Equations of - Max-Planck

HISTORY OF SCIENCE
QUANTUM MECHANICS
Max Planck founded quantum physics in 1900 – accidentally. In the years following
1920, many physicists developed modern quantum mechanics – through labyrinthine
paths and amid intense disputes. This complex history is being studied by a group of
historians working with JÜRGEN RENN, Director at the MAX PLANCK INSTITUTE
THE
HISTORY
OF
SCIENCE,
his colleague Christoph Lehner, and a group of physicists
working with MATTHIAS SCHEFFLER, Director at the FRITZ HABER INSTITUTE.
Equations of
Discontinuity
O
n December 14, 1900, Max Planck, who was born
150 years ago, gave a lecture before the Physical
Society of Berlin that is now considered to be the birth
of quantum theory. Planck had solved a problem that
had long perplexed many theoretical physicists. He had
found the right formula for the electromagnetic radiation spectrum that a perfect black body radiates. This
experiment played a key role at the time: the black body
is an idealized test object for the question of how matter
emits – and conversely, absorbs – electromagnetic radiation as a function of its temperature. Without this interplay of light and matter, our world would be pretty dark
because the Sun, for instance, wouldn’t shine.
Planck found the right formula for the temperaturedependent spectral behavior of black-body radiation because he was extremely well versed in thermodynamics.
Prior to this, he had significantly modernized the theory
of heat. However, to obtain his solution, the Berlin-based
physics professor had to postulate a completely new
variable h. At the time, this famous quantum of action
was purely a workaround for Planck. On that legendary
December day, neither he nor his audience suspected
that he had laid the foundation for quantum physics
with precisely this artifice.
“Planck essentially slid into this quantum story as the
father of quantum theory,” remarks Dieter Hoffmann:
“He was just as unaware of the consequences as other
physicists were.” Hoffmann is a historian at the Max
Planck Institute for the History of Science in Berlin and
is currently working on a biography of the accidental
father of quantum physics. “Planck didn’t entirely ac-
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cept the quantum story until 1908,” says Hoffmann,
which he by no means considers as detracting from
Planck’s outstanding achievement. Only the second great
pioneer of quantum physics, Albert Einstein (1879–
1955), suspected early on that black-body radiation harbored a revolution in physics. In his publication on the
light quantum hypothesis in 1905, his annus mirabilis,
Einstein clearly demonstrated the break with classical
physics that was hidden in Planck’s work.
The young research field experienced turbulent development until the early 1920s. Today, it is frequently referred to as the old quantum theory. After Planck’s initial spark, some of the important milestones include
Einstein’s pioneering publications in 1905 and 1907. In
these, the young physicist showed that Planck’s constant
has a truly fundamental significance. A further milestone was the 1913 Bohr model of the atom, with which
Danish physicist Niels Bohr (1885–1962) solved a pressing problem of earlier atomic models. According to them,
the electrons should orbit a positively charged nucleus,
much like planets around the Sun. But the laws of electrodynamics dictate that, by doing so, they would have
had to radiate their kinetic energy like small antennas
and crash into the nucleus. If the world worked like that,
atoms would have no chance of survival. Bohr saved
matter as we know it by introducing quantized paths on
which electrons can move without losing energy.
Bohr’s atomic model fit the experimental findings
well. However, it could not explain why the electron
paths are quantized. That is typical for the key weakness
of the old quantum theory. They were dealing purely
phenomenologically with quantum properties without
being able to explain what caused them. “These quantum conditions were postulated,” explains Christoph
Lehner, a historian at the Max Planck Institute for the
History of Science, “but it remained unclear why certain
physical variables, such as energy, are quantized.”
The brightest minds in physics were becoming more
and more dissatisfied with this situation. In the early
1920s, they committed themselves to creating a new,
more basic quantum theory. After many missteps, their
collective efforts led to success. Near the end of the
1920s, modern quantum mechanics had nearly taken on
the form in which it is still valid today, and which also
explains the quantization of energy. Some of its findings are now
more precisely substantiated by
experiment than any other fundamental theory of physics. At the
time, however, a heated debate
raged among physicists as to their
interpretation – that is, as to the
statements that could be derived
from them with respect to the nature of our world – and it still continues today.
Surprisingly, the creation story
of modern quantum mechanics is
far less researched than one would
expect, given its importance for
our culture. After all, we have long
since evolved into “quantum manipulators,” putting semiconductor
electronics and lasers to work for
us with the touch of a button.
However, this research subject is
no walk in the park for historians.
