Baggott J. The Quantum Story: A History in 40 Moments

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the quantum story

perform an observable random motion that is produced by thermal

motion. Such movement of suspended bodies has actually been observed

by physiologists who call it Brownian motion. The fourth paper is only

a rough draft at this point, and is an electrodynamics of moving bodies

which employs a modiﬁ cation of the theory of space and time.

Any one of these four papers would grant Einstein lasting recognition, even

fame. The ﬁ rst of these, written in March 1905, contains the bold assertion

that Planck’s radiation law only makes sense if radiation itself is thought to

be composed of discrete packets of energy which he called light-quanta.

The second and third, written in April and May, deal with the physical

reality of molecules and some observable consequences of molecular

motions.

In the fourth, which Einstein would complete in June, he derived

his special theory of relativity. In a subsequent ﬁ fth paper, a supplement to

the June special relativity paper which was published in September 1905,

he would go on to derive his most famous equation, E = mc

This was Einstein’s annus mirabilis, his miracle year. He was just 26 years old.

The problem with Planck’s derivation of his radiation law was that it was

inconsistent. Planck had ﬁ rst assumed energy to be continuously vari-

able, as the principles of classical physics demanded, and then he had

assumed precisely the opposite by distributing the ﬁ xed energy quanta

over the oscillators of the cavity material.

In June 1900 the English physicist Lord Rayleigh (William Strutt)

had published details of another theoretical model for cavity radiation

and had derived a radiation law that was somewhat different from both

Wien’s and Planck’s laws.

Rayleigh had applied the principles of classical

physics in ways that Planck had not. An error in Rayleigh’s calculation

was corrected by fellow Englishman James Jeans in June 1905. The result

is now known as the Rayleigh–Jeans law.

He submitted the second paper as a doctoral dissertation to the University of Zurich in

July 1905 (at that time the Zurich Polytechnic was unable to confer doctoral degrees). Einstein

maintained that his dissertation was ﬁ rst rejected because it was too short. Einstein added a

single sentence and resubmitted it, and it was accepted. Einstein became ‘Herr Doctor’. See

Isaacson, p. 103.

Rayleigh’s paper was published some months before Planck’s October paper on his radia-

tion law, but Planck had been unaware of Rayleigh’s result at the time.

annus mirabilis

Rayleigh’s reasoning and use of thermodynamic principles was both

logical and convincing, but the result was little short of disastrous.

The Rayleigh–Jeans law implies that the intensity of cavity radiation

should increase in proportion to the square of the radiation frequency

without limit. The law predicts that the total emitted energy should

quickly increase to absurd levels at high radiation frequencies, or short

wavelengths.

The Rayleigh–Jeans law provided a relatively close ﬁ t to

the experimental data at low frequencies, where Wien’s law failed, and

Wien’s law provided a reasonable ﬁ t to the data at high frequencies,

where the Rayleigh–Jeans law failed. Both were clearly limiting cases of

Planck’s law, which ﬁ t the data at all frequencies.

It seemed that a truly classical derivation produced laws that failed.

Only Planck’s mysterious non-classical derivation could produce a law

that worked.

Einstein proved this much to himself, independently deriving the

correct form for the Rayleigh–Jeans law. This was puzzling enough, but

Einstein was in pursuit of a bigger prize. He opened his March 1905 paper

with the statement:

A profound formal difference exists between the theoretical concepts that

physicists have formed about gases and other ponderable bodies, and

Maxwell’s theory of electromagnetic processes in so-called empty space.

While we consider the state of a body to be completely determined by the

positions and velocities of an indeed very large yet ﬁ nite number of atoms

and electrons, we make use of continuous spatial functions to determine

the electromagnetic state of a volume of space, so that a ﬁ nite number of

variables cannot be considered as sufﬁ cient for the complete determina-

tion of the electromagnetic state of space.

The evidence for atoms and molecules was becoming overwhelming and

this particulate view of matter was now in the ascendancy. Einstein was

pointing to the dichotomy created by models of particulate matter on the

one hand and waves of radiation described by Maxwell’s electromagnetic

ﬁ eld theory on the other.

