Philosophers
Mortimer Adler Rogers Albritton Alexander of Aphrodisias Samuel Alexander William Alston G.E.M.Anscombe Anselm Louise Antony Thomas Aquinas Aristotle David Armstrong Harald Atmanspacher Robert Audi Augustine J.L.Austin A.J.Ayer Alexander Bain Mark Balaguer Jeffrey Barrett William Belsham Henri Bergson Isaiah Berlin Bernard Berofsky Robert Bishop Max Black Susanne Bobzien Emil du BoisReymond Hilary Bok Laurence BonJour George Boole Émile Boutroux F.H.Bradley C.D.Broad Michael Burke C.A.Campbell Joseph Keim Campbell Rudolf Carnap Carneades Ernst Cassirer David Chalmers Roderick Chisholm Chrysippus Cicero Randolph Clarke Samuel Clarke Anthony Collins Antonella Corradini Diodorus Cronus Jonathan Dancy Donald Davidson Mario De Caro Democritus Daniel Dennett Jacques Derrida René Descartes Richard Double Fred Dretske John Dupré John Earman Laura Waddell Ekstrom Epictetus Epicurus Herbert Feigl John Martin Fischer Owen Flanagan Luciano Floridi Philippa Foot Alfred Fouilleé Harry Frankfurt Richard L. Franklin Michael Frede Gottlob Frege Peter Geach Edmund Gettier Carl Ginet Alvin Goldman Gorgias Nicholas St. John Green H.Paul Grice Ian Hacking Ishtiyaque Haji Stuart Hampshire W.F.R.Hardie Sam Harris William Hasker R.M.Hare Georg W.F. Hegel Martin Heidegger R.E.Hobart Thomas Hobbes David Hodgson Shadsworth Hodgson Baron d'Holbach Ted Honderich Pamela Huby David Hume Ferenc Huoranszki William James Lord Kames Robert Kane Immanuel Kant Tomis Kapitan Jaegwon Kim William King Hilary Kornblith Christine Korsgaard Saul Kripke Andrea Lavazza Keith Lehrer Gottfried Leibniz Leucippus Michael Levin George Henry Lewes C.I.Lewis David Lewis Peter Lipton John Locke Michael Lockwood E. Jonathan Lowe John R. Lucas Lucretius Ruth Barcan Marcus James Martineau Storrs McCall Hugh McCann Colin McGinn Michael McKenna Brian McLaughlin John McTaggart Paul E. Meehl Uwe Meixner Alfred Mele Trenton Merricks John Stuart Mill Dickinson Miller G.E.Moore C. Lloyd Morgan Thomas Nagel Friedrich Nietzsche John Norton P.H.NowellSmith Robert Nozick William of Ockham Timothy O'Connor David F. Pears Charles Sanders Peirce Derk Pereboom Steven Pinker Plato Karl Popper Porphyry Huw Price H.A.Prichard Hilary Putnam Willard van Orman Quine Frank Ramsey Ayn Rand Michael Rea Thomas Reid Charles Renouvier Nicholas Rescher C.W.Rietdijk Richard Rorty Josiah Royce Bertrand Russell Paul Russell Gilbert Ryle JeanPaul Sartre Kenneth Sayre T.M.Scanlon Moritz Schlick Arthur Schopenhauer John Searle Wilfrid Sellars Alan Sidelle Ted Sider Henry Sidgwick Walter SinnottArmstrong J.J.C.Smart Saul Smilansky Michael Smith Baruch Spinoza L. Susan Stebbing George F. Stout Galen Strawson Peter Strawson Eleonore Stump Francisco Suárez Richard Taylor Kevin Timpe Mark Twain Peter Unger Peter van Inwagen Manuel Vargas John Venn Kadri Vihvelin Voltaire G.H. von Wright David Foster Wallace R. Jay Wallace W.G.Ward Ted Warfield Roy Weatherford William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Scientists Michael Arbib Bernard Baars Gregory Bateson John S. Bell Charles Bennett Ludwig von Bertalanffy Susan Blackmore Margaret Boden David Bohm Niels Bohr Ludwig Boltzmann Emile Borel Max Born Satyendra Nath Bose Walther Bothe Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Donald Campbell Anthony Cashmore Eric Chaisson JeanPierre Changeux Arthur Holly Compton John Conway John Cramer E. P. Culverwell