Turn up the frequency and watch electrons burst from the metal the instant hf crosses the threshold — turning up the brightness alone never does it.
Shine light on a clean metal surface and, sometimes, electrons fly out of it. Whether that happens at all depends on one thing the wave model of light can't explain: frequency, not brightness.
Classical wave theory predicted that shining any frequency of light onto a metal for long enough, or brightly enough, should eventually free electrons — the wave just needed time to deliver enough energy. Experiments showed the opposite: below a certain threshold frequency, no electrons are ever emitted, no matter how intense the light or how long you wait.
Einstein's explanation was that light delivers its energy in discrete packets called photons, each carrying energy E = hf. A single electron absorbs a single photon in one go — so if that one photon doesn't carry enough energy, no amount of extra photons arriving alongside it will help.
Φ is the work function — the minimum energy needed to pull one electron free of the metal's surface. Different metals hold onto their electrons more or less tightly, so each has its own work function and its own threshold frequency f₀ = Φ/h.
The same photon model that explains this also cuts the other way: particles like electrons, which we normally think of as tiny bits of matter, show wave-like behaviour too — diffracting through a crystal lattice just as light diffracts through a slit. That's wave–particle duality: light and matter each show properties of both waves and particles, depending on how you look.
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