavelength *increase* (redshift) in scattered X-rays, with the shift depending on scattering angle — exactly as predicted by treating the photon as a particle carrying momentum p = h/λ and applying conservation of energy and momentum. Classical wave theory predicted zero wavelength shift (the electron oscillates at the driving frequency and re-radiates at the same frequency). The fact that the shift was independent of incident wavelength but depended only on scattering angle was unmistakable evidence for particle-like photon behavior.

Question 2

What is the wavelength shift Δλ when X-rays undergo Compton scattering at θ = 90°?

• A. Zero — no energy is transferred in perpendicular scattering
• B. λ_C = h/mₑc ≈ 2.43 × 10⁻¹² m (the Compton wavelength)
• C. 2λ_C ≈ 4.86 pm — twice the Compton wavelength
• D. It depends on the incident wavelength

Answer: (B)

The Compton formula is Δλ = (h/mₑc)(1 − cos θ). At θ = 90°, cos 90° = 0, so Δλ = h/mₑc = λ_C ≈ 2.43 pm. The maximum shift occurs at θ = 180° (backscatter): Δλ = 2λ_C. At θ = 0° (forward scatter), Δλ = 0 — no energy transfer. A critical feature: Δλ depends only on θ, not on the incident wavelength. This angle-only dependence is a characteristic fingerprint of particle collisions and has no natural explanation in classical wave theory.

Question 3

The Compton wavelength shift Δλ is larger for shorter-wavelength (higher-energy) incident photons than for longer-wavelength ones, at the same scattering angle.

• T. True
• F. False

Answer: False

This is the most tempting misconception. The formula Δλ = (h/mₑc)(1 − cos θ) contains no dependence on the incident wavelength λ — only on the scattering angle θ and fundamental constants. At a given angle, the absolute wavelength shift is always the same (≈ 2.43 pm at θ = 90°) regardless of whether the incident photon is a hard X-ray or a soft one. The *fractional* shift Δλ/λ is larger for short-wavelength photons, which is why the effect is measurable with X-rays but negligible for visible light.

Question 4

Compton scattering provides experimental evidence that photons carry momentum, not just energy.

• T. True
• F. False

Answer: True

The Compton formula is derived by applying conservation of *both* energy and momentum to the photon-electron collision, assigning the photon momentum p = h/λ. If photons carried no momentum, there would be no recoil of the electron, no energy transfer, and no wavelength shift — the classical Thomson scattering result. The fact that the observed wavelength shift matches exactly the prediction from momentum conservation confirms that photons carry momentum h/λ and exchange it with electrons in a collision. This was a cornerstone result in establishing the quantum picture of light.

Question 5

Why did Compton use X-rays rather than visible light in his experiment, and why does deriving the Compton formula require relativistic mechanics?

Model answer: Compton used X-rays because the wavelength shift Δλ = λ_C(1 − cos θ) is a fixed value (at most ~4.86 pm). For visible light (λ ~ 500 nm), this shift is less than 0.001% of the wavelength — far too small to measure. For X-rays (λ ~ 10–100 pm), the shift is a significant fraction of the wavelength and measurable. Relativistic mechanics is required because the recoiling electron can gain kinetic energy comparable to its rest mass energy (mₑc² ≈ 0.511 MeV) for energetic photons, and the relativistic energy-momentum relation E² = (pc)² + (mₑc²)² must be used for the electron. Non-relativistic treatment gives incorrect results.

The choice of X-rays was not arbitrary — it was dictated by the scale of the effect. The Compton wavelength λ_C = h/mₑc ≈ 2.43 pm sets the natural scale; the effect is only measurable when the incident wavelength is comparable to λ_C. The relativistic requirement is a deeper point: it shows that even 'low-energy' photon scattering can impart relativistic recoil velocities to electrons, making Compton scattering one of the earliest phenomena demanding special relativity for its explanation.