Charles Darwin Terrence Deacon Louis de Broglie Max Delbrück Abraham de Moivre Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Paul Ehrenfest Albert Einstein Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher Joseph Fourier Lila Gatlin Michael Gazzaniga GianCarlo Ghirardi J. Willard Gibbs Nicolas Gisin Paul Glimcher Thomas Gold A.O.Gomes Brian Goodwin Joshua Greene Jacques Hadamard Patrick Haggard Stuart Hameroff Augustin Hamon Sam Harris Hyman Hartman JohnDylan Haynes Martin Heisenberg Werner Heisenberg John Herschel Jesper Hoffmeyer E. T. Jaynes William Stanley Jevons Roman Jakobson Pascual Jordan Ruth E. Kastner Stuart Kauffman Simon Kochen Stephen Kosslyn Ladislav Kovàč Rolf Landauer Alfred Landé PierreSimon Laplace David Layzer Benjamin Libet Seth Lloyd Hendrik Lorentz Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau James Clerk Maxwell Ernst Mayr Ulrich Mohrhoff Jacques Monod Emmy Noether Howard Pattee Wolfgang Pauli Massimo Pauri Roger Penrose Steven Pinker Colin Pittendrigh Max Planck Susan Pockett Henri Poincaré Daniel Pollen Ilya Prigogine Hans Primas Adolphe Quételet Juan Roederer Jerome Rothstein David Ruelle Erwin Schrödinger Aaron Schurger Claude Shannon David Shiang Herbert Simon Dean Keith Simonton B. F. Skinner Roger Sperry Henry Stapp Tom Stonier Antoine Suarez Leo Szilard William Thomson (Kelvin) Peter Tse Heinz von Foerster John von Neumann John B. Watson Daniel Wegner Steven Weinberg Paul A. Weiss John Wheeler Wilhelm Wien Norbert Wiener Eugene Wigner E. O. Wilson H. Dieter Zeh Ernst Zermelo Wojciech Zurek Presentations Biosemiotics Free Will Mental Causation James Symposium 
The Quantum Postulate and the
Recent Development of Atomic Theory [Note: The text below is an annotated HTML version of the PDF version of the Nature paper. It is easily searched and allows cross references (hyperlinks) to other Bohr works defining "complementarity" that led to the "standard" and "orthodox" version of quantum physics called the Copenhagen Interpretation starting with an article by Werner Heisenberg in the 1950's. The "Quantum Postulate" refers in part to Bohr's "old quantum theory." Bohr postulated a discontinuous "quantum jump" of an electron between "stationary states" with the emission or absorption of radiation of frequency ν, in accordance with Planck's postulate E = hν and his "quantum of action" h. But note that for Bohr the radiation emitted or absorbed is continuous. He never endorsed Einstein's lightquantum hypothesis, although the Bohr atom is taught today as emitting a photon when the electron jumps between energy levels.]
The Quantum Postulate and the Recent Development of Atomic Theory.^{1}
By Prof. N. BOHR, For. Mem. R.S. IN connexion with the discussion of the physical interpretation of the quantum theoretical methods developed during recent years, I should like to make the following general remarks regarding the principles underlying the description of atomic phenomena, which I hope may help to harmonise the different views, apparently so divergent, concerning this subject. 1. QUANTUM POSTULATE AND CAUSALITY. The quantum theory is characterised by the acknowledgment of a fundamental limitation in the classical physical ideas when applied to atomic phenomena. The situation thus created is of a peculiar nature, since our interpretation of the experimental material rests essentially upon the classical concepts.
