1. 6. 5. Atoms - Physics and philosophy


^ 1. 6. 5. Atoms
Until the twentieth century, the development of the atomic theory of matter was pursued by scientists who are often more closely identified with chemistry than with physics. In 1789 Antoine Lavoisier published his 1. 6. 5. Atoms - Physics and philosophy Elements of Chemistry. In this work, he emphasized the need for quantitative methods in chemistry.

Then in 1802 John Dalton, an English schoolmaster, revived the theory of atoms. It was known by this time 1. 6. 5. Atoms - Physics and philosophy that gases always combine in fixed ratios by mass. For example one gram of hydrogen burns with eight grams of oxygen to produce nine grams of water. Dalton proposed that 1. 6. 5. Atoms - Physics and philosophy these ratios of whole numbers could be explained if the gases were formed of atoms whose masses were, themselves, in the ratio of simple integers.

In 1869 Dimitri Mendeleev of Russia, combining Dalton's atomic description 1. 6. 5. Atoms - Physics and philosophy with the fact that certain groups of elements had similar chemical properties, constructed the first periodic table. He pointed out that the gaps in this table should correspond to 1. 6. 5. Atoms - Physics and philosophy as-yet-undiscovered elements, and was able to predict their properties and atomic masses. Armed with this knowledge, scientists very quickly discovered most of the missing elements.


1. 7. Modern Physics: relativity and quantum physics

1. 7. 1. Relativity 1. 6. 5. Atoms - Physics and philosophy

By the end of the nineteenth century, most physicists were feeling quite smug. They seemed to have theories in place that would explain all physical phenomena. There was clearly a lot of cleaning up 1. 6. 5. Atoms - Physics and philosophy to do, but it looked like a fairly mechanical job: turn the crank on the calculator until the results come out. Apart from a few niggling problems like those lines in the 1. 6. 5. Atoms - Physics and philosophy light emitted by gas discharges, and the apparent dependence of the mass of high-speed electrons on their velocity ....

Twenty-five years later, this complacency had been completely destroyed by the 1. 6. 5. Atoms - Physics and philosophy invention of three entirely new theories: special relativity, general relativity, and quantum mechanics. The outstanding figure of this period was Albert Einstein. His name became a household word for his development 1. 6. 5. Atoms - Physics and philosophy, virtually single-handedly, of the theory of relativity, and he мейд a major contribution to the development of quantum mechanics in his explanation of the photoelectric effect.

Einstein was a clerk in 1. 6. 5. Atoms - Physics and philosophy a Swiss patent office when he published his special theory of relativity in 1905. He claimed in later life that the need for this theory emerged out of Maxwell's equations. Those equations 1. 6. 5. Atoms - Physics and philosophy changed their form when one rewrote them from the conventional perspective of a person moving at constant velocity. On the other хэнд, our experience tells us that we cannot tell if 1. 6. 5. Atoms - Physics and philosophy we are moving as long as our velocity is constant: you can throw a ball back and forth in a rapidly moving train car just as you can when the train is still. It 1. 6. 5. Atoms - Physics and philosophy is only when it accelerates -- slows down or speeds up -- that one experiences a change. Moreover, Maxwell's equations indicated that the speed of light did not depend on the speed 1. 6. 5. Atoms - Physics and philosophy of the person measuring this speed, whereas if one throws a stone while running, the speed of the runner contributes to the speed of the stone. To overcome these apparent difficulties 1. 6. 5. Atoms - Physics and philosophy with Maxwell's theory, which Einstein believed to describe reality correctly, he considered the effect of two postulates. The first was that all physical phenomena must obey the same equations for people moving 1. 6. 5. Atoms - Physics and philosophy at different constant velocities (the principle of relativity), and the second was that the speed, c, measured for light does not depend on the speed of the "observer" (the person carrying out the 1. 6. 5. Atoms - Physics and philosophy measurement).

These two postulates led directly to almost unbelievable results. They showed that the measurement of space and time depended on each other (that the time you measured for an occurrence 1. 6. 5. Atoms - Physics and philosophy depended on your position), and also depended on the speed of the observer. One immediate result is that "simultaneity" is relative to the observer. Two "events" that occur at the same time for 1. 6. 5. Atoms - Physics and philosophy one observer occur at different times as seen by an observer in motion relative to the first, provided that the events occur at different spatial locations; the concept of absolute time and 1. 6. 5. Atoms - Physics and philosophy space which had underpinned mechanics for two centuries lay in shatters. Einstein's theory also showed that the measured mass of an object depended on its velocity, and that mass (m 1. 6. 5. Atoms - Physics and philosophy) could be converted to energy (E) according to E=mc2, the principle behind the atomic bomb and nuclear power plants.

One of the beauties of Einstein's theory was that, as you let a 1. 6. 5. Atoms - Physics and philosophy body's speed become small compared to the speed of light, the equations would reduce to those of Newtonian mechanics. This requirement of physics, that a more general theory must reduce 1. 6. 5. Atoms - Physics and philosophy in some limit to more restrictive theories, is called the "correspondence principle". Thus we see that the development of the special theory of relativity in no way diminishes the stature of Newton 1. 6. 5. Atoms - Physics and philosophy. Although his concept of absolute space and time were incorrect, his genius remains: Newton's mechanics is still correct except for bodies whose speeds approach that of light.

It is important to discuss 1. 6. 5. Atoms - Physics and philosophy the fact that the results of the special theory contradict "common sense": we know that we do not have to correct our watches after we have been in a car 1. 6. 5. Atoms - Physics and philosophy, and that people who are running do not appear thinner than when at rest. The problem here is that our common sense is, by definition, the sense of how the common world works 1. 6. 5. Atoms - Physics and philosophy. However, the effects predicted by the special theory are significant only at a speed approaching that of light, and none of us has ever moved at such a speed relative to another 1. 6. 5. Atoms - Physics and philosophy object with which we can interact. Therefore, we must not assume that our low-speed common sense also applies at very high speeds. Similarly, we will see that the mechanics governing 1. 6. 5. Atoms - Physics and philosophy sub-microscopic bodies such as atoms is quite different to the mechanics describing 60-kg human beings.

In 1887 the Americans Albert Michelson and Edward Morley had attempted to measure the speed of 1. 6. 5. Atoms - Physics and philosophy the Earth through the ether by measuring the difference in the speed of light travelling in two perpendicular directions. A difference was expected, for the same reason that the speed of a 1. 6. 5. Atoms - Physics and philosophy water wave relative to you depends on whether you are travelling in the same direction as the wave or otherwise. They found no dependence on the direction of motion of the light, and 1. 6. 5. Atoms - Physics and philosophy interpreted this null result by claiming that the Earth dragged the ether with it. But if the ether interacted with matter in this way, why could it not be detected directly? Moreover 1. 6. 5. Atoms - Physics and philosophy, the observation by James Bradley in 1725 of stellar aberation rules out the hypothesis of ether drag. (Stellar aberation is the apparent movement of the stars in a small ellipse over the 1. 6. 5. Atoms - Physics and philosophy course of a year, because the Earth is moving and it takes some time for the light of the stars to reach Earth.) In 1892, Hendrik Lorentz and G.F. Fitzgerald independently hypothesized 1. 6. 5. Atoms - Physics and philosophy that the size of Michelson and Morley's measuring device must depend on its velocity so as to contract in the direction of motion exactly enough to give the null result.

Einstein's second postulate 1. 6. 5. Atoms - Physics and philosophy presented yet another possibility: the measured speed of light was intrinsically independent of the speed of the observer. However, it went much beyond interpreting the Michelson -Morley result and explained, for 1. 6. 5. Atoms - Physics and philosophy example, the experimental observation that an electron's mass depended on its velocity. In fact, Henri Poincaré, a renowned physicist, had suggested a year before Einstein's publication that a 1. 6. 5. Atoms - Physics and philosophy whole new mechanics might be required, in which mass depended on velocity. Einstein's theory cleared up so many outstanding problems that it was quite quickly accepted by most physicists.

Before leaving special 1. 6. 5. Atoms - Physics and philosophy relativity it is important to discuss briefly Einstein's role in the development of nuclear weapons. Nuclear fission had been discovered in Germany in 1938, just after the invasion of Austria by Hitler's 1. 6. 5. Atoms - Physics and philosophy forces. In 1939, faced with the threat that Germany would develop a nuclear bomb, Einstein was convinced by physicist Leo Szilard to write to President Roosevelt, pointing out the possibility and encouraging American 1. 6. 5. Atoms - Physics and philosophy research in this direction. In spite of this, Einstein actively opposed further development of nuclear weapons following the Second World War. In fact, he and British philosopher/mathematician Bertrand Russell founded 1. 6. 5. Atoms - Physics and philosophy the Pugwash organization, named after its first meeting in Pugwash, Nova Scotia, in 1954. This organization of leading scientists throughout the world, and its student wing, still meet regularly to discuss issues concerning the 1. 6. 5. Atoms - Physics and philosophy impact of science on society, and to prepare position papers for presentation to governments and the United Nations.

