Why Beauty is Truth Page 24
Max met Marie Merck, the sister of a friend, and in 1887 they married and rented an apartment. In all, they had four children: Karl, twins Emma and Grete, and Erwin.
In 1889, the year the twins were born, Max was appointed to Kirchhoff’s position in Berlin, becoming a full professor in 1892. The family moved to a villa in the Grunewald region of Berlin, close to a number of other leading academics. One, the theologian Adolf von Harnack, became a close friend. The Plancks were sociable, and famous intellectuals visited their house regularly. These included Einstein and the physicists Otto Hahn and Lise Meitner, who later made fundamental discoveries about nuclear fission, part of the long development leading to the atomic bomb and nuclear power stations. At these events the Plancks continued a tradition of playing music, started by Helmholtz.
For a time life was rosy, but Marie contracted a lung disease, possibly tuberculosis, and died in 1909. A year and a half later, at 52, Max remarried, this time to Marga von Hoesslin, with whom he had a third son, Hermann.
In 1894, a local electrical company was trying to develop a more efficient light bulb, so Max started some industrial contract research. Theoretically, the analysis of a light bulb was a standard physics problem known as “blackbody radiation”—how light would be emitted by a perfectly nonreflective body. Such a body, when heated, emits light of all frequencies, but the intensity of the light, or equivalently its energy, varies with the frequency. A fundamental question was, how does the frequency affect the intensity? Without such basic data, it would be difficult to invent a better light bulb.
There were good experimental results, and one theoretical law, the Rayleigh–Jeans law, had been derived from basic principles of classical physics. Unfortunately, this law disagreed with experiment at high frequencies. In fact, it predicted something impossible: as the light’s frequency increases, its energy should become infinitely large. This impossible result became known as the “ultraviolet catastrophe.” Further experiments led to a new law, which fitted the observations for high-frequency radiation, known as Wien’s law after its discoverer, Wilhelm Wien.
However, Wien’s law went wrong for low-frequency radiation.
Physicists were faced with two laws: one working at low frequencies but not at high ones, the other doing the exact opposite. Planck hit on the idea of interpolating between the two: that is, writing down a mathematical expression that approximated the Rayleigh–Jeans law at low frequencies and Wien’s law at high frequencies. The resulting formula is now called the Planck law for blackbody radiation.
This new law was deliberately designed to match experiments beautifully, across the entire spectrum of electromagnetic radiation, but it was purely empirical—derived from experiments, not from any basic physical principle. Planck, pursuing his avowed intention to understand known physics better, was dissatisfied, and he devoted much effort searching for physical principles that would lead to his formula.
Eventually, in 1900, Planck noticed a curious feature of his formula. He could derive it by much the same calculation that Rayleigh and Jeans had employed, provided he made one tiny change. The classical derivation had assumed that for any given frequency, the energy of electromagnetic radiation could in principle take any value whatsoever. In particular, it could get as close to zero as you wished. Planck realized that this assumption was the cause of the ultraviolet catastrophe, and that if he made a different assumption, that troublesome infinity disappeared from the calculation.
The assumption, though, was radical. The energy of radiation of a given frequency had to come as a whole number of “packets” of fixed size. In fact, the size of each packet had to be proportional to the frequency—that is, equal to the frequency multiplied by some constant, which we now call Planck’s constant and write using the symbol h.
These energy packets were called quanta (singular: quantum). Planck had quantized light.
All very well, but why had experimentalists never noticed that the energy was always a whole number of quanta? By comparing his calculations with the observed energies, Planck was able to calculate the size of his constant, and it turned out to be very, very small. In fact, h is roughly 6 × 10–34 joule-seconds. Roughly speaking, to notice the “gaps” in the possible range of energies—the values that classical physics permitted but quantum physics did not—you had to make observations that were accurate to the 34th decimal place. Even today, very few physical quantities can be measured to more than six or seven decimal places, and in those days three was asking a lot. Direct observation of quanta required absurd levels of accuracy.