Even just the tangled web of relationships among the many players
back then is very complex. “There
Attendees at the Solvay Conference
of 1927 – front row: Max Planck (2nd
from left), Albert Einstein (5th from
left); middle row: Niels Bohr (far right),
Louis de Broglie (3rd from right),
Paul Dirac (5th from right); top row:
Werner Heisenberg (3rd from right),
Erwin Schrödinger (6th from right).
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stance, Einstein and Planck demonstrates. And without his great
charisma, he would hardly have been
able to have so many extramarital affairs, as Lehner remarks with a grin.
With his wave mechanics, Erwin
Schrödinger was pursuing an alternative to matrix mechanics, a research field that was dominated primarily by the group working with
the renowned theorist Max Born
(1882–1970) at the University of
Göttingen. One of Born’s very talented students was Werner Heisenberg
(1901–1976), who formulated the
original idea of matrix mechanics.
“Matrix mechanics was essentially
inductive physics attempting to deEinstein could not get rid of the ghosts that he had summoned: He did not believe in "spooky
velop the formalism of a new quanaction at a distance” between entangled particles – even though he was the first to describe it.
tum theory based on existing knowlThis behavior contradicted the classical theory. Planck
edge,” is how Lehner characterizes this major current. It
conjectured that a quantum effect was playing a role,
was based primarily on Bohr’s model of the atom and
and tried to use a trick to reach a new statistical theory
the previous attempts to make it fit the measured data.
of gases. He was the first to account for the fact that, in
Matrices put numbers in table format. Heisenberg enthe quantum world, individual particles – such as gas
tered measurable variables in such matrices – basically
molecules – have no individual properties. Consequentthe light frequencies that the atom radiates. This made it
ly, particles of the same kind in the same physical state
possible to express non-measurable variables, such as
are indistinguishable: they can be switched around like
where the electrons are located. Surprisingly, such mashells in a shell game and the overall physical system
trices can also be used to determine all other physical
will not change at all. A statistical theory must take this
properties of an atom. Classical mechanics can be reininto account.
terpreted to matrix mechanics by expressing all physical
variables using such matrices.
Schrödinger, on the other hand, reaches quantum
physics by describing electrons as standing waves. On
the one hand, this approach is more deductive, beginning with a fundamental principle and developing the
conclusions. On the other hand, wave mechanics is a
De Broglie’s relation (1924)
much more descriptive approach than matrix mechanics,
Planck tried to do this, but his approach was too simple.
and was thus more intuitive for most physicists, and
The quantum pioneer had the right instinct, but it was a
easier to apply to physical problems. It was soon discovyoung Indian physicist who found the solution. Satyenered that both approaches produced the same results.
dranath Bose (1894–1974) developed a statistics that
However, Schrödinger’s hope that this would mean that
systematically counted indistinguishable quantum parti“this damned quantum jumping” could again be replaced
cles in the same quantum state as a single state. Bose’s
with a continuous physics of classical waves and fields
theory allowed him to describe the behavior of photons,
was not to be fulfilled.
or light quanta, correctly. The Indian published this work
Schrödinger got his idea from Max Planck’s attempts
in 1921 and sparked Einstein’s interest. Einstein recogto develop a quantum theory of gases, and from Einstein’s
nized that Bose’s statistical approach could be successpublications in the early 1920s. Planck had responded to
fully applied to other matter particles in, for example,
the recent findings of the newly emerged low-temperathe gaseous state.
ture physics before the First World War. It holds that very
Erwin Schrödinger studied the new Bose-Einstein stacold gases behave quite differently than classical gas thetistics. He tested it to see what statistical gas theory would
ory predicts. “The experiments showed that entropy – the
result if he described the gas molecules as waves instead
ability of gases to absorb heat – approached zero as their
of as particles. “He counted the wave states and, in doing
temperature decreased,” explains Lehner.
P HOTO :
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Italian Daniela Monaldi, for example, studies the early history of
Bose-Einstein condensation, which
has become an important tool of
experimental quantum physics, and
whose development has, as yet,
hardly been researched at all. In addition, from a historical perspective,
the genesis of the modern quantum
theory of solids is largely unknown,
even though it is precisely this –
specifically its application in semiconductor technology – that revolutionized our culture. This is what
physicist Christian Joas focuses on:
he is investigating how solid state
physics successfully adopted methods from the so-called quantum
Working visit in Copenhagen: Max Planck traveled to see Niels Bohr (left) in 1930. Bohr played
field theory in the 1950s.
a key role in developing Planck’s discovery that energy is quantized into modern quantum mechanics.