The General Theory of Relativity extended Einstein's ideas to bodies which are accelerating 1. 6. 5. Atoms - Physics and philosophy, rather than moving at constant velocity. Einstein showed that spacetime near masses could not be described by Euclidean geometry, but rather that a geometry invented by Riemann must be used. In this 1. 6. 5. Atoms - Physics and philosophy way, gravitation was shown to be a result of the curvature of spacetime in the vicinity of mass. The general theory allowed Einstein to predict the amount of the deflection 1. 6. 5. Atoms - Physics and philosophy of light in the eclipses of 1919 and 1921, a value which agreed with that measured. However, Einstein's theory of general relativity was not the last word on the subject. General relativity is still an active 1. 6. 5. Atoms - Physics and philosophy area of research today, partly because it provides us with much evidence on the evolution of the universe including such questions as, "Will the universe someday begin to collapse back upon itself 1. 6. 5. Atoms - Physics and philosophy under its gravitational attraction?"


^ 1. 7. 2. Quantum Physics

Einstein's theories of relativity were developed in a way close to Descartes' mathematical-deductive method. The special theory came from an attempt to harmonize electromagnetic theory 1. 6. 5. Atoms - Physics and philosophy with the principle of relativity. The general theory evolved from trying to reconcile the fact that inertial mass, the "resistance" to the force in the equation F=ma, has the same 1. 6. 5. Atoms - Physics and philosophy value as gravitational mass, even though the two are totally unrelated in Newtonian mechanics.

Quantum physics, on the other хэнд, emerged from attempts to explain experimental observations. In the late 1890s a 1. 6. 5. Atoms - Physics and philosophy major area of research centred on the explanation of "blackbody" radiation: a black object such as a fireplace poker, when heated until it begins to glow, emits light whose intensity depends on 1. 6. 5. Atoms - Physics and philosophy wavelength in a way which depends largely on the temperature of the body and little on its material of construction. Because of the universal nature of this phenomenon, it was apparent that 1. 6. 5. Atoms - Physics and philosophy it must depend on fundamental physical principles. In 1900 Max Planck used a "lucky guess" to obtain a mathematical equation which fitted the experimental data accurately. Three months later he derived the 1. 6. 5. Atoms - Physics and philosophy expression theoretically. To do this he assumed that a blackbody contained many small oscillators which emitted the light, much the way the oscillations of electrons along a transmission antenna emit radio waves. However 1. 6. 5. Atoms - Physics and philosophy, he had to allow these oscillators to emit energy only at certain frequencies rather than with a continuous range of frequencies, as would be expected from classical electromagnetism. Planck had no physical basis 1. 6. 5. Atoms - Physics and philosophy for this assumption; it was just the only way that he could fit the data.

Einstein used Planck's idea in his explanation of the photoelectric effect, in which electrons are 1. 6. 5. Atoms - Physics and philosophy ejected from a metal when it is exposed to light whose frequency exceeds a certain value. Einstein extended Planck's ideas on the emission of light from a blackbody to the general statement 1. 6. 5. Atoms - Physics and philosophy that light, itself, came in packets of energy, or quanta (called "photons" from the Greek "photos" meaning "light").

Each quantum has an energy E=hf, where f is the frequency 1. 6. 5. Atoms - Physics and philosophy of light and h is "Planck's constant". This was a bold move, since the work of Young and Fresnel had seemed to establish beyond all doubt that light acted as a wave 1. 6. 5. Atoms - Physics and philosophy, and Maxwell's theory did not include any mention of a particle nature to light. However, Einstein's assumption explained the fact that even an intense light below a certain frequency could not 1. 6. 5. Atoms - Physics and philosophy cause the emission of electrons: if each incoming light quantum gave all its energy to an electron in the metal, the electron could not escape if this energy was 1. 6. 5. Atoms - Physics and philosophy less than the binding energy of the electron. This explanation dismayed Planck, who never expected his suggestion to be applied so broadly.

In 1911 Ernest Rutherford fired very small particles, emitted in radioactive decay, at 1. 6. 5. Atoms - Physics and philosophy a thin film of gold. From the scattering pattern of the particles, he determined that the atom consisted of a small, heavy, positively charged nucleus surrounded by very light electrons. Niels Bohr 1. 6. 5. Atoms - Physics and philosophy used this model and the quantum ideas of Planck and Einstein in 1913 to explain why the light from gas discharges was emitted at only a few, discrete frequencies; this light 1. 6. 5. Atoms - Physics and philosophy formed emission "lines" of different colours when the light was passed through a slit and dispersed by a prism. Bohr suggested that the electrons in an atom were only allowed to occupy certain orbits 1. 6. 5. Atoms - Physics and philosophy of definite radius r around the nucleus, namely orbits whose angular momentum was given by mvr=nwhere m and v are the mass and velocity of the electron, and n is an 1. 6. 5. Atoms - Physics and philosophy integer. When an electron gained energy and was "excited" to a higher orbit during the gas discharge, it could lose this energy only by falling back to one of the 1. 6. 5. Atoms - Physics and philosophy lower allowed orbits, with its energy loss E being carried off by the emission of a quantum of light of energy f=E/h. The predicted frequencies for hydrogen matched the experimental values.

Beginning 1. 6. 5. Atoms - Physics and philosophy with the claim that mechanical models such as Bohr's were inappropriate because they tried to use the mechanics which had been developed for macroscopic bodies in situations where it might 1. 6. 5. Atoms - Physics and philosophy not apply, Werner Heisenberg in 1925 derived a purely mathematical theory that incorporated directly the empirical data, such as the wavelengths of spectral lines. The same year, Louis de Broglie argued that if 1. 6. 5. Atoms - Physics and philosophy light could act both as a wave and as a particle (photon) with definite energy, then perhaps material particles such as electrons could as well. He suggested that such a particle should 1. 6. 5. Atoms - Physics and philosophy have a wavelength given by =h/mv, where m is the particle's mass and v is its velocity.

By the next year, de Broglie's hypothesis had been used by 1. 6. 5. Atoms - Physics and philosophy Erwin Schrödinger to explain the quantization of Bohr's orbits. Moreover, Schrödinger showed that his wave mechanics was equivalent to Heisenberg's theory. By 1927, C.J. Davisson and L.H 1. 6. 5. Atoms - Physics and philosophy. Germer had confirmed de Broglie's hypothesis directly by producing a diffraction pattern by scattering electrons from the ordered atoms on the surface of a nickel sample, much like the two-slit interference 1. 6. 5. Atoms - Physics and philosophy pattern used by Thomas Young to prove that light behaved as a wave. This result is impossible if we consider the electron as a classical particle: it means that the electron 1. 6. 5. Atoms - Physics and philosophy must scatter off more than one nickel atom simultaneously or, in the two-slit analogy, go through both slits at the same time!

Rather than placing the electrons in the atom in definite 1. 6. 5. Atoms - Physics and philosophy orbits as envisioned by Bohr, Schrödinger's wave mechanics, as interpreted by Max Born, treated the square of the particle's wave amplitude as giving the probability that the electron was 1. 6. 5. Atoms - Physics and philosophy at a particular place in space, with the most probable positions corresponding to Bohr's orbits. From this discussion it is clear that we are treating the electron both as a 1. 6. 5. Atoms - Physics and philosophy particle and a wave. Consider Young's two-slit experiment again, but using electrons instead of light as the incident radiation. Suppose we position a fluorescent screen behind the two holes, and 1. 6. 5. Atoms - Physics and philosophy decrease the intensity of the electron beam until only one electron hits the screen at a time. Experimentally we see that each electron produces a tiny flash on the screen, as though it were struck 1. 6. 5. Atoms - Physics and philosophy by a particle rather than a wave. However, the number of particles arriving in a given region of the screen is greater where the diffraction pattern has its maxima. The electron 1. 6. 5. Atoms - Physics and philosophy acts like a particle when we demand a particle-like response, but like a wave when we demand a wave-like response. This is the conclusion come to by Bohr 1. 6. 5. Atoms - Physics and philosophy, in establishing his "principle of complementarity": the wave and particle descriptions of matter (or electromagnetic radiation) are complementary, in the sense that our experiments can test for one or the other, but never for 1. 6. 5. Atoms - Physics and philosophy both properties at the same time.