In 1911 the Austrian physicist Paul Ehrenfest called this problem the ‘Rayleigh–Jeans catas-

trophe in the ultraviolet’, now commonly known as the ultraviolet catastrophe.

the quantum story

Einstein was about to make a very bold move, one that he later told his

friend Habicht was ‘very revolutionary’. His problem was that this was a

move that could not be justiﬁ ed from within the framework of the classical

theory, from within the constraints that led to the Rayleigh–Jeans law.

He solved this problem by introducing what he called the ‘heuristic

principle’. He suggested, purely as an unproven hypothesis, that:

If monochromatic radiation (of sufﬁ ciently low density) behaves, as con-

cerns the dependence of its entropy on volume, as though the radiation

were a discontinuous medium consisting of energy quanta of magnitude

[hn], then it seems reasonable to investigate whether the laws governing

the emission and transformation of light are also constructed such as if

light consisted of such energy quanta.

Two hundred years after Newton, Einstein was ready to return to a ‘cor-

puscular’ theory of light. He was ready to assume that: ‘. . . in the propaga-

tion of a light ray emitted from a point source, the energy is not distributed

continuously over ever-increasing volumes of space, but consists of a ﬁ nite

number of energy quanta localised at points of space that move without

dividing, and can be absorbed or generated only as complete units.’

Einstein was not proposing to abandon the wave theory of radia-

tion completely. There was simply too much experimental evidence for

the wave properties of light—phenomena such as light diffraction and

interference could only be explained in terms of a wave model. Einstein

proposed to reconcile these two contrasting descriptions by recognizing

that wave phenomena were the result of time-averaged observations. So,

wave interference reﬂ ects not the instantaneous ‘snapshot’ of the motion

of individual, localized light-quanta, but rather the collective motions of

many light-quanta statistically averaged over time.

Although unproven, Einstein proposed to use his light-quantum hypoth-

esis to solve other problems in physics unconnected with cavity radiation

and Planck’s law. He now turned his attention to the photoelectric effect.

This was another phenomenon that had been puzzling physicists

for some time. It was known that shining light on metal surfaces could

result in the ejection of electrons from these surfaces. In the wave model

of light, the light energy is proportional to the amplitude (or intensity)

of the light wave. So, if the intensity of the light is increased, it seemed

annus mirabilis

The slope of such a graph actually gives the value of Planck’s constant.

sensible to assume that the energy of the ejected electrons would increase

as a consequence. But this is not what was found in the laboratory.

Only the number of ejected electrons was found to increase with increas-

ing light intensity, not their individual energies. Instead, the energy of

the electrons was found to increase with increasing light frequency, com-

pletely at odds with accepted theories of electromagnetic radiation.

Einstein solved this problem by suggesting that a light-quantum inci-

dent on the surface transfers all its energy to a single electron. That elec-

tron is ejected with an energy equal to the energy of the light-quantum

less an amount expended by escaping the surface, and which is therefore

characteristic of the metal (a property now known as the work function).

Einstein now took Planck’s equation, e = hn, and invested it with a com-

pletely new meaning. Instead of representing a convenient mathematical

relationship between the ﬁ xed energy quanta and the frequency of the

oscillators, as Planck had done, Einstein assumed it to represent the

relationship between the energy of the light-quantum and its (radiation)

frequency. In doing this Einstein divorced Planck’s relationship completely

from the problem of cavity radiation. He made it a fundamental property

of light relevant in all situations.

By imparting an energy hn to an electron in the metal, each light-quantum

would produce an ejected electron whose energy could be expected to

increase with increasing radiation frequency, n. Increasing the intensity

of the radiation would increase the number of light quanta incident on

the metal surface, increasing the number of ejected electrons, but not

their energies.