The "quantum postulate" is Bohr's assumption of "stationary states" and the discontinuous "quantum jump" of an electron between states, with the emission of radiation of frequency ν, following Planck's 1909 postulate of discrete oscillators with energy
Notwithstanding the difficulties
which hence are involved in the formulation
of the quantum theory, it seems, as we shall see,
that its essence may be expressed in the socalled
quantum postulate, which attributes to any atomic
process an essential discontinuity, or rather individuality,
completely foreign to the classical
theories and symbolised by Planck's quantum of
action.
E = hν, with h Planck's famous constant for the "quantum of action" . This postulate implies a renunciation as regards the causal spacetime coordination of atomic processes. Indeed, our usual description of physical phenomena is based entirely on the idea that the phenomena concerned may be observed without disturbing them appreciably. This appears, for example, clearly in the theory of relativity, which has been so fruitful for the elucidation of the classical theories. As emphasised by Einstein, every observation or measurement ultimately rests on the coincidence of two independent events at the same spacetime point. Just these coincidences will not be affected by any differences which the spacetime coordination of different observers otherwise may exhibit.
Bohr denied an "independent reality" to quantum phenomena; observation involves subjective elements dependent on both human senses and theoretical concepts.
Now the quantum postulate
implies that any observation of atomic
phenomena will involve an interaction with the
agency of observation not to be neglected.
Accordingly,
an independent reality in the ordinary
physical sense can neither be ascribed to the
phenomena nor to the agencies of observation.
After all, the concept of observation is in so far
arbitrary as it depends upon which objects are
included in the system to be observed. Ultimately
every observation can of course be reduced to our
sense perceptions. The circumstance, however,
that in interpreting observations use has always
to be made of theoretical notions, entails that for
every particular case it is a question of convenience
at what point the concept of observation involving
the quantum postulate with its inherent
'irrationality' is brought in.
The "point" at which the observation involves the quantum is called the Heisenberg "cut". Bohr later regretted his 'irrationality' comment.
If definition of a system state requires
This situation has farreaching consequences.
On one hand, the definition of the state of
a physical system, as ordinarily understood,
claims the elimination of all external disturbances.
But in that case, according to the quantum
postulate, any observation will be impossible,
and, above all, the concepts of space and time
lose their immediate sense. On the other hand,
if in order to make observation possible we permit
certain interactions with suitable agencies
of measurement, not belonging to the system,
an unambiguous definition of the state of the
system is naturally no longer possible, and
there can be no question of causality in the
ordinary sense of the word.
no disturbance, then observation is impossible.
If an observation interacts with the system,
Spacetime
coordination and the claim of causality are complementary.
The very nature of the
quantum theory thus forces us to regard the spacetime
coordination and the claim of causality, the
union of which characterises the classical theories,
as complementary but exclusive features of the
description, symbolising the idealisation of observation
and definition respectively. Just as the relativity
theory has taught us that the convenience
of distinguishing sharply between space and time
rests solely on the smallness of the velocities
ordinarily met with compared to the velocity of
light, we learn from the quantum theory that the
appropriateness of our usual causal spacetime
description depends entirely upon the small value
of the quantum of action as compared to the
actions involved in ordinary sense perceptions.
Indeed, in the description of atomic phenomena,
the quantum postulate presents us with
the task of developing a 'complementarity' theory
the consistency of which can be judged only by
weighing the possibilities of definition and observation.
They "symbolize" observation
Relativity has a limit Quantum mechanics
The quantum mechanics of particles (1925) and the competing "wave mechanics" (1926) had just been developed. Bohr saw these as complementary, the waves in spacetime, the particles needing causal interactions
This view is already clearly brought out by the
muchdiscussed question of the nature of light and
the ultimate constituents of matter. As regards
light, its propagation in space and time is adequately
expressed by the electromagnetic theory.
Especially the interference phenomena in vacuo
and the optical properties of material media are
completely governed by the wave theory superposition
principle. Nevertheless, the conservation
of energy and momentum during the interaction between radiation and matter, as evident in the photoelectric and Compton effect, finds its adequate expression just in the light quantum idea put forward by Einstein. The BohrKramersSlater denial of energy and momentum conservation (1924) had failed, and Einstein's "lightquantum hypothesis" just been confirmed by the BotheGeiger experiments.