In 1927 Heisenberg proved that it was impossible to determine both a particle's position and momentum with arbitrary precision; if one is known very accurately 1. 6. 5. Atoms - Physics and philosophy, then the uncertainty in the other becomes large. This "Uncertainty Principle" showed that there are theoretical limits on a person's ability to describe the world. The limits are not a serious 1. 6. 5. Atoms - Physics and philosophy consideration for large bodies, but become very important for bodies the size of an atom or smaller. The uncertainty principle also makes it clear that the presence of the experimenter always 1. 6. 5. Atoms - Physics and philosophy affects the results of an experiment at some level. For example, if we try to determine the position of a small particle very accurately we must, in principle, change its momentum by the very act 1. 6. 5. Atoms - Physics and philosophy of observing it.

Quantum mechanics has now been extended to explain a wide range of phenomena at the sub-microscopic level, including the structure of the atomic nucleus. Experimentally, this structure 1. 6. 5. Atoms - Physics and philosophy has been determined in a manner similar in principle to Rutherford's scattering experiment, using accelerators which produce incident particles of very high energy.

Philosophically, the developments of quantum mechanics 1. 6. 5. Atoms - Physics and philosophy were far-reaching. Like relativity, they again showed that humans could not assume that the physical laws which seem to govern a 60-kg person moving at speeds up to several hundred kilometres 1. 6. 5. Atoms - Physics and philosophy per hour also applied to bodies far from this regime. They also brought into question the assumption of the perfectly deterministic world proposed by Laplace. Clearly it was impossible to predict the position and 1. 6. 5. Atoms - Physics and philosophy velocity of every body for all future times if you could not even know these coordinates accurately at a single instant in time. This conclusion has even been used as 1. 6. 5. Atoms - Physics and philosophy the basis of the claim that humans have free will, that all is not predetermined as would seem to be the case in a purely mechanistic, deterministic world governed by the laws of physics 1. 6. 5. Atoms - Physics and philosophy. These ideas are still heavily debated today, as in a recent article by Roger Penrose in the book Quantum Implications.

Indeed, Einstein himself was never able to accept fully the uncertainty implied in 1. 6. 5. Atoms - Physics and philosophy quantum mechanics, declaring that he did not believe that God played dice (Clark, pp.414, 415). In an attempt to show that quantum theory was at variance with the real world, he helped 1. 6. 5. Atoms - Physics and philosophy develop the Einstein-Podolsky-Rosen (EPR) paradox, a "thought experiment" which shows that quantum mechanical theory must lead to what seems like an impossible situation: what you do to one particle 1. 6. 5. Atoms - Physics and philosophy can affect a second, even if they are sufficiently separated in space that a light signal could not pass from the first to the second fast enough to cause the observed 1. 6. 5. Atoms - Physics and philosophy effect. That is, either the knowledge of the event can travel between the particles faster than the speed of light, or the two particles really are not separate but remain interconnected in some fundamental 1. 6. 5. Atoms - Physics and philosophy sense. It was the latter option which was under debate.

An experiment designed to test this hypothesis was carried out by ^ D. Aspect and coworkers in 1981 [Physical Review Letters 47,460 (1981) and 49, 91 (1982)] and was 1. 6. 5. Atoms - Physics and philosophy shown to confirm what was predicted: the two particles really were connected over large distances by "non-local" forces acting instantaneously. That is, the EPR paradox, rather than showing a 1. 6. 5. Atoms - Physics and philosophy basic inconsistency in quantum theory, actually points to one more aspect of nature that contravenes common sense.

^ 1. 8. The unification of physical phenomena

The work of Maxwell represents the first great theoretical unification 1. 6. 5. Atoms - Physics and philosophy of physical phenomena, in this case the integration of magnetic, electrical and optical theory into one all-encompassing framework. Again, this must be seen as desirable under Ockham's Razor, which argues for economy of 1. 6. 5. Atoms - Physics and philosophy understanding. Such economy is the strength of modern analytical science, which emphasizes the logical description of a vast range of physical phenomena from a few basic principles, rather than 1. 6. 5. Atoms - Physics and philosophy the memorization of a large number of isolated facts or formulae. The former approach enables the user to predict effects not seen previously, to invent, whereas the latter restricts one to what already 1. 6. 5. Atoms - Physics and philosophy is known.

Other great unifications that have taken place in physics include the integration of classical mechanics, quantum physics and heat in the development of statistical mechanics. This subject assumes that the properties of 1. 6. 5. Atoms - Physics and philosophy large systems, such as gases or solids, can be calculated by working out the average of the properties of all their constituent particles. For example, the relationship between the temperature and pressure 1. 6. 5. Atoms - Physics and philosophy of a gas can be calculated by treating the gas as being мейд up of a very large number of independent molecules, and calculating the average force they produce 1. 6. 5. Atoms - Physics and philosophy as they collide with the container walls, using Newtonian mechanics for the particles.

This approach was followed for gases by ^ Maxwell and Ludwig Boltzmann (1844-1906). Boltzmann also showed that Clausius' entropy could be interpreted 1. 6. 5. Atoms - Physics and philosophy as a measure of the disorder of a system. In particular, he proved that the value for entropy can be obtained from a knowledge of the total number of different states in 1. 6. 5. Atoms - Physics and philosophy which a system can be found. That, in turn, depends on the number of different potential configurations of all the particles which comprised the system. This statistical approach has led to the development 1. 6. 5. Atoms - Physics and philosophy of "quantum statistics", the application of statistical mechanics to quantum phenomena.

Perhaps the greatest such unification that has taken place in this century is the integration of electromagnetism and quantum 1. 6. 5. Atoms - Physics and philosophy mechanics, in quantum electrodynamics (QED). This feat earned Richard Feynman, Julian Schwinger, and Sin-itiro Tomonaga the Nobel Prize for physics in 1965. It is capable of predicting the spin g-factor of the electron 1. 6. 5. Atoms - Physics and philosophy with a numerical accuracy of 1 part in 1010!

In 1979, Sheldon Glashow, Abdus Salam, and Stephen Weinberg were given the Nobel Prize for their "electroweak theory" that unified the electromagnetic and weak nuclear forces. Attempts 1. 6. 5. Atoms - Physics and philosophy have also been мейд to form a quantum theory of the strong nuclear force. Because of its similarity to QED, it has been called quantum chromodynamics (QCD). "Chromo" comes 1. 6. 5. Atoms - Physics and philosophy from the Greek word for colour, and refers to the fact that the quarks that make up neutrons and protons come in several varieties that have been given the names red, blue 1. 6. 5. Atoms - Physics and philosophy and green, and their antiparticles. (These names have been chosen in analogy to light. These three colours can be combined to give white light; the three quarks combine to give a "colourless 1. 6. 5. Atoms - Physics and philosophy" particle.) The combination of electroweak theory and QCD comprises what is called the "Standard Model". Attempts are still under way to integrate QCD and electroweak theory into a single "Grand Unified Theory" (GUT).

Much effort 1. 6. 5. Atoms - Physics and philosophy has also gone into trying to unify electromagnetism and gravitation. In fact, Einstein spent most of the latter part of his life trying to create a quantum form of 1. 6. 5. Atoms - Physics and philosophy the general theory of relativity. As can be seen from these few examples, the nineteenth-century belief that the main theoretical work of physicists was over could not have been further from the 1. 6. 5. Atoms - Physics and philosophy truth!

Bibliography

Butterfield, H., The Origins of Modern Science, 1300-1800

(Clarke-Irwin, Toronto) 1977. A good discussion of the interplay between science and society.

Capra, F., The Turning Point (Simon and Schuster, New York) 1982. Reductionist vs 1. 6. 5. Atoms - Physics and philosophy. holistic science, from a physicist's perspective.

Clark, R.W., Einstein, The Life and Times (Avon, New York) 1971.

Cline, B.L., Men who Мейд a New Physics (previously entitled The 1. 6. 5. Atoms - Physics and philosophy Questioners) (Signet, New York) 1965. A very readable account of the origins of quantum physics and relativity.

Cole, M.D., The Maya, 3rd ed. (Thames and Hudson, London) 1984.

Dijksterhuis, E.J., The 1. 6. 5. Atoms - Physics and philosophy Mechanization of the World Picture (Oxford University) 1961.

Drake, S., Telescopes, Tides and Tactics: A Galilean Dialogue about the Starry Messenger and Systems of the World (University of Chicago Press, Chicago) 1983. This book 1. 6. 5. Atoms - Physics and philosophy includes a translation of Galileo's description of his first astronomical observations, and MUST be read. It contains copies of Galileo's original sketches of the appearance of the Moon and of the moons of 1. 6. 5. Atoms - Physics and philosophy Jupiter.

