22.3 Wave–particle duality

Photoemission is evidence that light transfers energy in discrete photons, each with energy E=hf.

The threshold frequency, near-instant emission and intensity–frequency separation are difficult to explain with a purely continuous-wave energy supply.

A low-intensity beam above threshold can eject electrons immediately, whereas a bright beam below threshold cannot in the ideal model.

This evidence supports quantised energy transfer; it does not mean every classical wave description is useless in every context.

When electrons pass through a thin crystal or narrow spacing, they can form diffraction patterns rather than only particle-like spots.

The pattern is explained by a wavelength associated with the moving electrons; changing electron momentum changes the spacing of maxima and minima.

A ring pattern from a polycrystalline film is evidence of constructive interference from many crystal orientations.

Diffraction does not say an electron is a classical water wave; it reveals wave-like interference in the probability description.

A particle with momentum p has de Broglie wavelength λ=h/p, so greater momentum means a shorter wavelength.

For a non-relativistic electron p=mv; use the momentum actually delivered by the apparatus, not simply the particle’s rest mass.

Accelerating an electron through a larger potential difference increases its speed and reduces λ, changing the diffraction scale.

The wavelength is not a path the particle visibly traces; it predicts the scale of interference and diffraction effects.

Use λ=h/p with h=6.63×10⁻³⁴ J s and momentum p in kg m s⁻¹.

If speed is known and v is far below c, calculate p=mv first; keep units consistent before comparing λ with a slit or lattice spacing.

For a 9.11×10⁻³¹ kg electron moving at 2.0×10⁶ m s⁻¹, λ is about 0.36 nm, comparable with atomic spacings.

Do not substitute kinetic energy directly for p unless you first use the appropriate relation; rounding p too early can distort λ.