As is well known, the
doubts regarding the validity of the superposition
principle on one hand and of the conservation laws
on the other, which were suggested by this apparent
contradiction, have been definitely disproved
through direct experiments. This situation would
seem clearly to indicate the impossibility of a
causal spacetime description of the light phenomena.
On one hand, in attempting to trace
the laws of the timespatial propagation of light
according to the quantum postulate, we are confined
to statistical considerations. On the other hand,
the fulfilment of the claim of causality for the
individual light processes, characterised by the
quantum of action, entails a renunciation as regards
the spacetime description.
Once again, spacetime and causality are complementary views of classical concepts.
Of course, there can
be no question of a quite independent application
of the ideas of space and time and of causality.
The two views of the nature of light are rather to
be considered as different attempts at an interpretation
of experimental evidence in which the
limitation of the classical concepts is expressed in
complementary ways.
The "waveparticle duality" first seen by Einstein in 1909 for light was applied to matter by Louis de Broglie in 1923, following Einstein's ideas.
The problem of the nature of the constituents of
matter presents us with an analogous situation.
The individuality of the elementary electrical
corpuscles is forced upon us by general evidence.
Nevertheless, recent experience, above all the
discovery of the selective reflection of electrons
from metal crystals, requires the use of the
wave theory superposition principle in accordance
with the original ideas of L. de Broglie. Just
as in the case of light, we have consequently in
the question of the nature of matter, so far as we
adhere to classical concepts, to face an inevitable
dilemma, which has to be regarded as the very
expression of experimental evidence.
Waves of radiation in free space and individual material particles are
In fact, here
again we are not dealing with contradictory but with
complementary pictures of the phenomena, which
only together offer a natural generalisation of the
classical mode of description. In the discussion
of these questions, it must be kept in mind that,
according to the view taken above, radiation
in free space as well as isolated material particles
are abstractions, their properties on the
quantum theory being definable and observable
only through their interaction with other systems.
Nevertheless, these abstractions are, as we shall
see, indispensable for a description of experience
in connexion with our ordinary spacetime
view.
complementary ideal abstractions. Isolation is impossible. The difficulties with which a causal spacetime description is confronted in the quantum theory, and which have been the subject of repeated discussions, are now placed into the foreground by the recent development of the symbolic methods.
The "symbolic methods" are HeisenbergBornJordan "quantum mechanics."
An important contribution to the problem of a
consistent application of these methods has been
made lately by Heisenberg (Zeitschr. f. Phys.,
43, 172; 1927). In particular, he has stressed the
peculiar reciprocal uncertainty which affects all
measurements of atomic quantities. Before we
enter upon his results it will be advantageous to
show how the complementary nature of the description
appearing in this uncertainty is unavoidable
already in an analysis of the most elementary
concepts employed in interpreting experience.
The "uncertainty" principle may only be epistemological limits on complementary observations? 2. QUANTUM OF ACTION AND KINEMATICS. The fundamental contrast between the quantum of action and the classical concepts is immediately apparent from the simple formulas which form the common foundation of the theory of light quanta and of the wave theory of material particles. If Planck's constant be denoted by h, as is well known,
E τ = I λ = h, . . . (1)
where E and I are energy and momentum respectively, τ and λ the corresponding period of vibration and wavelength. In these formulae the two notions of light and also of matter enter in sharp contrast. While energy and momentum are associated with the concept of particles, and hence may be characterised according to the classical point of view by definite spacetime coordinates, the period of vibration and wavelength refer to a plane harmonic wave train of unlimited extent in space and time. Only with the aid of the superposition principle does it become possible to attain a connexion with the ordinary mode of description. Indeed, a limitation of the extent of the wavefields in space and time can always be regarded as resulting from the interference of a group of elementary harmonic waves. As shown by de Broglie (Thèse, Paris, 1924), the translational velocity of the individuals associated with the waves can be represented by just the socalled groupvelocity. Let us denote a plane elementary wave by
A cos 2π ( νt  xσ_{x}  yσ_{y}  zσ_{z} + δ ),
where A and δ are constants determining respectively the amplitude and the phase. The quantity ν = 1/τ is the frequency, σ_{x}, σ_{y}, σ_{z} the wave numbers in the direction of the coordinate axes, which may be regarded as vector components of the wave number σ = l / λ. in the direction of propagation. While the wave or phase velocity is given by ν / σ, the group  velocity is defined by dν / dσ. Now according to the relativity theory we have for a particle with the velocity v :
I = ( v / c^{2} ) E and vdI = dE,
where c denotes the velocity of light. Hence by equation (1) the phase velocity is c^{2} / v and the groupvelocity v. The circumstance that the former is in general greater than the velocity of light emphasises the symbolic character of these considerations. At the same time, the possibility of identifying the velocity of the particle with the groupvelocity indicates the field of application of spacetime pictures in the quantum theory.