Drake, S., The Role of Music in Galileo's Experiments Scientific American, p. 98, June 1975.

Finocchiaro, M.A., The Galileo Affair, A Documentary History (University of California Press, Berkeley) 1989. Gives the 1. 6. 5. Atoms - Physics and philosophy context for Galileo's trial, and a translation of a number of the original documents.

French, M., Beyond Power (Ballantine, New York) 1985. A feminist perspective on patriarchal society.

Hawking, S.W., A Brief 1. 6. 5. Atoms - Physics and philosophy History of Time (Bantam, 1988). A discussion of modern cosmology for the layperson, from one of the world's experts.

Horgan, J., Quantum Philosophy, Scientific American, July 1992, p.94. A discussion 1. 6. 5. Atoms - Physics and philosophy of recent investigations of the EPR paradox.

Hiley, B.J. and Peat, F.D. (editors), Quantum Implications - Essays in Honour of David Bohm (Routledge, New York) 1987. An excellent but fairly mathematical consideration of 1. 6. 5. Atoms - Physics and philosophy the implications of quantum theory.

Kramer, E., Nature and Growth of Modern Mathematics, (Princeton University Press, New York) 1982.

Jammer, M., The Conceptual Development of Quantum Mechanics, (McGraw-Hill, New York) 1966. This book 1. 6. 5. Atoms - Physics and philosophy is quite mathematical.

Mason, S.F., A History of the Sciences (Collier, New York), 1962. An excellent general history, very complete.

Rossiter, M.W., Women Scientists in America: Struggles and Strategies to 1940, (John 1. 6. 5. Atoms - Physics and philosophy Hopkins University Press, Baltimore) 1982.

Schneer, C.J., The Evolution of Physical Science (Grove Press, New York) 1960. Greeks to modern physical science.

Tuana, N. (editor), Feminism and Science (Indiana University Press, Bloomington) 1989. Addresses gender bias 1. 6. 5. Atoms - Physics and philosophy in science.

Whitehead, A.N., Science and the Modern World, (Cambridge University Press) 1933.

Williams, L.P., The Origins of Field Theory (Random House, Toronto) 1966. (Not in Trent Library).

Chapter2. The Uncertainty 1. 6. 5. Atoms - Physics and philosophy Principle

2. 1. Introduction

2. 2. Heisenberg

2. 2. 1 Matrix mechanis and wave mechanics

2. 2. 2 Heisenberg's argument

2. 2. 3 The interpretation of Heisenberg's relation

2. 2. 4 Uncertainty relations or uncertainty principle?

2. 3. Bohr

Bohr's view on the uncertainty relations

2. 4. The Minimal 1. 6. 5. Atoms - Physics and philosophy Interpretation



Introductory Remarks




The transition from classical to quantum physics marks a genuine revolution in our understanding of the physical world.

One striking aspect of the difference between classical and quantum physics is 1. 6. 5. Atoms - Physics and philosophy that whereas classical mechanics presupposes that one can assign exact simultaneous values to the position and momentum of a particle, quantum mechanics denies this possibility. Instead, according to quantum mechanics, the more 1. 6. 5. Atoms - Physics and philosophy precisely the position of a particle is given, the less precisely one can say what its momentum is. This is (a simplistic and preliminary formulation of) the quantum mechanical uncertainty principle. This principle played 1. 6. 5. Atoms - Physics and philosophy an important role in many discussions on the philosophical implications of quantum mechanics and on the consistency of the interpretation endorsed by the founding fathers Heisenberg and Bohr, the so 1. 6. 5. Atoms - Physics and philosophy-called Copenhagen interpretation.

This, of course, should not suggest that the uncertainty principle is the only aspect in which classical and quantum physics differ conceptually. In particular the implications of quantum mechanics for 1. 6. 5. Atoms - Physics and philosophy notions such as (non)-locality, entanglement and identity play no less havoc with classical intuitions.




^ 2. 1. Introduction

The uncertainty principle is certainly one of the most famous and important aspects of quantum mechanics. Often 1. 6. 5. Atoms - Physics and philosophy, it has even been regarded as the most distinctive feature in which this theory differs from a classical conception of the physical world. Roughly speaking, the uncertainty principle states that one cannot assign 1. 6. 5. Atoms - Physics and philosophy exact simultaneous values to the position and momentum of a quantum mechanical system. Rather, we can only determine such quantities with some characteristic ‘uncertainties’, which cannot both become arbitrarily small at 1. 6. 5. Atoms - Physics and philosophy the same time. But what exactly is the meaning of this uncertainty principle? And indeed, is it really a principle of quantum mechanics? In particular, what does it mean that a quantity 1. 6. 5. Atoms - Physics and philosophy is determined only up to some uncertainty? These are the main questions we will explore in the following, focussing on the views of Heisenberg and Bohr.

In many expositions of the subject, the 1. 6. 5. Atoms - Physics and philosophy ‘uncertainty’ may refer sometimes to a lack of knowledge of a quantity by an observer, or to the experimental inaccuracy with which a quantity is measured, or to some ambiguity in 1. 6. 5. Atoms - Physics and philosophy the definition of a quantity, or to a statistical spread in some ensemble of similarly prepared systems. Corresponding to this confusing multitude of different meanings, there are many different names 1. 6. 5. Atoms - Physics and philosophy for these ‘uncertainties’. For example, apart from those already mentioned (inaccuracy, spread) one finds imprecision, indefiniteness, indeterminateness, indeterminacy, latitude, etc. Even Heisenberg and Bohr did not decide on a single terminology. Forestalling a discussion about 1. 6. 5. Atoms - Physics and philosophy which name is the most appropriate, we mention here that we use the name ‘uncertainty principle’ simply because it seems the most common one in the literature.


^ 2. 2. Heisenberg

2. 2. 1 Matrix mechanics 1. 6. 5. Atoms - Physics and philosophy and wave mechanics

Heisenberg introduced his now famous relations in an article of 1927, entitled "Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". A (partial) translation of this title is: "On the anschaulich 1. 6. 5. Atoms - Physics and philosophy content of quantum theoretical kinematics and mechanics". Here, the term anschaulich is particularly notable. Apparently, it is one of those German words that defy an unambiguous translation into other languages. Heisenberg's title 1. 6. 5. Atoms - Physics and philosophy is translated as "On the physical content …" by Wheeler and Zurek (1983). His collected works (Heisenberg, 1984) translate it as "On the perceptible content …", while Cassidy's biography of Heisenberg (Cassidy, 1992), refers to 1. 6. 5. Atoms - Physics and philosophy the paper as "On the perceptual content …". Literally, the closest translation of the term anschaulich is ‘visualizable’. But, as in most languages, words that make reference to vision are not always intended literally 1. 6. 5. Atoms - Physics and philosophy. Seeing is widely used as a metaphor for understanding, especially for immediate understanding. Hence, anschaulich also means ‘intelligible’ or ‘intuitive’.[1]

Why was this issue of the Anschaulichkeit of quantum mechanics 1. 6. 5. Atoms - Physics and philosophy such a prominent concern to Heisenberg? This question has already been considered by a number of commentators. For the answer, it turns out, we must go back a little in time. In 1. 6. 5. Atoms - Physics and philosophy 1925 Heisenberg had developed the first coherent mathematical formalism for quantum theory. His leading idea was that only those quantities that are in principle observable should play a role in the theory, and that all 1. 6. 5. Atoms - Physics and philosophy attempts to form a picture of what goes on inside the atom should be avoided. In atomic physics the observational data were obtained from spectroscopy and associated with atomic transitions 1. 6. 5. Atoms - Physics and philosophy. Thus, Heisenberg was led to consider the ‘transition quantities’ as the basic ingredients of the theory. Max Born, later that year, realized that the transition quantities obeyed the rules of matrix calculus. In 1. 6. 5. Atoms - Physics and philosophy a famous series of papers Heisenberg, Born and Jordan developed this idea into the matrix mechanics version of quantum theory.

Formally, matrix mechanics remains close to classical mechanics. The central idea 1. 6. 5. Atoms - Physics and philosophy is that all physical quantities must be represented by infinite self-adjoint matrices (later identified with operators on a Hilbert space). It is postulated that the matrices q and p representing the canonical position and 1. 6. 5. Atoms - Physics and philosophy momentum variables of a particle satisfy the so-called canonical commutation rule


qp − pq = i (2-1)


Where = h/2π, h denotes Planck's constant, and boldface type is used to represent matrices. The new 1. 6. 5. Atoms - Physics and philosophy theory scored spectacular empirical success by encompassing nearly all spectroscopic data known at the time, especially after the concept of the electron spin was included in the theoretical framework.

