Compare photon and matter wavelengths without conflating their energies

Begin with evidence

A diffraction pattern can reveal a wavelength for electrons as well as for light. A photon has zero rest mass and no electric charge. In vacuum its speed is c; propagation through a material is described here by the optical speed c/n rather than by a free-vacuum trajectory. For a photon, momentum magnitude is p=h/λ and energy is E=pc=hf. For a nonrelativistic electron, the same de Broglie wavelength relation λ=h/p applies, but kinetic energy is p²/(2m), not pc. Equal wavelength therefore means equal momentum magnitude, not equal kinetic energy. The particle and wave descriptions constrain different observations rather than describing a tiny classical ball carrying a literal transverse wiggle.

Connect the model

An electron accelerated from rest through a potential difference of magnitude V gains kinetic energy eV if other transfers are negligible. Combining that energy with K=p²/(2m) gives p=√(2meV) and λ=h/√(2meV). Quadrupling voltage doubles momentum and halves wavelength. A photon at half its original wavelength instead doubles both momentum and energy. Ratios make the different dependencies visible without hiding them behind small SI numbers.

Interpret the ledger

The workbench limits electron voltages to 10–100 V; its computed speed stays below 0.02c, supporting the nonrelativistic approximation here. The photon control spans 100–500 nm. These are mathematical packets, not instructions for an electron gun, ultraviolet lamp or electrical supply. The separate ledgers state units; a plotted wavelength is not a particle trajectory.

Test the boundary

A single count at a detector is localized, while the distribution of many similarly prepared detections can show interference. Neither a classical wave carrying arbitrary divisible energy nor an unqualified classical-particle path accounts for every quantum observation. State which evidence your model explains. Wavelength alone does not determine the probability distribution for an arbitrary experimental apparatus.

State what is quantized in the model

Bound atomic energy states are discrete, and the historical Bohr model imposes discrete orbital angular momentum. Do not turn that statement into a universal assertion that every linear-momentum component of every bound quantum state can only take discrete values. A localized bound-state wavefunction can have a continuous momentum-measurement distribution. Compare the two claims: a hydrogen atom in a stated stationary level has a definite allowed energy, but that does not give its electron a single definite classical velocity. The standing-wave construction in the next lesson is a constrained historical model, not a simultaneous position-and-momentum trajectory.

Worked original case

At 40 V, the electron has K=40 eV, p≈3.41693×10⁻²⁴ kg m/s and λ≈0.193919 nm. A 300 nm photon has E≈4.13281 eV and p≈2.20869×10⁻²⁷ kg m/s. Their wavelengths and energy laws differ.

Predict, compare and retain evidence

Does four times the electron accelerating voltage reduce wavelength by four? Capture electron voltages 10, 40 and 90 V at one photon wavelength. Predict the wavelength ratios first; then vary only photon wavelength and explain why its energy scaling differs.

Changed-case practice

Two nonrelativistic electrons begin at rest and gain 25 eV and 100 eV. Find the second wavelength divided by the first; compare photon wavelengths at those two energies.

Check the reasoning after your attempt

Electron ratio is √(25/100)=1/2. Photon ratio is 25/100=1/4 because E=hc/λ. The energies do not imply the same dispersion relation.

Reflect and source-check

What additional assumption is needed before using p²/(2m) for a particle?

Original prose and supplied numerical evidence; checked 30 October 2026 against the current AP Physics 2 framework, fall 2026 clarifications and OpenStax. Virtual-only: no physical source, exposure or apparatus is requested. Public examples and model captures are not independent mastery or supervised laboratory participation.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.

Supplemental quantum interpretation: MIT energy versus momentum eigenstates, checked 31 October 2026. No Fourier calculation is required for this qualitative caution.