The Universe Within (8 page)

In 1913, the upheaval continued when Niels Bohr, working at Manchester under Ernest Rutherford, published a paper titled “On the Constitution of Atoms and Molecules.” Much as Planck had done for light, Bohr invoked quantization to explain the orbits of electrons in atoms. Just before Bohr's work, Rutherford's experiments had revealed the atom's inner structure, showing it to be like a miniature solar system, with a tiny, dense nucleus at its centre and electrons whizzing around it.

Rutherford used the mysterious alpha particles, which Marie and Pierre Curie had observed to be emitted from radioactive material, as a tool to probe the structure of the atom. He employed a radioactive source to bombard a thin sheet of gold foil with alpha particles, and he detected how they scattered. He was amazed to find that most particles went straight through the metal but a few bounced back. He concluded that the inside of an atom is mostly empty space, with a tiny object — the atomic nucleus — at its centre. Legend has it that the morning after Rutherford made the discovery, he was scared to get out of bed for fear he would fall through the floor.
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Rutherford's model of the atom consisted of a tiny, positively charged nucleus orbited by negatively charged electrons. Since unlike charges attract, the electrons are drawn into orbit around the nucleus. However, according to Maxwell's theory of electromagnetism, as the charged electrons travelled around the nucleus they would cause changing electric and magnetic fields and they would continually emit electromagnetic waves. This loss of energy would cause the electrons to slow down and spiral inward to the nucleus, causing the atom to collapse. This would be a disaster every bit as profound as the ultraviolet catastrophe: it would mean that every atom in the universe would collapse in a very short time. The whole situation was very puzzling.

Niels Bohr, working with Rutherford, was well aware of the puzzle. Just as Planck had quantized electromagnetic waves, Bohr tried to quantize the orbits of the electron in Rutherford's model. Again, he required that a quantity with the same units as Hamilton's action — in Bohr's case, the momentum of the electron times the circumference of its orbit — came in whole-number multiples of Planck's constant. A hydrogen atom is the simplest atom, consisting of just one electron in orbit around a proton, the simplest nuclear particle. One quantum gave the innermost orbit, two the next orbit, and so on. As Bohr increased the number of quanta, he found his hypothesis predicted successive orbits, each one farther from the nucleus. In each orbit, the electron has a certain amount of energy. It could “jump” from one orbit to another by absorbing or emitting electromagnetic waves with just the right amount of energy.

Experiments had shown that atoms emitted and absorbed light only at certain fixed wavelengths, corresponding through Planck's rule to fixed packets of energy. Bohr found that with his simple quantization rule, he could accurately match the wavelengths of the light emitted and absorbed by the hydrogen atom.

· · ·

PLANCK, EINSTEIN, AND BOHR'S
breakthroughs had revealed the quantum nature of light and the structure of atoms. But the quantization rules they imposed were
ad hoc
and lacked any principled basis. In 1925, all that changed when Heisenberg launched a radically new view of physics with quantization built in from the start. His approach was utterly ingenious. He stepped back from the classical picture, which had so totally failed to make sense of the atom. Instead, he argued, we must build the theory around the only directly observable quantities — the energies of the light waves emitted or absorbed by the orbiting electrons. So he represented the position and momentum of the electron in terms of these emitted and absorbed energies, using a technique known as “Fourier analysis in time.”

At the heart of Fourier's method is a strange number called
i
, the imaginary number, the square root of minus one. By definition,
i
times
i
is minus one. Calling
i
“imaginary” makes it sound made up. But within mathematics
i
is every bit as definite as any other number, and the introduction of
i,
as I shall explain, makes the numbers more complete than they would otherwise be. Before Heisenberg, physicists thought of
i
as merely a convenient mathematical trick. But in Heisenberg's work,
i
was far more central. This was the first indication of reality's imaginary aspect.

The imaginary number
i
entered mathematics in the sixteenth century, during the Italian Renaissance. The mathematicians of the time were obsessed with solving algebraic equations. Drawing on the results of Indian, Persian, and Chinese mathematicians before them, they started to find very powerful formulae. In 1545, Gerolamo Cardano summarized the state of the art in algebra, in his book
Ars Magna
(
The Great Art
). He was the first mathematician to make systematic use of negative numbers. Before then, people believed that only positive numbers made sense, since one cannot imagine a negative number of objects or a negative distance or negative time. But as we all now learn in school, it is often useful to think of numbers as lying on a number line, running from minus infinity to plus infinity from left to right, with zero in the middle. Negative numbers can be added, subtracted, multiplied, or divided just as well as positive numbers can.

Cardano and others had found general solutions to algebraic equations, but sometimes these solutions involved the square root of a negative number. At first sight, they discarded such solutions as meaningless. Then Scipione del Ferro invented a secret method of pretending these square roots made sense. He found that by manipulating the formulae he could sometimes get these square roots to cancel out of the final answer, allowing him to find many more solutions of equations.

There was a great deal of intrigue over this trick, because the mathematicians of the time held public contests, sponsored by wealthy patrons, in which any advantage could prove lucrative. But eventually the trick was published, first by Cardano and then more completely by Rafael Bombelli. In his 1572 book, simply titled
Algebra
,

Bombelli systematically explained how to extend the rules of arithmetic to include
i
.
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You can add, subtract, multiply, or divide it with any ordinary number. When you do, you will obtain numbers like
x
+
iy
, where
x
and
y
are ordinary numbers. Numbers like this, which involve
i
, are called “complex numbers.” Just as we can think of the ordinary numbers as lying on a number line running from negative to positive values, we can think of the complex numbers as lying in a plane, where
x
and
y
are the horizontal and vertical coordinates. Mathematicians call this the “complex plane.” The number zero is at the origin and any complex number has a squared length, given by Pythagoras's rule as
x
2
+
y
2
.