It 1. 6. 5. Atoms - Physics and philosophy came as a great surprise, therefore, when one year later, Erwin Schrödinger presented an alternative theory, which became known as wave mechanics. Schrödinger assumed that an electron could be represented as 1. 6. 5. Atoms - Physics and philosophy an oscillating charge cloud, evolving continuously in space and time according to a wave equation. The discrete frequencies in the atomic spectra were not due to discontinuous transitions (quantum jumps) but 1. 6. 5. Atoms - Physics and philosophy to a resonance phenomenon. Further, Schrödinger argued that the two theories were equivalent.

Even so, the two approaches differed greatly in interpretation and spirit. Whereas Heisenberg eschewed the use of visualizable pictures, and 1. 6. 5. Atoms - Physics and philosophy accepted discontinuous transitions as a primitive notion, Schrödinger claimed as an advantage of his theory that it was anschaulich. In Schrödinger's vocabulary, this meant that the theory represented the 1. 6. 5. Atoms - Physics and philosophy observational data by means of continuously evolving causal processes in space and time. He considered this condition of Anschaulichkeit to be an essential requirement on any acceptable physical theory. In fact 1. 6. 5. Atoms - Physics and philosophy, Schrödinger was not alone in appreciating this aspect of his theory. Many other leading physicists were attracted to wave mechanics for the same reason.

For a while in 1926, before 1. 6. 5. Atoms - Physics and philosophy it emerged that wave mechanics has serious problems of its own, Schrödinger's approach seemed to gather more support in the physics community than matrix mechanics.

Understandably, Heisenberg was unhappy about this development. In 1. 6. 5. Atoms - Physics and philosophy a letter of 8 June 1926 to Pauli he confessed that Schrödinger's approach disgusted him, and in particular: "What Schrödinger writes about the Anschaulichkeit of his theory, … I consider Mist (Pauli 1. 6. 5. Atoms - Physics and philosophy, 1979, p. 328)". Again, this last German term is translated differently by various commentators: as "junk" (Miller, 1982) "rubbish" (Beller 1999) "crap" (Cassidy, 1992), and perhaps more literally, as "bullshit" (de Regt, 1997). Nevertheless, in published writings, Heisenberg 1. 6. 5. Atoms - Physics and philosophy voiced a more balanced opinion. In a paper he summarized the peculiar situation which the simultaneous development of two competing theories had brought about. Although he argued that Schrödinger's 1. 6. 5. Atoms - Physics and philosophy interpretation was untenable, he admitted that matrix mechanics did not provide the Anschaulichkeit which мейд wave mechanics so attractive. He concluded: "to obtain a contradiction-free anschaulich interpretation, we still lack some 1. 6. 5. Atoms - Physics and philosophy essential feature in our image of the structure of matter". The purpose of his 1927 paper was to provide exactly this lacking feature.


^ 2. 2. 2 Heisenberg's argument

Let us now look at the 1. 6. 5. Atoms - Physics and philosophy argument that led Heisenberg to his uncertainty relations. He started by redefining the notion of Anschaulichkeit. Whereas Schrödinger associated this term with the provision of a causal space-time picture of the phenomena, Heisenberg 1. 6. 5. Atoms - Physics and philosophy, by contrast, declared:

We believe we have gained anschaulich understanding of a physical theory, if in all simple cases we can grasp the experimental consequences qualitatively and see that 1. 6. 5. Atoms - Physics and philosophy the theory does not lead to any contradictions. (Heisenberg, 1927, p. 172)

His goal was, of course, to show that, in this new sense of the word, matrix mechanics could lay the same claim to Anschaulichkeit as 1. 6. 5. Atoms - Physics and philosophy wave mechanics.

To do this, he adopted an operational assumption: terms like ‘the position of a particle’ have meaning only if one specifies a suitable experiment by which ‘the 1. 6. 5. Atoms - Physics and philosophy position of a particle’ can be measured. In general, there is no lack of such experiments, even in the domain of atomic physics. However, experiments are never completely accurate. We should be prepared 1. 6. 5. Atoms - Physics and philosophy to accept, therefore, that in general the meaning of these quantities is also determined only up to some characteristic inaccuracy.

As an example, he considered the measurement of the position of 1. 6. 5. Atoms - Physics and philosophy an electron by a microscope. The accuracy of such a measurement is limited by the wave length of the light illuminating the electron. Thus, it is possible, in principle, to make such 1. 6. 5. Atoms - Physics and philosophy a position measurement as accurate as one wishes, but only by using light of a very short wave length, e.g., γ-rays. But for γ-rays, the Compton effect cannot be ignored: the 1. 6. 5. Atoms - Physics and philosophy interaction of the electron and the illuminating light should then be considered as a collision of at least one photon with the electron. In such a collision, the electron suffers a recoil which 1. 6. 5. Atoms - Physics and philosophy disturbs its momentum. Moreover, the shorter the wave length, the larger is this change in momentum. Thus, at the moment when the position of the particle is accurately known, Heisenberg argued, its momentum cannot 1. 6. 5. Atoms - Physics and philosophy be accurately known.

At the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous 1. 6. 5. Atoms - Physics and philosophy change in momentum. This change is the greater the smaller the wavelength of the light employed, i.e., the more exact the determination of the position. At the instant at which the 1. 6. 5. Atoms - Physics and philosophy position of the electron is known, its momentum therefore can be known only up to magnitudes which correspond to that discontinuous change; thus, the more precisely the position is determined, the less precisely 1. 6. 5. Atoms - Physics and philosophy the momentum is known, and conversely (Heisenberg, 1927, p. 174-5).

This is the first formulation of the uncertainty principle. In its present form it is an epistemological principle, since it limits 1. 6. 5. Atoms - Physics and philosophy what we can know about the electron. From "elementary formulae of the Compton effect" Heisenberg estimated the ‘imprecisions’ to be of the order


δpδq  h (2-2)


He continued: "In this circumstance we see the 1. 6. 5. Atoms - Physics and philosophy direct anschaulich content of the relation qp − pq = i ".

He went on to consider other experiments, designed to measure other physical quantities and obtained analogous relations for time and energy:


δt δE 1. 6. 5. Atoms - Physics and philosophy  h (2-3)



and action J and angle w


δw δJ  h (2-4)



which he saw as corresponding to the "well-known" relations


tE − Et = ior wJ − Jw = i (2-5)


However, we must say that these generalisations 1. 6. 5. Atoms - Physics and philosophy did not turn out as straightforward as Heisenberg suggested.

The first mathematically exact formulation of the uncertainty relations is due to Kennard. He proved in 1927 the theorem that for all normalized state vectors 1. 6. 5. Atoms - Physics and philosophy |ψ> the following inequality holds:


Δψp Δψq ≥ /2 (2-6)


Here, Δψp and Δψq are standard deviations of position and momentum in the state vector |ψ>, i.e.,


(Δψp)² = ψ − (

ψ)², (Δψq)² = ψ − (ψ)². (2-7)


where ψ = denotes the expectation 1. 6. 5. Atoms - Physics and philosophy value in state |ψ>. This inequality (7) was generalized in 1929 by Robertson who proved the result that for all observables (self-adjoint operators) A and B


ΔψA ΔψB ≥ ½|<[A,B]>ψ| (2-8)


where [A, B] = AB − BA denotes the commutator. This 1. 6. 5. Atoms - Physics and philosophy relation was in turn strengthened by Schrödinger (1930), who obtained:


(ΔψA)² (ΔψB)² ≥ ¼|<[A,B]>ψ|² + ¼|<{A−ψ, B−ψ}>ψ|² (2-9)


where {A, B} = (AB + BA) denotes the anti-commutator.

Since the above inequalities have the virtue of being exact 1. 6. 5. Atoms - Physics and philosophy and general, in contrast to Heisenberg's original semi-quantitative formulation, it is tempting to regard them as the exact counterpart of Heisenberg's relations (2)-(4). Indeed, such was Heisenberg 1. 6. 5. Atoms - Physics and philosophy's own view. In his Chicago Lectures (Heisenberg 1930, pp. 15-19), he presented Kennard's derivation of relation (6) and claimed that "this proof does not differ at all in mathematical content" from the semi-quantitative argument he 1. 6. 5. Atoms - Physics and philosophy had presented earlier, the only difference being that now "the proof is carried through exactly".