The use of a wave description reduces sharpness in definitions
Here the complementary
character of the description appears,
since the use of wavegroups is necessarily accompanied
by a lack of sharpness in the definition of
period and wavelength, and hence also in the definition
of the corresponding energy and momentum
as given by relation (1).
Δt Δν = Δx Δσ_{x} = Δy Δσ_{y} = Δz Δσ_{z} = 1,
where Δt, Δx, Δy, Δz denote the extension of the wavefield in time and in the directions of space corresponding to the coordinate axes. These relations — well known from the theory of optical instruments, especially from Rayleigh's investigation of the resolving power of spectral apparatus — express the condition that the wavetrains extinguish each other by interference at the spacetime boundary of the wavefield. They may be regarded also as signifying that the group as a whole has no phase in the same sense as the elementary waves. From equation (1) we find thus:
Δt ΔE = Δx ΔI_{x} = Δy ΔI_{y} = Δz ΔI_{z} = h, . . (2)
as determining the highest possible accuracy in the definition of the energy and momentum of the individuals associated with the wavefield. In general, the conditions for attributing an energy and a momentum value to a wavefield by means of formula (1) are much less favourable. Even if the composition of the wavegroup corresponds in the beginning to the relations (2), it will in the course of time be subject to such changes that it becomes less and less suitable for representing an individual. It is this very circumstance which gives rise to the paradoxical character of the problem of the nature of light and of material particles. The limitation in the classical concepts expressed through relation (2) is, besides, closely connected with the limited validity of classical mechanics, which in the wave theory of matter corresponds to the geometrical optics, in which the propagation of waves is depicted through 'rays.' Only in this limit can energy and momentum be unambiguously defined on the basis of spacetime pictures. For a general definition of these concepts we are confined to the conservation laws, the rational formulation of which has been a fundamental problem for the symbolical methods to be mentioned below. In the language of the relativity theory, the content of the relations (2) may be summarised in the statement that according to the quantum theory a general reciprocal relation exists between the maximum sharpness of definition of the spacetime and energymomentum vectors associated with the individuals.
Bohr may still hope to "reconcile" conservation laws by claiming spacetime points are "unsharp" (reminiscent of his BKS statistical conservation ideas).
This circumstance may be
regarded as a simple symbolical expression for the
complementary nature of the spacetime description
and the claims of causality. At the same time,
however, the general character of this relation
makes it possible to a certain extent to reconcile
the conservation laws with the spacetime coordination
of observations, the idea of a coincidence
of welldefined events in a spacetime point being
replaced by that of unsharply defined individuals
within finite spacetime regions.
This circumstance permits us to avoid the wellknown paradoxes which are encountered in attempting to describe the scattering of radiation by free electrical particles as well as the collision of two such particles.
BohrKramersSlater failed to combine instantaneous and discontinuous electron jumps with continuous radiation. Here Bohr hopes the electron can be spread out in a finite spacetime region just as the radiation is?