There are a number of science hisare so incredibly many sources,” says Lehner, “and these
tory publications on the convoluted path that ended in
sources are not particularly well analyzed.” Furthermore,
modern quantum mechanics, but “ultimately, not much
the research subject itself – quantum physics with all of
new has been added in the past approximately 30 years,”
its philosophical implications – is very difficult to digest.
concludes Lehner. The first key articles and books came
It is thus no wonder that science historian Lehner, like
out in the 1960s and 1970s. “They came from the eyemany of his colleagues, studied physics.
witness generation – physicists who experienced this deTwo years ago, Jürgen Renn, Director at the Max
velopment firsthand,” explains Lehner.
Planck Institute for the History of Science, and Matthias
“But eyewitness accounts aren’t always reliable,”
Scheffler, Director at the Berlin-based Fritz Haber Instisays the historian: “This is particularly noticeable in
tute of the Max Planck Society, launched an initiative
the case of such chaotic histories as that of quantum
that is being financed by the Innovation Fund of the
mechanics.” The problem is that the physicists of that
President of the Max Planck Society. The project aims to
generation who were still alive at the time told the stothoroughly research the genesis of modern quantum mery looking back across a span of decades. “So they were
chanics from today’s perspective.
now seeing what had been done in the 1920s through
the glasses of their much more modern knowledge
of quantum mechanics,” the historian explains. That is
why they presented the long-past events in a slightly
altered form – usually unintentionally. “That’s just how
human memory works,” says Lehner. To make matters
worse, physicists have a natural tendency to present
Planck’s distribution law (1900)
subjects in a form that is idealized for teaching and
that leaves out confusing meanderings. So other sourcThere are now ten historians working on it at the insties, such as letters and notebooks from those days, are
tute in Berlin alone, says Lehner, who coordinates the
much more reliable for the historians than the reports
project. “For our profession, that’s huge.” In addition,
of contemporaries.
there are partnerships around the world with other hisLehner himself and his colleagues are currently concentorians – and, as a special treat, also with physicists who
trating on the critical phase of the mid-1920s. Their presare active researchers. Foremost among the collaboratent focus centers on the notebooks of a man who providing Max Planck scientists is Scheffler with a few of his
ed important stimulation from “an outsider position,” says
colleagues. Also on board are researchers from the Max
Lehner: Erwin Schrödinger (1887–1961), who was already
Planck Institute for Gravitational Physics (Albert Einover 40 years old at the time, came to quantum physics
stein Institute) in Potsdam. What these physicists bring
via a circuitous route. However, this Viennese bohemian
to the table is, above all, the perspective of today’s user
is anything but an outsider-type. He was indeed taken seof quantum mechanics.
riously as a physicist, as his correspondence with, for in-
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Protagonists of quantum physics (from left): Erwin Schrödinger, Louis de Broglie, Werner Heisenberg.
so, elegantly obtained the Bose-Einstein gas theory,” explains Lehner. Inspired by this success, Schrödinger came
up with his wave equation in 1926. This Schrödinger
equation is one of the most famous formulas in physics
today. A further reason why Schrödinger had a keen eye
for wave phenomena was because he was familiar with
the work of another outsider. The French theoretician
Louis de Broglie (1892–1987) had boldly extended the
wave-particle dualism that had been known for light
quanta since Einstein’s publications in 1905 to all matter
particles. De Broglie, in his 1924 dissertation, was the first
to derive a wave formula for electrons. It was clear that
photons were not a special case: it appeared that, in the
quantum world, all particles also possessed wave properties. Schrödinger delved into this.
Schrödinger equation (1926)
Using the physicist’s notebooks from 1925 and 1926,
Christoph Lehner and a few of his colleagues are tracing the method with which Erwin Schrödinger developed his wave mechanics. In his formalism, Schrödinger described how electrons move around an atomic
nucleus like waves. His wave equation elegantly yielded the solution of quantized paths that Niels Bohr still
needed to postulate in his atomic model: Schrödinger
and de Broglie showed that the permissible electron
paths in the atom behave like taut strings that can vibrate only in certain notes. Schrödinger was able to use
his equation to elegantly and clearly calculate the
quantum states of the hydrogen atom.
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However, there is a speed limit for Schrödinger’s wave
equation. It becomes imprecise as soon as the electrons
become so fast that they enter the domain of relativistic
effects. That does indeed happen in atoms. “For
Schrödinger, this equation was thus merely a stopgap
solution,” says Lehner. As he noticed when studying the
notebooks, the Viennese physicist was desperately
searching for a generally applicable, relativistic formulation and viewed his now famous equation as an unloved byproduct.
