1.1Where classical physics ran out

Around 1900 physics looked close to finished. Newton’s mechanics dealt with matter, Maxwell’s equations with light, and the statistics of huge numbers of molecules with heat. Then someone worked out how much energy there should be in the glow inside a hot oven, and the answer came out as infinity.

The picture that was about to fail

The physics of 1900 is now called classical physics, and it rests on a picture so natural that it hardly seems like an assumption. A particle is somewhere, and it is moving somehow. At every instant it has a position, written $q$, and a momentum, $p$ (its mass times its velocity). These are facts about the particle whether or not anyone knows them.

When there are too many particles to follow, as in a gas, you give up on the individual values and use averages. But in this picture statistics only ever stands for ignorance. Every molecule still has its exact $q$ and $p$; we just haven’t been told them. Keep hold of that idea. Most of this course is about why it had to go.

A box of light

Heat an empty box, an oven, to a steady temperature $T$. Its walls glow, and the inside fills with radiation. How much energy does that radiation hold?

Light trapped in a box behaves like sound trapped in an organ pipe or a wave on a guitar string: only certain patterns fit, each with its own frequency. One such pattern is called a mode. Each mode can hold more or less energy independently of the others, like a separate little oscillator.

The statistical theory of heat has a simple and well-tested rule for oscillators: left at temperature $T$ for long enough, every one of them ends up with the same average energy, $kT$, where $k$ is a fixed constant. And that is the disaster. A guitar string has a highest note it can usefully play, but there is no shortest wave of light: ever shorter waves, of ever higher frequency, always fit in the box. There are infinitely many modes, each owed $kT$. The glowing walls would go on pouring heat into them for ever and never settle down.

Real ovens do settle down, of course, and the light inside them has a perfectly finite energy, with a peak at a frequency that climbs as the oven gets hotter. That is why a poker glows red before it glows white.

Energy in lumps

In 1900 Max Planck found a way to get the measured curve. He had to assume something for which he had no justification at all: that a mode of frequency $\nu$ cannot take or give energy in any amount, but only in whole lumps of size

The constant $h$ is now called Planck’s constant. To see why lumps cure the infinity, compare the price of one lump with what the heat has to spend, which is roughly $kT$. For low-frequency modes a lump is cheap, the lumpiness hardly shows, and the old rule holds. For high-frequency modes one lump costs far more than $kT$, so the mode can almost never afford even one and stays empty. The modes are all still there, infinitely many of them, but nearly all are frozen out.

Some numbers. A lump of green light, $\nu = 5.5\times10^{14}$ Hz, is $3.6\times10^{-19}$ J. At room temperature $kT$ is about $4\times10^{-21}$ J, ninety times too little, which is why the things in your room do not glow green. At the 5,800 K of the Sun’s surface $kT$ is $8\times10^{-20}$ J, and green lumps are affordable.

Light itself comes in lumps

Planck thought of the lumps as a quirk of how matter and light trade energy. Five years later Einstein took them seriously as a property of light, and used them to explain a puzzle about metals.

The lumps were later named photons. By the 1920s it was also clear that a photon carries momentum as well as energy, and bounces off an electron as one billiard ball bounces off another. For light of wavelength $\lambda$,

A lump that is also a wave

None of this made Maxwell’s waves go away. Light still bends round edges and still shows interference, which only waves do. So it is tempting to say that a photon is just a short burst of wave: a pulse. That idea runs straight into a fact about waves that has nothing to do with physics. It is pure mathematics.

A wave with one exact wavelength goes on for ever in both directions. To build a pulse that is confined to a short stretch, you have to add together waves of many different wavelengths, arranged so that they reinforce each other in one place and cancel everywhere else. The shorter the pulse, the wider the range of wavelengths you need.

Writing $\Delta x$ for the length of the pulse and $\Delta(1/\lambda)$ for the spread of its ingredients, the figure shows $\Delta x \cdot \Delta(1/\lambda)$ staying roughly equal to 1. Now put in $p = h/\lambda$, so that a spread in $1/\lambda$ is a spread in momentum divided by $h$:

A photon confined to a small region has, unavoidably, a spread of momenta. One with a sharply defined momentum is spread out in space. It cannot be given both an exact position and an exact momentum, because there is no such pulse to be.

And so is everything else

If this were only about light it would not have troubled anyone much. In 1900 nobody thought of light as particles in the first place. Two things made it everybody’s problem.

First, Heisenberg pointed out in 1927 what happens when you look at a particle, any particle. To see where an electron is you must bounce something off it, and the gentlest thing available is a photon. The electron’s position and momentum, as found this way, can be no sharper than those of the photon used to find them. So everything observed with light inherits the same limit.

Second, in the same year electrons fired at a crystal were found to produce the patterns that waves produce, with a wavelength given by the same formula, $\lambda = h/p$. Matter is wavy too.

The conclusion drawn at the time, and still standing, is that the classical particle with its exact $q$ and exact $p$ is a fiction. It is an extremely good fiction for cricket balls, because $h$ is so small: pin a one-gram bead down to a thousandth of a millimetre and the limit on its speed is about $10^{-24}$ metres a second. For an electron in an atom it is no good at all.

A warning before going on

What replaced the classical particle is a mathematical object called a state vector, or wave function, usually written $\psi$. You will meet it properly in Part III. There is a natural temptation, having lost $q$ and $p$, to give $\psi$ the job they used to do: to say that every particle has a wave function, a real property of it whether or not anyone knows it.

Peres opens his book by refusing that move. There is no experiment that supports it, and taking it literally is what produces the famous “paradoxes” of quantum theory. His approach, and the one this course follows, is to talk only about what is done in a laboratory and what is then seen. The next lesson starts doing exactly that, with a lamp and a crystal.

This lesson follows §1-1 of the book, which covers the same ground in two pages and assumes you have met it before. If the oven argument is new to you, the term to look up is “black-body radiation”.