It may seem strange that a mathematical difference so tiny that it can never be seen could have such a huge effect on the radiation law. But the calculation of the law involves adding up all the contributions to the energy from all possible frequencies. The result is a collective effect of all possible quanta. From the Moon you can’t spot an individual grain of sand on Earth. But you can see the Sahara. If sufficiently many very tiny units combine, the result can be huge.
Planck’s physics thrived, but his personal life was filled with tragedy. His son Karl was killed in action during the First World War. His daughter Grete died in childbirth in 1917, and Emma suffered the same fate in 1919, having married Grete’s widower. Much later, Erwin was executed by the Nazis for taking part in the unsuccessful 1944 attempt to assassinate Adolf Hitler.
By 1905, new evidence had turned up that supported Planck’s radical proposal, in the form of Einstein’s work on the photoelectric effect. Recall that this is the discovery that light can be converted into electricity. Einstein was aware that electricity comes in discrete packages. Indeed, by then physicists knew that electricity is the motion of tiny particles called electrons. From the photoelectric effect, Einstein deduced that the same must be true of light. This not only verified Planck’s ideas about light quanta, it explained what the quanta are: light waves, like electrons, must be particles.
How can a wave be a particle? Yet that was the unequivocal message of the experiments. The discovery of particles of light, or photons, quickly led to the quantum picture of the world in which particles are really waves, behaving sometimes like one, sometimes like the other.
The physics community started to take quanta more seriously. The great Danish physicist Niels Bohr came up with a quantized model of the atom, in which electrons move in circular orbits around a central nucleus, with the size of the circle being limited to discrete quanta. The French physicist Louis de Broglie reasoned that since photons can be both waves and particles, and electrons are emitted by suitable metals when they are impacted by photons, then electrons must also be both waves and particles. Indeed, all matter must possess this dual existence—sometimes solid particle, sometimes undulating wave. That’s why experiments can indicate either form.
Neither “particle” nor “wave” really describes matter at extremely tiny scales. The ultimate constituents of matter are a bit of both—wavicles. De Broglie invented a formula to describe wavicles.
Now came a key step, essential to our story. Erwin Schrödinger took de Broglie’s formula and turned it into an equation that describes how wavicles move. Just as Newton’s laws of motion were fundamental to classical mechanics, Schrödinger’s equation became fundamental to quantum mechanics.
Erwin was born in Vienna in 1886, the offspring of a mixed marriage. His father, Rudolf Schrödinger, manufactured cerecloth, a waxy cloth used to make shrouds for the dead; he was also a botanist. Rudolf was a Catholic, while Erwin’s mother, Georgine Emilia Brenda, was a Lutheran. From 1906 to 1910 Erwin studied physics in Vienna under Franz Exner and Friedrich Hasenöhrl, becoming Exner’s assistant in 1911. He gained his habilitation in 1914, at the start of World War I, and spent the war as an officer in the Austrian artillery. Two years after the war ended, he married Annemarie Bertel. In 1920 he became the equivalent of an associate professor in Stuttgart, and by 1921 he was a full professor in Breslau, now the city Wrocław, in Poland.
He publi
shed the equation that is now named after him in 1926, in a paper showing that it gives the correct energy levels for the spectrum of the hydrogen atom. This was quickly followed by three other major papers on quantum theory. In 1927, he joined Planck in Berlin, but in 1933, upset by the anti-Semitism of the Nazis, he left Germany for Oxford, where he was made a fellow of Magdalen College. Not long after he arrived, he and Paul Dirac were awarded the Nobel Prize in physics.
Schrödinger maintained a scandalously unorthodox lifestyle, living with two women, and this offended the tender sensibilities of the Oxford dons. Within a year he had moved again, this time to Princeton, where he was offered a permanent position, but he decided not to accept—possibly because his attachment to both wife and mistress in the same household didn’t go down any better in Princeton than it did in Oxford. Eventually, he settled in Graz, Austria, in 1936, and ignored the opinions of straitlaced Austrians.