Then it turns out, rather beautifully, that any complex number raised to the power of any other complex number is also a complex number. There are no more problems with square roots or cube roots or any other roots. In this sense, the complex numbers are
complete
: once you have added
i
, and any multiple of
i
, to the ordinary numbers, you do not need to add anything else. And later on, mathematicians proved that when you use complex numbers,
every
algebraic equation has a solution. This result is called the “fundamental theorem of algebra.” To put it simply, the inclusion of
i
makes algebra a far more beautiful subject than it would oterhwise be.

And from this idea came an equation that Richard Feynman called “the most remarkable formula in mathematics.”
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It was discovered by Leonhard Euler, one of the most prolific mathematicians of all time. Euler was the main originator and organizer of the field of analy­sis — the collection of mathematical techniques for dealing with infinities. One of his many innovations was his use of the number
e
, which takes the numerical value 2.71828 . . . and which arises in many areas of mathematics. It describes exponential growth and is used in finance for calculating compound interest or the cumulative effects of economic inflation, in biology for the multiplication of natural populations, in information science, and in every area of physics. What Euler found is that
e
raised to
i
times an angle gives the two basic trigonometric functions, the sine and cosine. His formula connects algebra and analysis to geometry. It is used in electrical engineering for the flow of AC currents and in mechanical engineering to study vibrations; it is also used in music, computer science, and even in cosmology. In Chapter Four, we shall find Euler's formula at the heart of our unified description of all known physics.

Heisenberg used Euler's formula (in the form of a Fourier series in time) to represent the position of an electron as a sum of terms involving the energy states of the atom. The electron's position became an infinite array of complex numbers, with every number representing a connection coefficient between two different energy states of the atom.

The appearance of Heisenberg's paper had a dramatic effect on the physicists of the time. Suddenly there was a mathematical formalism that explained Bohr's rule for quantization. However, within this new picture of physics, the position or velocity of the electron was a complex matrix, without any familiar or intuitive interpretation. The classical world was fading away.

Not long after Heisenberg's discovery, Schrödinger published his famous wave equation. Instead of trying to describe the electron as a point-like particle, Schrödinger described it as a wave smoothly spread out over space. He was familiar with the way in which a plucked guitar string or the head of a drum vibrates in certain specific wave-like patterns. Developing this analogy, Schr
ö
dinger found a wave equation whose solutions gave the quantized energies of the orbiting electron in the hydrogen atom, just as Heisenberg's matrices had done. Heisenberg's and Schrödinger's pictures turned out to be mathematically equivalent, though most physicists found Schrödinger's waves more intuitive. But, like Heisenberg's matrices, Schrödinger's wave was a complex number. What on earth could it represent?

Shortly before the Fifth Solvay Conference, Max Born proposed the answer: Schrödinger's wavefunction was a “probability wave.” The probability to find the particle at any point in space is the squared length of the wavefunction in the complex plane, given by the Pythagorean theorem. In this way, geometry appeared at the heart of quantum theory, and the weird complex numbers that Heisenberg and then Schrödinger had introduced became merely mathematical tools for obtaining probabilities.

This new view of physics was profoundly dissatisfying to physicists like Einstein, who wanted to visualize concretely how the world works. In the run-up to the Solvay meeting, all hope of that was dashed. Heisenberg published his famous uncertainty principle, showing that, within quantum theory, you could not specify the position and velocity of a particle at the same time. As he put it, “The more precisely the position [of an electron] is determined, the less precisely the momentum is known in this instant, and vice versa.”
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If you know
exactly
where a particle is now, then you cannot say
anything
about where it will be a moment later. The very best you can hope for is a fuzzy view of the world, one where you know the position and velocity approximately.

Heisenberg's arguments were based on general principles, and they applied to any object, even large ones like a ball or a planet. For these large objects, the quantum uncertainty represents only a tiny ambiguity in their position or velocity. However, as a matter of principle, the uncertainty is always there. What Heisenberg's uncertainty principle showed is that, in quantum theory, nothing is as definite as Newton, or Maxwell, or any of the pre-quantum physicists had supposed it to be. Reality is far more slippery than our classical grasp of it would suggest.

ONE OF THE MOST
beautiful illustrations of the quantum nature of reality is the famous “double-slit experiment.” Imagine placing a partition with two narrow, parallel slits in it, between a source of light of one colour — like a green laser — and a screen. Only the light that falls on a slit will pass through the partition and travel on to the screen. The light from each slit spreads out through a process called “diffraction,” so that each slit casts a broad beam of light onto the screen. The two beams of light overlap on the screen
(click to see image)
.

However, the distance the light has to travel from either slit to each point on the screen is in general different, so that when the light waves from both slits arrive at the screen, they may add or they may cancel. The pattern of light formed on the screen is called an “interference pattern”: it consists of alternate bright and dark stripes at the locations where the light waves from the two slits add or cancel.
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Diffraction and interference are clas
sic examples of wave-like behaviour, seen not only in light but in water waves, sound waves, radio waves, and so on.

Now comes the quantum part. If you dim the light source and replace the screen with a detector, like a digital camera sensitive enough to detect individual photons — Planck's quanta of light — then you can watch the individual photons arrive. The light does not arrive as a continuous beam with a fixed intensity. Instead, the photons arrive as a random string of energy packets, each one announcing its arrival at the camera with a flash. The pattern of flashes still forms interference stripes, indicating that even though each photon of light arrived in only one place as an energy packet, the photons travelled through both slits and interfered as waves.