Photon and quantum relations
E = hfPlanck relationde Broglie wavelengthphotoelectric equationquantum formulas
The small family of relations built on Planck's constant — photon energy and momentum, matter waves, the photoelectric effect and the Bohr levels.
Photon Energy (E = hf)
The energy of a single photon: Planck's constant times the light's frequency.
Photon Energy from Wavelength (E = hc/λ)
Photon energy written in terms of wavelength: shorter waves, more energetic photons.
Photon Momentum (p = h/λ)
Massless but not momentum-less: a photon carries h divided by its wavelength.
De Broglie Wavelength
Every moving particle has a wavelength: Planck's constant over its momentum mv.
Photoelectric Effect
Maximum kinetic energy of an ejected electron: photon energy (hf) minus the work function.
Bohr Model Energy Levels
Energy of the hydrogen atom's nth level: −13.606 eV divided by n².
Wien's Displacement Law
The peak wavelength of thermal radiation, with b = 2.8978 × 10⁻³ m·K.
How they fit together
Planck's constant is the thread. Energy comes in lumps proportional to frequency, so E = hf; substitute c = fλ and you have the wavelength version. Momentum p = h/λ turns the relation around, and de Broglie's leap was to read it backwards — if light with momentum has a wavelength, then electrons with momentum have one too, which is why electron microscopes work and why the hydrogen atom has discrete levels at all.
Choose by what you are given: frequency picks E = hf, wavelength picks E = hc/λ, and they are the same equation. The photoelectric equation is the one with a threshold in it — a photon either carries more than the work function or it ejects nothing, no matter how intense the beam, and that stubborn fact is what Einstein explained in 1905 and what actually earned him the Nobel Prize. The universal practical trap is units: h in joule-seconds gives joules, and spectroscopy is done in electron-volts, so a factor of 1.602×10⁻¹⁹ separates an answer that looks sensible from one that does not.