According to
the classical concepts, the description of the
scattering requires a finite extent of the radiation
in space and time, while in the change
of the motion of the electron demanded by the
quantum postulate one seemingly is dealing with
an instantaneous effect taking place at a definite
point in space. Just as in the ease of radiation,
however, it is impossible to define momentum and
energy for an electron without considering a finite
spacetime region. Furthermore, an application
of the conservation laws to the process implies
that the accuracy of definition of the energy
momentum vector is the same for the radiation
and the electron. In consequence, according to
relation (2), the associated spacetime regions can
be given the same size for both individuals in
interaction.
A similar remark applies to the collision between two material particles, although the significance of the quantum postulate for this phenomenon was disregarded before the necessity of the wave concept was realised. Here this postulate does indeed represent the idea of the individuality of the particles which, transcending the spacetime description, meets the claim of causality. While the physical content of the light quantum idea is wholly connected with the conservation theorems for energy and momentum, in the case of the electrical particles the electric charge has to be taken into account in this connexion. It is scarcely necessary to mention that for a more detailed description of the interaction between individuals we cannot restrict ourselves to the facts expressed by formulae (1) and (2), but must resort to a procedure which allows us to take into account the coupling of the individuals, characterising the interaction in question, where just the importance of the electric charge appears. As we shall see, such a procedure necessitates a further departure from visualisation in the usual sense. 3. MEASUREMENTS IN THE QUANTUM THEORY.
Heisenberg was upset that Bohr so strongly adopted Schrödinger's wavemechanical views. Although wave mechanics and matrix mechanics were equivalent formulations of quantum mechanics, Bohr was emphasizing the wavelike properties, and embarrassing Heisenberg by pointing out the mistaken "disturbance" explanation of uncertainty in Heisenberg's γray microscope.
In his investigations already mentioned on the
consistency of the quantum theoretical methods,
Heisenberg has given the relation (2) as an expression
for the maximum precision with which
the spacetime coordinates and momentumenergy
components of a particle can be measured
simultaneously. His view was based on the
following consideration: On one hand, the coordinates
of a particle can be measured with any
desired degree of accuracy by using, for example,
an optical instrument, provided radiation of
sufficiently short wavelength is used for illumination.
According to the quantum theory, however,
the scattering of radiation from the object is always
connected with a finite change in momentum,
which is the larger the smaller the wavelength of
the radiation used. The momentum of a particle,
on the other hand, can be determined with any
desired degree of accuracy by measuring, for
example, the Doppler effect of the scattered radiation,
provided the wavelength of the radiation
is so large that the effect of recoil can be neglected,
but then the determination of the space coordinates
of the particle becomes correspondingly less
accurate.
The essence of this consideration is the inevitability of the quantum postulate in the estimation of the possibilities of measurement. A closer investigation of the possibilities of definition would still seem necessary in order to bring out the general complementary character of the description. Indeed, a discontinuous change of energy and momentum during observation could not prevent us from ascribing accurate values to the spacetime coordinates, as well as to the momentumenergy components before and after the process. The reciprocal uncertainty which always affects the values of these quantities is, as will be clear from the preceding analysis, essentially an outcome of the limited accuracy with which changes in energy and momentum can be defined, when the wavefields used for the determination of the spacetime coordinates of the particle are sufficiently small.
Ironically, Max Born (My Life, p.213) says that Heisenberg could not answer Wien's question on resolving power and nearly failed the oral exam for his doctorate.
In using an optical instrument for determinations
of position, it is necessary to remember that
the formation of the image always requires a
convergent beam of light. Denoting by λ the
wavelength of the radiation used, and by ε the
socalled numerical aperture, that is, the sine of
half the angle of convergence, the resolving power
of a microscope is given by the wellknown expression
λ / 2ε. Even if the object is illuminated by
parallel light, so that the momentum h / λ of the
incident light quantum is known both as regards
magnitude and direction, the finite value of the
aperture will prevent an exact knowledge of the
recoil accompanying the scattering. Also, even if
the momentum of the particle were accurately
known before the scattering process, our knowledge
of the component of momentum parallel to
the focal plane after the observation would be
affected by an uncertainty amounting to 2εh / λ.