Einstein’s theory was very simple, and yet it made a number of impor-

tant, testable predictions. He veriﬁ ed that the voltage of photo-electricity

produced in such an experiment was of the same order of magnitude as

voltages already observed in experiments carried out by German physi-

cist Phillipp Lenard in 1902. He then went on to predict that a graph of

the voltage of photo-ejected electrons versus the frequency of the incident

light would be a straight line, with a slope that would be independent of

the nature of the metal used in the experiments.

the quantum story

The physics community was not at all sure what to make of this. Ein-

stein’s genius, revealed in his ﬁ ve 1905 papers, was clear. Planck became

an enthusiastic supporter of the special theory of relativity, but baulked at

the idea that light energy could be quantized. As far as he was concerned,

the counting procedure he had used in his derivation of the radiation law

was a mathematical convenience, with no grounding in reality. Einstein

had gone too far.

When Einstein was subsequently recommended for membership of

the prestigious Prussian Academy of Sciences, its leading members—

Planck among them—acknowledged his remarkable contributions to

physics. But they also acknowledged what they took to be errors in his

judgement, errors that could be forgiven. ‘That he may sometimes have

missed the target in his speculations, as, for example, in his hypothesis

of light-quanta, cannot be really held against him, for it is not possible

to introduce really new ideas even in the most exact sciences without

sometimes taking a risk.’

Planck and Einstein entered into a lively correspondence. Planck was pre-

pared to concede that the appearance of ﬁ xed energy quanta might be the

result of some as yet unknown internal properties of the atoms in the cavity

material. In other words, the ﬁ xed energy elements did not arise because

light is composed of such elements, they arise because only ﬁ xed energy

elements could be emitted by the atoms that formed the cavity itself.

Planck had lit a slow-burning fuse in December 1900. Five years later

the quantum revolution was begun, but it had begun with the merest

whisper. Reaction to the light-quantum hypothesis had been strongly

negative. Einstein was very aware of the potential for conﬂ ict between the

light-quantum and the wave theory of light, and this made him cautious,

although he remained unrepentant.

In 1906 Einstein was promoted, to ‘technical expert, second class’.

Einstein’s vision was broader even than the light-quantum. He anticipated that the quan-

tization of energy might be a much more universal phenomenon and in 1907 went on to

develop a quantum theory of the heat capacities of crystalline solids. Although nobody

took this work very seriously at the time, by 1909 new experimental results were forcing

the physics community to sit up and take notice.

In 1910 the eminent German scientist Walther Nernst chose to pay Einstein a visit (by

this time Einstein was back in Zurich, at the ETH,

but was still a relative unknown). The

visit by Nernst encouraged greater respect for Einstein and his work, with one Zurich col-

league remarking: ‘This Einstein must be a clever fellow, if the great Nernst comes so far

from Berlin to Zurich to talk to him.’ Nernst was instrumental in drawing attention to

Einstein’s quantum approach and from early 1911 a growing number of scientists began

to cite Einstein’s papers and embrace quantum ideas.

In the meantime, atoms had evolved from hypothetical entities, dismissed by some

physicists as the result of wildly speculative metaphysics, into the objects of detailed labo-

ratory studies. The discovery of the negatively charged electron by English physicist Joseph

John Thomson in 1897 implied that atoms, indivisible for more than 2000 years, now

had to be recognized as having some kind of internal structure.

A Little Bit of Reality

Manchester, April 1913

The Eidgenossische Technische Hochschule. This was the same Zurich Polytechnic where

Einstein had completed his graduate studies, by now restructured as a university and able to

confer PhD degrees. It was named the ETH in 1911.

the quantum story

Alpha particles are the result of a certain type of radioactivity. They are the positively

charged nuclei of helium atoms, consisting of two protons and two neutrons (although this

was not understood at the time of Rutherford’s experiments).

If the dimensions of the atom are compared to those of our solar system, and Pluto (admit-

tedly, no longer classiﬁ ed as a planet) were to be thought of as the atom’s outermost electron,

then the atomic nucleus would have a radius about a tenth of the radius of the sun.