Hitler’s occupation of Austria caused severe difficulties for Schrödinger, a known Nazi opponent. He publicly rejected his earlier views (and much later apologized to Einstein for doing so). The ploy didn’t work: he lost his job because he was politically unreliable, and had to flee to Italy.
Schrödinger finally settled in Dublin. The year 1944 saw the publication of What Is Life? an intriguing but flawed attempt to apply quantum physics to the problem of living creatures. He based his ideas on the concept of “negentropy,” the tendency of life to disobey—or somehow subvert—the second law of thermodynamics. Schrödinger emphasized that the genes of living creatures must be some kind of complicated molecule, containing coded instructions. We now call this molecule DNA, but its structure was discovered only in 1953, by Francis Crick and James Watson—inspired, in part, by Schrödinger.
In Ireland, Schrödinger retained his relaxed attitude to sexuality, getting involved with students and fathering two children by different mothers. He died of tuberculosis in Vienna in 1961.
Schrödinger is best known for his cat. Not a real cat, but one that appeared in a thought experiment. It is generally interpreted as a reason for not considering Schrödinger’s waves to be real physical things. Instead, they are thought of as a behind-the-scenes description that can never be verified experimentally but that has the right consequences. However, this interpretation is controversial—if the waves do not exist, why do their consequences all work out so nicely?
Anyway, back to the cat. According to quantum mechanics, wavicles can interfere with each other, piling up on top of each other and reinforcing when peak meets peak, and canceling each other out when peak meets trough. This type of behavior is called “superposition,” so quantum wavicles can superpose—implying that they can contain a variety of potential states without fully existing in any of them. Indeed, according to Bohr and the famous “Copenhagen interpretation” of quantum theory, that is the natural state of affairs. Only when we observe some physical quantity do we force it out of some quantum superposition and into a single “pure” state.
This interpretation works well for electrons, but Schrödinger wondered what it would imply for a cat. In his thought experiment, a cat locked in a box can be in a superposition of the states alive and dead. When you open the box, you observe the cat and force it into either one state or the other. As Pratchett noticed in Maskerade, cats aren’t like that. Greebo, a hypermacho cat, emerges from a box in a third state: absolutely bloody furious.
Schrödinger also knew that cats aren’t like that, though for different reasons. An electron is a submicroscopic entity, and it behaves like something on the quantum level. It possesses (when we measure it) a particular position or velocity or spin that can be described relatively simply. A cat is macroscopic, and it doesn’t. You can superpose electron states, but not cats. My wife and I have two cats, and when they try to superpose, the result is flying fur and two highly indignant cats. The jargon term here is “decoherence,” which explains why large quantum systems like cats look like the familiar “classical” systems in our daily lives. Decoherence tells us that the cat contains so many wavicles that they all get tangled up together and ruin the superposition in less time than light can travel the diameter of an electron. So cats, being macroscopic systems composed of an absolutely gigantic number of quantum particles, behave like cats. They can be alive, or dead, but not both at once.
Nonetheless, on suitably small scales—and we are talking very small stuff here, not anything you can see in a normal microscope—the universe behaves just as quantum physics says it does, and it can do two different things at the same time. And that changes everything.
Just how strange the quantum world must be emerged from the research of Werner Heisenberg. Heisenberg was a brilliant theoretical physicist, but his grasp of experiments was so poor that during his examination for the doctorate he couldn’t answer simple questions about telescopes and microscopes. He didn’t even know how a battery worked.
August Heisenberg married Anna Wecklein in 1899. He was a Lutheran, she a Catholic, and she converted to his religion to make the marriage possible. They had much in common: he was a teacher and an expert classicist specializing in ancient Greek, while she was a head teacher’s daughter and an expert on the Greek tragedies. Their first son, Erwin, was born in 1900 and became a chemist. Their second, Werner, was born in 1901 and changed the world.