But it may be useful to point out that both in status and intended role there 1. 6. 5. Atoms - Physics and philosophy is a subtle difference between Kennard's inequality and Heisenberg's previous formulation (2). The inequalities discussed here are not statements of empirical fact, but theorems of the formalism. As such, they presuppose the 1. 6. 5. Atoms - Physics and philosophy validity of this formalism, and in particular the commutation relation (1), rather than elucidating its intuitive content or to create ‘room’ or ‘freedom’ for the validity of this relation. At best, one should 1. 6. 5. Atoms - Physics and philosophy see the above inequalities as showing that the formalism is consistent with Heisenberg's empirical principle.

This situation is similar to that arising in other theories of principle where, as 1. 6. 5. Atoms - Physics and philosophy would be noted, one often finds that, next to an empirical principle, the formalism also provides a corresponding theorem. And similarly, this situation should not, by itself, cast doubt on the question 1. 6. 5. Atoms - Physics and philosophy whether Heisenberg's relation can be regarded as a principle of quantum mechanics.

There is a second notable difference between (2) and (6). Heisenberg did not give a general definition for the ‘uncertainties 1. 6. 5. Atoms - Physics and philosophy’ δp and δq. The most definite remark he мейд about them was that they could be taken as "something like the mean error". In the discussions of thought experiments, he and Bohr would always 1. 6. 5. Atoms - Physics and philosophy quantify uncertainties on a case-to-case basis by choosing some parameters which happened to be relevant to the experiment at хэнд. By contrast, the inequalities (6)-(9) employ a single specific 1. 6. 5. Atoms - Physics and philosophy expression as a measure for ‘uncertainty’: the standard deviation. This choice is not unnatural, given that this expression is well-known and widely used in error theory and the description of statistical 1. 6. 5. Atoms - Physics and philosophy fluctuations.

Heisenberg summarized his findings in a general conclusion: all concepts used in classical mechanics are also well-defined in the realm of atomic processes. But, as a pure fact of experience experiments that serve 1. 6. 5. Atoms - Physics and philosophy to provide such a definition for one quantity are subject to particular indeterminacies, obeying relations (2)-(4) which prohibit them from providing a simultaneous definition of two canonically conjugate quantities. Note that 1. 6. 5. Atoms - Physics and philosophy in this formulation the emphasis has slightly shifted: he now speaks of a limit on the definition of concepts, i.e. not merely on what we can know, but what we 1. 6. 5. Atoms - Physics and philosophy can meaningfully say about a particle.


^ 2. 2. 3 The interpretation of Heisenberg's relation

The relations Heisenberg had proposed were soon considered to be a cornerstone of the Copenhagen interpretation of quantum mechanics. Just a 1. 6. 5. Atoms - Physics and philosophy few months later, Kennard (1927) already called them the "essential core" of the new theory. Taken together with Heisenberg's contention that they provided the intuitive content of the theory and their prominent role 1. 6. 5. Atoms - Physics and philosophy in later discussions on the Copenhagen interpretation, a dominant view emerged in which they were regarded as a fundamental principle of the theory.

The interpretation of these relations has often been 1. 6. 5. Atoms - Physics and philosophy debated.

Do Heisenberg's relations express restrictions on the experiments we can perform on quantum systems, and, therefore, restrictions on the information we can gather about such systems; or

Do they express restrictions on 1. 6. 5. Atoms - Physics and philosophy the meaning of the concepts we use to describe quantum systems? Or else,

Are they restrictions of an ontological nature, i.e., do they assert that a quantum system simply does not 1. 6. 5. Atoms - Physics and philosophy possess a definite value for its position and momentum at the same time?

The difference between these interpretations is partly reflected in the various names by which the relations are known, e 1. 6. 5. Atoms - Physics and philosophy.g. as ‘inaccuracy relations’, or: ‘uncertainty’, ‘indeterminacy’ or ‘unsharpness relations’, etc. The debate between these different views has been addressed by many authors, but it has never been settled completely 1. 6. 5. Atoms - Physics and philosophy. Let it suffice here to make only two general observations.

First, it is clear that in Heisenberg's own view, all the above questions stand or fall together. Indeed, we have seen that he adopted 1. 6. 5. Atoms - Physics and philosophy an operational "measurement=meaning" principle according to which the meaningfulness of a physical quantity was equivalent to the existence of an experiment purporting to measure that quantity. Similarly, his "measurement 1. 6. 5. Atoms - Physics and philosophy=creation" principle allowed him to attribute physical reality to such quantities. Hence, Heisenberg's discussions moved rather freely and quickly from talk about experimental inaccuracies to epistemological or ontological issues and back 1. 6. 5. Atoms - Physics and philosophy again.

However, ontological questions seemed to be of somewhat less interest to him. For example, there is a passage (Heisenberg, 1927, p. 197), where he discusses the idea that, behind our observational data 1. 6. 5. Atoms - Physics and philosophy, there might still exist a hidden reality in which quantum systems have definite values for position and momentum, unaffected by the uncertainty relations. He emphatically dismisses this conception as an unfruitful and meaningless 1. 6. 5. Atoms - Physics and philosophy speculation, because, as he says, the aim of physics is only to describe observable data. Similarly in the Chicago Lectures (Heisenberg 1930, p. 11) he warns against the fact that the human language 1. 6. 5. Atoms - Physics and philosophy permits the utterance of statements which have no empirical content at all, but nevertheless produce a picture in our imagination. He notes, "One should be especially careful in using the words ‘reality 1. 6. 5. Atoms - Physics and philosophy’, ‘actually’, etc., since these words very often lead to statements of the type just mentioned." So, Heisenberg also endorsed an interpretation of his relations as rejecting a reality in which particles have simultaneous 1. 6. 5. Atoms - Physics and philosophy definite values for position and momentum.

The second observation is that although for Heisenberg experimental, informational, epistemological and ontological formulations of his relations were, so to say, just different sides of the 1. 6. 5. Atoms - Physics and philosophy same coin, this does not hold for those who do not share his operational principles or his view on the task of physics. Alternative points of view, in which e.g. the ontological reading 1. 6. 5. Atoms - Physics and philosophy of the uncertainty relations is denied, are therefore still viable. The statement, often found in the literature of the thirties, that Heisenberg had proved the impossibility of associating a definite 1. 6. 5. Atoms - Physics and philosophy position and momentum to a particle is certainly wrong. But the precise meaning one can coherently attach to Heisenberg's relations depends rather heavily on the interpretation one favors for quantum 1. 6. 5. Atoms - Physics and philosophy mechanics as a whole. And in view of the fact that no agreement has been reached on this latter issue, one cannot expect agreement on the meaning of the uncertainty relations either.


^ 2. 2. 4. Uncertainty 1. 6. 5. Atoms - Physics and philosophy relations or uncertainty principle?

Let us now move to another question about Heisenberg's relations: do they express a principle of quantum theory? Probably the first influential author to call these relations a 1. 6. 5. Atoms - Physics and philosophy ‘principle’ was Eddington, who referred to them as the ‘Principle of Indeterminacy’. In the English literature the name uncertainty principle became most common. It is used both by Condon and Robertson in 1. 6. 5. Atoms - Physics and philosophy 1929, and also in the English version of Heisenberg's Chicago Lectures (Heisenberg, 1930), although, remarkably, nowhere in the original German version of the same book (see also Cassidy, 1998). Indeed, Heisenberg never seems to 1. 6. 5. Atoms - Physics and philosophy have endorsed the name ‘principle’ for his relations. His favourite terminology was ‘inaccuracy relations’ (Ungenauigkeitsrelationen) or ‘indeterminacy relations’ (Unbestimmtheitsrelationen). We know only one passage (Heisenberg, 1958, p. 43), where he mentioned that his 1. 6. 5. Atoms - Physics and philosophy relations "are usually called relations of uncertainty or principle of indeterminacy". But this can well be read as his yielding to common practice rather than his own preference.

But does the relation 1. 6. 5. Atoms - Physics and philosophy (2) qualify as a principle of quantum mechanics? Several authors, foremost Karl Popper (1967), have contested this view. Popper argued that the uncertainty relations cannot be granted the status of a principle, on the grounds 1. 6. 5. Atoms - Physics and philosophy that they are derivable from the theory, whereas one cannot obtain the theory from the uncertainty relations. (The argument being that one can never derive any equation, say, the Schrödinger 1. 6. 5. Atoms - Physics and philosophy equation, or the commutation relation (1), from an inequality.)

Popper's argument is, of course, correct but we think it misses the point. There are many statements in physical theories which are called principles 1. 6. 5. Atoms - Physics and philosophy even though they are in fact derivable from other statements in the theory in question. A more appropriate departing point for this issue is not the question of logical priority but 1. 6. 5. Atoms - Physics and philosophy rather Einstein's distinction between ‘constructive theories’ and ‘principle theories’.