The product of the least inaccuracies with which
the positional coordinate and the component of
momentum in a definite direction can be ascertained
is therefore just given by formula (2). One
might perhaps expect that in estimating the accuracy
of determining the position, not only the
convergence but also the length of the wavetrain
has to be taken into account, because the particle
could change its place during the finite time of
illumination. Due to the fact, however, that the
exact knowledge of the wavelength is immaterial
for the above estimate, it will be realised that for
any value of the aperture the wavetrain can
always be taken so short that a change of position
of the particle during the time of observation may
be neglected in comparison to the lack of sharpness
inherent in the determination of position due to
the finite resolving power of the microscope.
Heisenberg looked up the answers to all the questions he could not answer, and the optical formula for resolution became the basis for his most famous work just a few years later. But not before Bohr pointed out a mistake in Heisenberg's first draft suggesting that a "disturbance" was the source of the uncertainty. Heisenberg says he was "brought to tears." In measuring momentum with the aid of the Doppler effect—with due regard to the Compton effect—one will employ a parallel wavetrain. For the accuracy, however, with which the change in wavelength of the scattered radiation can be measured the extent of the wavetrain in the direction of propagation is essential. If we assume that the directions of the incident and scattered radiation are parallel and opposite respectively to the direction of the position coordinate and momentum component to be measured, then c λ/ 2l can be taken as a measure of the accuracy in the determination of the velocity, where l denotes the length of the wavetrain. For simplicity, we here have regarded the velocity of light as large compared to the velocity of the particle. If m represents the mass of the particle, then the uncertainty attached to the value of the momentum after observation is cmλ/ 2l. In this case the magnitude of the recoil, 2h / λ, is sufficiently well defined in order not to give rise to an appreciable uncertainty in the value of the momentum of the particle after observation. Indeed, the general theory of the Compton effect allows us to compute the momentum components in the direction of the radiation before and after the recoil from the wavelengths of the incident and scattered radiation. Even if the positional coordinates of the particle were accurately known in the beginning, our knowledge of the position after observation nevertheless will be affected by an uncertainty. Indeed, on account of the impossibility of attributing a definite instant to the recoil, we know the mean velocity in the direction of observation during the scattering process only with an accuracy 2h / λ. The uncertainty in the position after observation hence is 2hl / mcλ. Here, too, the product of the inaccuracies in the measurement of position and momentum is thus given by the general formula (2). Just as in the case of the determination of position, the time of the process of observation for the determination of momentum may be made as short as is desired if only the wavelength of the radiation used is sufficiently small. The fact that the recoil then gets larger does not, as we have seen, affect the accuracy of measurement. It should further be mentioned, that in referring to the velocity of a particle as we have here done repeatedly, the purpose has only been to obtain a connexion with the ordinary spacetime description convenient in this ease. As it appears already from the considerations of de Broglie mentioned above, the concept of velocity must always in the quantum theory be handled with caution. It will also be seen that an unambiguous definition of this concept is excluded by the quantum postulate. This is particularly to be remembered when comparing the results of successive observations. Indeed, the position of an individual at two given moments can be measured with any desired degree of accuracy; but if, from such measurements, we would calculate the velocity of the individual in the ordinary way, it must be clearly realised that we are dealing with an abstraction, from which no unambiguous information concerning the previous or future behaviour of the individual can be obtained. According to the above considerations regarding the possibilities of definition of the properties of individuals, it will obviously make no difference in the discussion of the accuracy of measurements of position and momentum of a particle if collisions with other material particles are considered instead of scattering of radiation. In both cases we see that the uncertainty in question equally affects the description of the agency of measurement and of the object. In fact, this uncertainty cannot be avoided in a description of the behaviour of individuals with respect to a coordinate system fixed in the ordinary way by means of solid bodies and unperturbable clocks. The experimental devices—opening and closing of apertures, etc.— are seen to permit only conclusions regarding the spacetime extension of the associated wavefields. In tracing observations back to our sensations, once more regard has to be taken to the quantum postulate in connexion with the perception of the agency of observation, be it through its direct action upon the eye or by means of suitable auxiliaries such as photographic plates, Wilson clouds, etc. It is easily seen, however, that the resulting additional statistical element will not influence the uncertainty in the description of the object. It might even be conjectured that the arbitrariness in what is regarded as object and what as agency of observation would open up a possibility of avoiding this uncertainty altogether. In connexion with the measurement of the position of a particle, one might, for example, ask whether the momentum transmitted by the scattering could not be determined by means of the conservation theorem from a measurement of the change of momentum of the microscope — including light source and photographic plate — during the process of observation.