Further secrets of this structure were revealed by New Zealander Ernest Rutherford in

1909–11. Together with his research associates Hans Geiger and Ernest Marsden in Man-

chester, he had been conducting experiments in which energetic alpha-particles,

emit-

ted with high velocity through the breakdown of certain radioactive elements, were ﬁ red

through thin gold foil. To their astonishment, they found that about one in 8000 alpha-

particles was deﬂ ected by the foil, sometimes by more than ninety degrees. This was no less

startling than observing high velocity machine-gun bullets deﬂ ected by tissue paper.

These results were subsequently interpreted by Rutherford to mean that most of the

atom’s mass is concentrated in a small central nucleus, with the much lighter electrons

orbiting the nucleus much like the planets orbit the sun. According to this model, the

atom is largely empty space.

As a visual image of the internal structure of the atom,

Rutherford’s planetary model remains compelling to this day.

It was the theory of the electron that occupied the thoughts of the 25-year

old Danish physicist Niels Bohr in September 1911 as he left Denmark

on the ferry through the Storebælt, the strait between the islands of Zea-

land and Fyn. He was bound for England. He had left behind Margrethe

Nørlund, whom he had ﬁ rst met in 1909 and to whom he had become

engaged in the summer of 1910. Clutching a poor English translation of

his PhD thesis and a stipend from the Carlsberg Foundation, Bohr was

heading for Cambridge to work in J.J. Thomson’s laboratory.

In the early years of the twentieth century, Cambridge was one of the

leading centres of theoretical and experimental physics and Thomson

was universally admired not only for his contributions to science but

also for his irrepressible enthusiasm. He had won the 1906 Nobel Prize

for physics for his discovery of the electron and had since immersed

himself in the theory of atomic structure. His often disastrous encoun-

ters with experimental apparatus effectively ruled him out of any form

of ‘hands-on’ experimentation (he once declared all the glass in his

laboratory to be bewitched).

a little bit of reality

Thomson was determined to explain the behaviour of atoms and mol-

ecules in terms of the particle he had discovered. The result of his endeav-

ours was a theoretical model of atoms consisting of uniform sphere of

weightless positive charge in which hundreds of negatively charged elec-

trons are embedded, like currants in a cake. In this model, the electrons

are largely responsible for the mass of the atom.

The model was not without its problems. Whilst he could devise some

stable conﬁ gurations of motionless electrons embedded in the posi-

tively charged medium, he suspected that it was the motion of electrons

inside atoms that was responsible for the properties of magnetic materi-

als. However, any model involving moving electrons was predicted to be

inherently unstable.

Thomson had been forced to re-think this model when in 1910 experi-

ments at the Cambridge laboratory showed that he had greatly over-

estimated the number of electrons present in each atom. There were

not hundreds, as he had originally proposed. There were considerably

fewer.

Bohr went to Cambridge because he judged it to be the centre of

physics and Thomson a wonderful man. Unfortunately, their relation-

ship got off to a poor start from which it seems never to have recovered.

As a young postdoctoral student, Bohr’s grasp of the English language

was poor. Though always polite and courteous, Bohr’s manner could

sometimes appear brusque and was open to misinterpretation. His ﬁ rst

meeting with Thomson was not auspicious. He entered Thomson’s ofﬁ ce

with a copy of one of Thomson’s books on atomic structure, pointed to

a particular section and declared: ‘This is wrong.’ It is hardly surprising

that Thomson did not immediately warm to him.

Bohr struggled on, growing increasingly frustrated. He attended some

lectures, carried out some experiments under Thomson’s direction that

he thought were fairly pointless and tried hard to learn English. In Den-

mark, Bohr had enjoyed some success on the football ﬁ eld, as a goalkeeper

(his brother Harald had played a few games for the Danish national team

and had won a silver medal at the Olympic Games in London in 1908).

Bohr joined a local football club, but his physics was going nowhere. In

October 1911 he wrote to his brother complaining that Thomson was

very difﬁ cult to talk to, and seemed unable to accept criticism.

the quantum story

Bohr met Rutherford for the ﬁ rst time in early November 1911, during a visit

to Manchester. He resolved to transfer to Rutherford’s research team for the

ﬁ nal months of his postdoctoral year and learn about radioactivity. Bohr

was aware of Rutherford’s planetary model of the atom, but at this time his

principal interest was radioactivity and the Manchester laboratory was the

world’s leading centre for experimental studies of the phenomenon.