According to Einstein (Einstein, 1919), constructive theories are theories which postulate the existence of simple entities behind the phenomena. They endeavour to reconstruct 1. 6. 5. Atoms - Physics and philosophy the phenomena by framing hypotheses about these entities. Principle theories, on the other хэнд, start from empirical principles, i.e. general statements of empirical regularities, employing no or only a bare minimum 1. 6. 5. Atoms - Physics and philosophy of theoretical terms. The purpose is to build up the theory from such principles. That is, one aims to show how these empirical principles provide sufficient conditions for the introduction of further theoretical concepts 1. 6. 5. Atoms - Physics and philosophy and structure.

The prime example of a theory of principle is thermodynamics. Here the role of the empirical principles is played by the statements of the impossibility of various kinds of perpetual 1. 6. 5. Atoms - Physics and philosophy motion machines. These are regarded as expressions of brute empirical fact, providing the appropriate conditions for the introduction of the concepts of energy and entropy and their properties. (There is 1. 6. 5. Atoms - Physics and philosophy a lot to be said about the tenability of this view, but that is not the topic of this entry.)

Now obviously, once the formal thermodynamic theory is built, one can also 1. 6. 5. Atoms - Physics and philosophy derive the impossibility of the various kinds of perpetual motion. (They would violate the laws of energy conservation and entropy increase.) But this derivation should not misguide one into thinking that 1. 6. 5. Atoms - Physics and philosophy they were no principles of the theory after all. The point is just that empirical principles are statements that do not rely on the theoretical concepts (in this case entropy and energy) for their 1. 6. 5. Atoms - Physics and philosophy meaning. They are interpretable independently of these concepts and, further, their validity on the empirical level still provides the physical content of the theory.

A similar example is provided by special relativity 1. 6. 5. Atoms - Physics and philosophy, another theory of principle, which Einstein deliberately designed after the ideal of thermodynamics. Here, the empirical principles are the light postulate and the relativity principle. Again, once we have built up 1. 6. 5. Atoms - Physics and philosophy the modern theoretical formalism of the theory (the Minkowski space-time) it is straightforward to prove the validity of these principles. But again this does not count as an argument for claiming that 1. 6. 5. Atoms - Physics and philosophy they were no principles after all. So the question whether the term ‘principle’ is justified for Heisenberg's relations, should, in our view, be understood as the question whether they are conceived 1. 6. 5. Atoms - Physics and philosophy of as empirical principles.

One can easily show that this idea was never far from Heisenberg's intentions. We have already seen that Heisenberg presented them as the result of 1. 6. 5. Atoms - Physics and philosophy a "pure fact of experience". A few months after his 1927 paper, he wrote a popular paper with the title "Ueber die Grundprincipien der Quantenmechanik" ("On the fundamental principles of quantum mechanics") where 1. 6. 5. Atoms - Physics and philosophy he мейд the point even more clearly. Here Heisenberg described his recent break-through in the interpretation of the theory as follows: "It seems to be a general law of nature that we cannot 1. 6. 5. Atoms - Physics and philosophy determine position and velocity simultaneously with arbitrary accuracy". Now actually, and in spite of its title, the paper does not identify or discuss any ‘fundamental principle’ of quantum mechanics. So, it 1. 6. 5. Atoms - Physics and philosophy must have seemed obvious to his readers that he intended to claim that the uncertainty relation was a fundamental principle, forced upon us as an empirical law of nature, rather than a 1. 6. 5. Atoms - Physics and philosophy result derived from the formalism of this theory.

So, although Heisenberg did not originate the tradition of calling his relations a principle, it is not implausible to attribute the view to him 1. 6. 5. Atoms - Physics and philosophy that the uncertainty relations represent an empirical principle that could serve as a foundation of quantum mechanics. In fact, his 1927 paper expressed this desire explicitly: "Surely, one would like to be 1. 6. 5. Atoms - Physics and philosophy able to deduce the quantitative laws of quantum mechanics directly from their anschaulich foundations, that is, essentially, relation [(2)]" (ibid, p. 196). This is not to say that Heisenberg was successful in reaching this 1. 6. 5. Atoms - Physics and philosophy goal, or that he did not express other opinions on other occasions.

Let us conclude this section with three remarks.

First, if the uncertainty relation is to serve as an empirical principle 1. 6. 5. Atoms - Physics and philosophy, one might well ask what its direct empirical support is. In Heisenberg's analysis, no such support is mentioned. His arguments concerned thought experiments in which the validity of the theory, at least at a 1. 6. 5. Atoms - Physics and philosophy rudimentary level, is implicitly taken for granted. Jammer (1974, p. 82) conducted a literature search for high precision experiments that could seriously test the uncertainty relations and concluded they were still scarce 1. 6. 5. Atoms - Physics and philosophy in 1974. Real experimental support for the uncertainty relations in experiments in which the inaccuracies are close to the quantum limit have come about only more recently.

^ A second point is the question 1. 6. 5. Atoms - Physics and philosophy whether the theoretical structure or the quantitative laws of quantum theory can indeed be derived on the basis of the uncertainty principle, as Heisenberg wished. Serious attempts to build up quantum theory as 1. 6. 5. Atoms - Physics and philosophy a full-fledged Theory of Principle on the basis of the uncertainty principle have never been carried out. Indeed, the most Heisenberg could and did claim in this respect was that 1. 6. 5. Atoms - Physics and philosophy the uncertainty relations created "room" (Heisenberg 1927, p. 180) or "freedom" (Heisenberg, 1931, p. 43) for the introduction of some non-classical mode of description of experimental data, not that they uniquely lead to the 1. 6. 5. Atoms - Physics and philosophy formalism of quantum mechanics. A serious proposal to construe quantum mechanics as a theory of principle was provided only recently by Bub (2000). But, remarkably, this proposal does not use the uncertainty relation as one of 1. 6. 5. Atoms - Physics and philosophy its fundamental principles.

Third, it is remarkable that in his later years Heisenberg put a somewhat different gloss on his relations. In his autobiography, he described how he had 1. 6. 5. Atoms - Physics and philosophy found his relations inspired by a remark by Einstein that "it is the theory which decides what one can observe" -- thus giving precedence to theory above experience, rather than the other way around. Some 1. 6. 5. Atoms - Physics and philosophy years later he even admitted that his famous discussions of thought experiments were actually trivial since "… if the process of observation itself is subject to the laws of quantum theory, it must 1. 6. 5. Atoms - Physics and philosophy be possible to represent its result in the mathematical scheme of this theory" (Heisenberg, 1975, p. 6).


2. 3. Bohr

In spite of the fact that Heisenberg's and Bohr's views on quantum mechanics are often 1. 6. 5. Atoms - Physics and philosophy lumped together as (part of) ‘the Copenhagen interpretation’, there is considerable difference between their views on the uncertainty relations.

^ Bohr's view on the uncertainty relations

In his Como lecture, published in 1. 6. 5. Atoms - Physics and philosophy 1928, Bohr gave his own version of a derivation of the uncertainty relations between position and momentum and between time and energy. He started from the relations


E = hν and p = h/λ (2-10)



which 1. 6. 5. Atoms - Physics and philosophy connect the notions of energy E and momentum p from the particle picture with those of frequency ν and wavelength λ from the wave picture. He noticed that a wave packet of limited extension 1. 6. 5. Atoms - Physics and philosophy in space and time can only be built up by the superposition of a number of elementary waves with a large range of wave numbers and frequencies. Denoting the spatial and temporal 1. 6. 5. Atoms - Physics and philosophy extensions of the wave packet by Δx and Δt, and the extensions in the wave number σ = 1/λ and frequency by Δσ and Δν, it follows from Fourier analysis that in the most favorable case Δx Δσ ≈ Δt Δν ≈ 1, and, using 1. 6. 5. Atoms - Physics and philosophy (10), one obtains the relations


Δt ΔE ≈ Δx Δp ≈ h (2-11)



Note that Δx, Δσ, etc., are not standard deviations but unspecified measures of the size of a wave packet. (The original text has equality signs 1. 6. 5. Atoms - Physics and philosophy instead of approximate equality signs, but, since Bohr does not define the spreads exactly the use of approximate equality signs seems more in line with his intentions. Moreover, Bohr himself 1. 6. 5. Atoms - Physics and philosophy used approximate equality signs in later presentations.) These equations determine, according to Bohr: "the highest possible accuracy in the definition of the energy and momentum of the individuals associated with the wave field" (Bohr 1. 6. 5. Atoms - Physics and philosophy 1928, p. 571). He noted, "This circumstance may be regarded as a simple symbolic expression of the complementary nature of the space-time description and the claims of causality" (ibid).] We note a 1. 6. 5. Atoms - Physics and philosophy few points about Bohr's view on the uncertainty relations.

