Around 1900 physics looked nearly finished: Newton for matter, Maxwell for light. Then a few plain experiments refused to fit. Light knocked electrons out of metals in lumps rather than in a stream, atoms that ought to have collapsed in a flash sat stable and glowed only in certain colours, and single electrons fired at two slits built up the stripes of a wave.

Classical physics is a ramp, on which any height is allowed. The experiments kept finding a staircase: energy handed over in steps, light arriving in packets, atoms that can only stand on certain stairs. Quantum mechanics began as the theory of the staircase and turned out to be the theory of everything small.

The largest energy of an electron knocked out of a metal, in eV, against the frequency of the light in $10^{15}$ Hz. Below a threshold nothing comes out however bright the light; above it the energy climbs a straight line whose slope is Planck's constant $h$, the same for every metal.

Light of frequency $f$ exchanges energy with matter only in whole quanta of $hf$, where $h\approx6.626\times10^{-34}$ J s is Planck's constant. Matter, in turn, behaves as a wave whose wavelength is set by its momentum. Quantum mechanics is the theory that makes both statements consistent, and the rest of this course builds it.

• The photoelectric effect: an electron leaves a metal with energy at most $hf-W$, where $W$ is the work needed to free it. Brighter light frees more electrons, never faster ones. Einstein's reading, in 1905, was that light arrives in packets, later called photons.
• Atomic stability: a classical electron circling a nucleus is an accelerating charge, which radiates and spirals inward in about $10^{-11}$ s. Real atoms last for ever.
• Line spectra: a gas of hydrogen emits only sharp wavelengths, the visible ones at 656, 486, 434 and 410 nm. Each is the energy difference between two allowed levels of the atom.

Light of frequency $f$ is absorbed and emitted only in whole packets of energy, each worth $hf$. One packet frees at most one electron.

For light it is easier to know the wavelength than the frequency. Planck’s relation then reads $E=hc/\lambda$, and with the product $hc$ written in convenient units it is one division.

The photon hands all its energy to one electron. Part of it pays the work function $W$, the cost of leaving the metal, and the rest is the most kinetic energy the electron can leave with.

A brighter beam of the same colour carries more photons per second, each with the same energy. The number of electrons follows the brightness; their energy does not.

Classically an electron circling a nucleus is always changing direction, so it is accelerating, and an accelerating charge radiates. Losing energy, it spirals inward.

de Broglie’s relation gives every particle a wavelength fixed by its momentum. For an electron accelerated through a voltage $V$, the kinetic energy is $eV$, so the momentum grows as $\sqrt{V}$.

Great work! You now know how to:

• say which experiments broke classical physics, and why
• compute a photon energy from a wavelength with $hc\approx1240$ eV nm
• use $hf-W$ for the fastest photoelectron
• scale a de Broglie wavelength with momentum