Germany was still a monarchy at this time, and the teaching profession carried high social status, so the Heisenbergs were financially comfortable and could send their sons to good schools. In 1910, August was made professor of medieval and modern Greek at the University of Munich, to which city the family moved. In 1911, Werner started at the King Maximilian School in Munich, where Planck had also studied. Werner’s grandfather, Nikolaus Wecklein, was the school principal. The boy was bright and quick, partly because his father encouraged him to compete with his elder brother, and showed remarkable abilities in math and science. He had musical talent too, and learned the piano so well that by the age of 12 he was performing in school concerts.
Heisenberg later wrote that “both my interests in languages and in mathematics were awakened rather early.” He earned top grades in Greek and Latin and did well in mathematics, physics, and religion. His worst subjects were athletics and German. His mathematics teacher, Christoph Wolff, was excellent, and stretched Werner’s abilities by setting him special problems to solve. Soon the pupil had outclassed the teacher, and Heisenberg’s school report stated, “With his independent work in the mathematical-physical field he has come far beyond the demands of the school.” He taught himself relativity, preferring its mathematical content to its physical implications. When his parents asked him to tutor a local college student for her exams, he taught himself calculus, a subject not included in the school curriculum. He developed an interest in number theory, saying that “it’s clear, everything is so that you can understand it to the bottom.”
To help Werner improve his Latin, his father brought him some old papers on mathematics, written in that language. Among them was Kronecker’s dissertation on a topic (“complex units”) in algebraic number theory. Kronecker, a world-class number theorist, famously believed that “God created the integers—all else is the work of Man.” Heisenberg was inspired to have a go at proving Fermat’s Last Theorem. After nine years in the school he graduated at the top of his class and attended the University of Munich.
When World War I broke out, the Allies blockaded Germany. Food and fuel were in very short supply; the school had to be closed because it could not be heated, and on one occasion Werner was so weak from starvation that he fell off his bike into a ditch. His father and his teachers were fighting in the army; the young men who remained behind received military training and nationalistic indoctrination. The end of the war brought the end of the German monarchy as well, and Bavaria briefly had a socialist government along Soviet lines, but in 1919 German troops from Berlin kicked out the socialists and restored a more moderate social democracy.
Like most
of his generation, Werner was disillusioned by Germany’s defeat and blamed his elders for their military failure. He became the leader of a group associated with the New Boy Scouts, an extremist right-wing organization that aimed to restore the monarchy and dreamed of a Third Reich. Many branches of the New Boy Scouts were anti-Semitic, but Werner’s group included a number of Jewish boys. He spent a lot of time with his boys, camping and hiking and generally trying to recapture a romantic vision of Germany as it once had been, but these activities ended in 1933 when Hitler banned all youth organizations other than those he had set up himself.
In 1920, Werner went to the University of Munich, intending to become a pure mathematician until an interview with one of the pure math professors put him off the idea. He decided instead to study physics under Arnold Sommerfeld. Immediately recognizing Werner’s abilities, Sommerfeld allowed him to attend advanced classes. Soon Werner had done some original research on the quantum approach to atomic structure. His doctorate was awarded in 1923, breaking the university’s record for speed. In the same year, Hitler tried to overthrow the Bavarian government in the “beer hall putsch,” intended as a prelude to a march on Berlin, but the attempt failed. Hyperinflation was rampant; Germany was coming to pieces.
Werner continued working. He collaborated with many leading physicists, all of whom were thinking about quantum theory because that was where the action was. He worked with Max Born to devise a better theory of the atom. It occurred to Heisenberg to represent the state of an atom in terms of the frequencies observed in its spectrum—the kinds of light that it emitted. He boiled this idea down to a peculiar kind of mathematics involving lists of numbers. Born eventually realized that this kind of list was actually quite respectable: mathematicians called it a matrix. Happy that the ideas made sense, Born sent the paper off for publication. As the ideas developed, they matured into a new, systematic mathematics of quantum theory: matrix mechanics. It was seen as a competitor to Schrödinger’s wave mechanics.