^ First of all, Bohr does not refer to discontinuous changes in the relevant quantities during the measurement process. Rather, he emphasizes the possibility 1. 6. 5. Atoms - Physics and philosophy of defining these quantities. This view is markedly different from Heisenberg's.

Indeed, Bohr not only rejected Heisenberg's argument that these relations are due to discontinuous disturbances implied by the act 1. 6. 5. Atoms - Physics and philosophy of measuring, but also his view that the measurement process creates a definite result.

Nor did he approve of an epistemological formulation or one in terms of experimental inaccuracies 1. 6. 5. Atoms - Physics and philosophy.

Instead, Bohr always stressed that in his point of view the uncertainty relations are foremost an expression of complementarity. At first sight, this might seem odd, since, after all, complementarity corresponds to a dichotomic relation 1. 6. 5. Atoms - Physics and philosophy between two types of description. The uncertainty relations "express" this dichotomy in the informal sense that if we take energy and momentum to be perfectly well-defined, i.e., symbolically ΔE = Δp 1. 6. 5. Atoms - Physics and philosophy = 0, the position and time variables are completely undefined, Δx = Δt = ∞, and vice versa. However, by focussing on these extremes only, we leave out of consideration that the uncertainty relations also (and more 1. 6. 5. Atoms - Physics and philosophy properly) allow for an intermediate situation in which the mentioned uncertainties are all non-zero and finite.

However, Bohr never followed up on this suggestion that we might be able to strike 1. 6. 5. Atoms - Physics and philosophy a compromise between the two mutually exclusive modes of description in terms of unsharply defined quantities. Indeed, an attempt to do so, would take the formalism of quantum theory more 1. 6. 5. Atoms - Physics and philosophy seriously than the concepts of classical language, and this step Bohr refused to take. Instead, in his later writings he would be content with stating that the uncertainty relations simply defy an unambiguous interpretation 1. 6. 5. Atoms - Physics and philosophy in classical terms.

It must here be remembered that even in the indeterminacy relation [Δq Δp ≈ h] we are dealing with an implication of the formalism which defies unambiguous expression in 1. 6. 5. Atoms - Physics and philosophy words suited to describe classical pictures. Thus a sentence like "we cannot know both the momentum and the position of an atomic object" raises at once questions as to the 1. 6. 5. Atoms - Physics and philosophy physical reality of two such attributes of the object, which can be answered only by referring to the conditions for an unambiguous use of space-time concepts, on the one хэнд, and 1. 6. 5. Atoms - Physics and philosophy dynamical conservation laws on the other хэнд. (Bohr, 1949, p. 211)

Finally, on a more formal level, we note that Bohr's derivation does not rely on the commutation relations (1) and (5), but on Fourier analysis. To be 1. 6. 5. Atoms - Physics and philosophy sure, these two approaches are equivalent as far as the relationship between position and momentum is concerned. But this is not so for time and energy. This means that, for 1. 6. 5. Atoms - Physics and philosophy a derivation based on the commutation relations, the position-momentum and time-energy relations are not on an equal footing, which is contrary to Bohr's approach in terms of Fourier analysis 1. 6. 5. Atoms - Physics and philosophy (Hilgevoord, 1996 and 1998).


^ 2. 4. The minimal interpretation

In the previous two sections we have seen how both Heisenberg and Bohr attributed a far-reaching status to the uncertainty relations. They both argued that these relations place 1. 6. 5. Atoms - Physics and philosophy fundamental limits on the applicability of the usual classical concepts. Moreover, they both believed that these limitations were inevitable and forced upon us. However, we have also seen that they 1. 6. 5. Atoms - Physics and philosophy reached such conclusions by starting from radical and controversial assumptions. This entails, of course, that their radical conclusions remain unconvincing for those who reject these, or other assumptions. Indeed, the operationalist-positivist viewpoint 1. 6. 5. Atoms - Physics and philosophy adopted by these authors has long since lost its appeal among philosophers of physics.

So the question may be asked what alternative views of the uncertainty relations are still viable. Of course 1. 6. 5. Atoms - Physics and philosophy, this problem is intimately connected with that of the interpretation of the wave function, and hence of quantum mechanics as a whole. Since there is no consensus about the latter, one cannot expect consensus 1. 6. 5. Atoms - Physics and philosophy about the interpretation of the uncertainty relations either. Here we only describe a point of view, which we call the ‘minimal interpretation’, which seems to be shared by both the adherents 1. 6. 5. Atoms - Physics and philosophy of the Copenhagen interpretation and of other views.

In quantum mechanics a system is supposed to be described by its quantum state, also called its state vector. Given the state vector, one 1. 6. 5. Atoms - Physics and philosophy can derive probability distributions for all the physical quantities pertaining to the system such as its position, momentum, angular momentum, energy, etc. The operational meaning of these probability distributions is that they 1. 6. 5. Atoms - Physics and philosophy correspond to the distribution of the values obtained for these quantities in a long series of repetitions of the measurement. More precisely, one imagines a great number of copies of the system under 1. 6. 5. Atoms - Physics and philosophy consideration, all prepared in the same way. On each copy the momentum, say, is measured. Generally, the outcomes of these measurements differ and a distribution of outcomes is obtained. The 1. 6. 5. Atoms - Physics and philosophy theoretical momentum distribution derived from the quantum state is supposed to coincide with the hypothetical distribution of outcomes obtained in an infinite series of repetitions of the momentum measurement. The same holds, mutatis mutandis 1. 6. 5. Atoms - Physics and philosophy, for all the other physical quantities pertaining to the system. Note that no simultaneous measurements of two or more quantities are required in defining the operational meaning of the probability distributions.

Uncertainty 1. 6. 5. Atoms - Physics and philosophy relations can be considered as statements about the spreads of the probability distributions of the several physical quantities arising from the same state. For example, the uncertainty relation between the position 1. 6. 5. Atoms - Physics and philosophy and momentum of a system may be understood as the statement that the position and momentum distributions cannot both be arbitrarily narrow -- in some sense of the word "narrow" -- in any quantum 1. 6. 5. Atoms - Physics and philosophy state. Inequality (9) is an example of such a relation in which the standard deviation is employed as a measure of spread. From this characterization of uncertainty relations follows that a more detailed 1. 6. 5. Atoms - Physics and philosophy interpretation of the quantum state than the one given in the previous paragraph is not required to study uncertainty relations as such. In particular, a further ontological or linguistic interpretation of the 1. 6. 5. Atoms - Physics and philosophy notion of uncertainty, as limits on the applicability of our concepts given by Heisenberg or Bohr, need not be supposed.

Indeed, this minimal interpretation leaves open whether it makes sense 1. 6. 5. Atoms - Physics and philosophy to attribute precise values of position and momentum to an individual system. Some interpretations of quantum mechanics, e.g. Heisenberg and Bohr, deny this; while others, e.g. the interpretation of de Broglie 1. 6. 5. Atoms - Physics and philosophy and Bohm insist that each individual system has a definite position and momentum. The only requirement is that, as an empirical fact, it is not possible to prepare pure ensembles in which 1. 6. 5. Atoms - Physics and philosophy all systems have the same values for these quantities, or ensembles in which the spreads are smaller than allowed by quantum theory. Although interpretations of quantum mechanics, in which each system has 1. 6. 5. Atoms - Physics and philosophy a definite value for its position and momentum are still viable, this is not to say that they are without problems or, at least strange features, of their own. They do not imply 1. 6. 5. Atoms - Physics and philosophy a return to classical physics.

We end with a few remarks on this minimal interpretation. First, it may be noted that the minimal interpretation of the uncertainty relations is little more than filling 1. 6. 5. Atoms - Physics and philosophy in the empirical meaning of inequality (9), or an inequality in terms of other measures of width, as obtained from the standard formalism of quantum mechanics. As such, this view 1. 6. 5. Atoms - Physics and philosophy shares many of the limitations we have noted above about this inequality. Indeed, it is not straightforward to relate the spread in a statistical distribution of measurement results with the inaccuracy of this 1. 6. 5. Atoms - Physics and philosophy measurement, such as, e.g. the resolving power of a microscope. Moreover, the minimal interpretation does not address the question whether one can make simultaneous accurate measurements of position and momentum 1. 6. 5. Atoms - Physics and philosophy. As a matter of fact, one can show that the standard formalism of quantum mechanics does not allow such simultaneous measurements. But this is not a consequence of relation (9).

If one feels that 1. 6. 5. Atoms - Physics and philosophy statements about inaccuracy of measurement, or the possibility of simultaneous measurements, belong to any satisfactory formulation of the uncertainty principle, the minimal interpretation may thus be too minimal.

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