But someone else was to crack this difficult nut:
English physicist Paul Dirac (1902–1984) in 1928,
with his Dirac equation. The young genius from Cambridge University was one of the main actors in the
drama of the emerging quantum mechanics. Two years
prior, Dirac had already had a major breakthrough
that led to the currently valid mathematical formulation of modern quantum mechanics. “Before that,
wave mechanics and matrix mechanics essentially
constituted two half theories,” says Lehner. In 1926,
Dirac combined the two halves into a whole in his
transformation theory.
Then 1927 became what was possibly the most exciting year in the development of quantum mechanics.
Werner Heisenberg formulated his uncertainty relation
in Göttingen, and the fifth Solvay Conference took place
in October. “This conference, especially, was fantastic,”
raves Lehner. “Everyone who was anyone at the time
was there, and all of the various opinions collided.” At
this legendary physicists conference, which Belgian industrialist Ernest Solvay had brought into being in 1911,
the foundations would be laid for the now dominant interpretation of quantum mechanics. After the conference, Niels Bohr and Werner Heisenberg battled it out
and came to an agreement about the interpretation that
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Bohr then successfully propagated
in the physics community. That is
why, decades later, the name “Copenhagen interpretation” came to be
widely accepted.
Bohr’s adversary was Albert Einstein, who was becoming more and
more suspicious of the ghosts that he
himself had summoned. Particularly
Heisenberg’s new uncertainty relation became a point of contention
between Bohr and Einstein. It states
that the location and the momentum
– that is, the speed – of a particle
cannot both be precisely determined
at the same time. Heisenberg, Bohr
and other physicists accepted this
uncertainty as a natural, fundamenAgreeing to disagree: Niels Bohr and Albert Einstein had intense debates about quantum uncertainty.
tal limit of measuring precision.
Einstein was not willing to accept that. He saw unparadox. If two or more particles are in a joint quancertainty as an indication that quantum mechanics
tum state, then all other particles must immediately
was not a fundamental theory, but rather merely a
sense when one of the particles is measured. In quanstatistical approach. To prove this, Einstein devised
tum mechanics, this applies without restriction, even if
more and more thought experiments that Bohr, in rethe “entangled” particles are far apart from one anothturn, rebutted. Eyewitness Paul Ehrenfest (1880–1933)
er. Einstein called this effect “spooky action at a diswrote the following about the debate between the two
tance,” and it told him that there was a fundamental
titans: “[…] like a game of chess. Einstein all the time
flaw in quantum mechanics.
with new examples […] to break the uncertainty relation. Bohr from out of philosophical smoke clouds
constantly searching for the tools to crush one example after the other. Einstein like a jack-in-the-box;
jumping out fresh every morning. Oh, that was priceHeisenberg’s uncertainty relation (1927)
less. But I am almost without reservation pro Bohr
But nature does indeed have this strange property, as
contra Einstein.”
has since become clear. Viennese physicist Anton ZeilDespite a few attacks and counterproposals, the Coinger, who is also involved in the research project with
penhagen interpretation was and is accepted by most
the team in Berlin, and his group recently completed a
physicists as the best possible working hypothesis to
particularly spectacular experiment in which photons
date. The Copenhagen interpretation comprises, for exremained entangled across a distance of 144 kilometers.
ample, Max Born’s finding that God – contrary to EinSuch technological developments as quantum cryptogstein’s famous witticism – evidently does play dice:
raphy are already taking advantage of this phenomenon.
physical events occur only with a certain probability,
It is always possible to tell when a message encoded in
and quantum mechanics can precisely predict only this
entangled photons has been intercepted by comparing
probability. A further ingredient is Niels Bohr’s complethe properties of both light particles. It is thus relatively
mentarity principle. It states that an experiment can
easy to expose a spy.
demonstrate either the particle properties of the object of
But Erwin Schrödinger did not yet suspect any of this
interest or its wave properties, but never both at once.
when he first wrote the wave equation in his notebook.
However, the most recent experiments contradict Bohr’s
Christoph Lehner enthusiastically points to a copy of
strict verdict.
this page. A legal dispute is currently raging over the
Einstein would never let go of his criticism that
ownership of the original, between the University of Viquantum mechanics is incomplete. In 1935, he pubenna, where it is housed, and Schrödinger’s daughter
lished, together with Boris Podolsky (1896–1966) and
and sole heir. The manuscripts of the pioneers of quanNathan Rosen (1909–1995), a thought experiment that
tum mechanics have long since become coveted items.
was intended to highlight this problem of incompleteROLAND WENGENMAYR
ness. It is famous today as the Einstein-Podolsky-Rosen
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