In truth, Rutherford’s planetary model had not been given much con-

sideration by the physics community. Although the image might be

compelling, the planetary structure was also quite impossible, for the

same reason that moving electrons in Thomson’s model were impossible.

Unlike the sun and planets, electrons and atomic nuclei carry electrical

charge. It was known from Maxwell’s theory that electrical charges

moving in an electromagnetic ﬁ eld would radiate energy in the form of

waves. These waves could be expected to carry energy away from the

orbiting electrons, slowing them down and leaving them exposed to the

irresistible pull of the positively charged nucleus. The electrons in such a

planetary model would spiral down towards the nucleus as they lost their

energy and the atoms would collapse in on themselves within about one

hundred-millionth of a second.

Thomson, for one, didn’t believe a word of it.

Rutherford agreed to accept Bohr into his team for the remainder of

his postdoctoral year, provided Bohr could get Thomson to agree. Thom-

son did not object, and arrangements for Bohr’s transfer to Manchester

were made in December 1911. He started work there the following March.

‘The whole thing was very interesting in Cambridge,’ Bohr said years

later, ‘but it was absolutely useless.’

Bohr initially set to work on experiments on the absorption of alpha-

particles by aluminium. But experimental physics was not his forte,

and after a few weeks he asked Rutherford if he could instead work on

theoretical problems. Although Rutherford took a keen interest in the

various research programmes and the work of his associates, he was

busy writing a book on the physics of radioactive substances and had

little time for extensive discussions. Bohr learnt what he needed about

radioactivity from two Manchester colleagues, George von Hevesy and

Charles Darwin (whom Bohr would always introduce to others as the

grandson of the ‘real’ Darwin).

a little bit of reality

It was whilst working on some problems raised by Darwin that he was

led from radioactivity to atomic structure. He still had to deal with the fact

that a system of negatively charged electrons orbiting a positively charged

nucleus is inherently unstable in classical terms. Perhaps, he reasoned,

some progress could be made by employing quantum ideas. He had

become convinced that the inner electronic structure of the Rutherford

model was governed in some way by Planck’s quantum of action.

Like Einstein before him, Bohr now realized that to expect to resolve

the paradoxes of the atom from within an entirely classical description

was to expect too much. The classical picture said that atoms should not

exist. Yet atoms clearly do exist and so it was likely to be impossible to

deduce a theoretical description using only the mathematical methods

of classical mechanics.

Something else was needed.

Faced with this kind of impasse, in 1905 Einstein had invoked his

‘heuristic principle’. Bohr now did something very similar. He simply

hypothesised that, if atoms are stable, then this surely means that there

must exist certain stable conﬁ gurations of the electrons orbiting a central

nucleus. These stable orbits would be dependent in some, as yet unspeci-

ﬁ ed, way on Planck’s constant. ‘This seems to be nothing else than what

was to be expected as it seems rigorously proved that the mechanics

cannot explain the facts in problems dealing with single atoms.’

There was more. Planck’s derivation of his radiation law had suggested

that the cavity oscillators could absorb or emit energy only in integral

multiples of the energy element hn. Bohr now assumed that the electron

orbits would be organized similarly, the orbital energies increasing as

nhn, where n = 1, 2, 3, . . . The lowest energy, innermost stable orbit would

correspond to n = 1.

In June 1912 he wrote a letter to his brother Harald:

Perhaps I have found out a little about the structure of atoms. Don’t talk

about it to anybody, for otherwise I couldn’t write to you about it so soon.

If I should be right it wouldn’t be a suggestion of the nature of a possi-

bility (i.e., an impossibility, as J.J. Thomson’s theory) but perhaps a lit-

tle bit of reality . . . You understand that I may yet be wrong; for it hasn’t

been worked out fully yet ( but I don’t think so); also, I do not believe that

Rutherford thinks that it is completely wild . . . Believe me, I am eager to