Bohr famously defended the uncertainty principle against criticisms by Einstein in his "discussion with Einstein" at the 1927 Solvay conference.
A closer investigation shows,
however, that such a measurement is impossible,
if at the same time one wants to know the position
of the microscope with sufficient accuracy. In
fact, it follows from the experiences which have
found expression in the wave theory of matter,
that the position of the centre of gravity of a body
and its total momentum can only be defined
within the limits of reciprocal accuracy given by
relation (2).
Strictly speaking, the idea of observation belongs to the causal spacetime way of description. Due to the general character of relation (2), however, this idea can be consistently utilised also in the quantum theory, if only the uncertainty expressed through this relation is taken into account. As remarked by Heisenberg, one may even obtain an instructive illustration to the quantum theoretical description of atomic (microscopic) phenomena by comparing this uncertainty with the uncertainty, due to imperfect measurements, inherently contained in any observation as considered in the ordinary description of natural phenomena. He remarks on that occasion that even in the case of macroscopic phenomena we may say, in a certain sense, that they are created by repeated observations. It must not be forgotten, however, that in the classical theories any succeeding observation permits a prediction of future events with everincreasing accuracy, because it improves our knowledge of the initial state of the system. According to the quantum theory, just the impossibility of neglecting the interaction with the agency of measurement means that every observation introduces a new uncontrollable element.
Bohr seems ready to accept the idea that quantum theory is acausal. It makes clear the complementarity of spacetime descriptions (now unsharp)
Indeed, it follows from the above considerations
that the measurement of the positional coordinates
of a particle is accompanied not only by
a finite change in the dynamical variables, but also
the fixation of its position means a complete rupture
in the causal description of its dynamical behaviour,
while the determination of its momentum
always implies a gap in the knowledge of its
spatial propagation. Just this situation brings
out most strikingly the complementary character
of the description of atomic phenomena which
appears as an inevitable consequence of the contrast
between the quantum postulate and the distinction
between object and agency of measurement,
inherent in our very idea of observation.
and claims of causality (now acausal). 4. CORRESPONDENCE PRINCIPLE AND MATRIX THEORY. Hitherto we have only regarded certain general features of the quantum problem. The situation implies, however, that the main stress has to be laid on the formulation of the laws governing the interaction between the objects which we symbolise by the abstractions of isolated particles and radiation. Points of attack for this formulation are presented in the first place by the problem of atomic constitution. As is well known, it has been possible here, by means of an elementary use of classical concepts and in harmony with the quantum postulate, to throw light on essential aspects of experience. For example, the experiments regarding the excitation of spectra by electronic impacts and by radiation are adequately accounted for on the assumption of discrete stationary states and individual transition processes. This is primarily due to the circumstance that in these questions no closer description of the spacetime behaviour of the processes is required. Here the contrast with the ordinary way of description appears strikingly in the circumstance that spectral lines, which on the classical view would be ascribed to the same state of the atom, will, according to the quantum postulate, correspond to separate transition processes, between which the excited atom has a choice. Notwithstanding this contrast, however, a formal connexion with the classical ideas could be obtained in the limit, where the relative difference in the properties of neighbouring stationary states vanishes asymptotically and where in statistical applications the discontinuities may be disregarded. Through this connexion it was possible to a large extent to interpret the regularities of spectra on the basis of our ideas about the structure of the atom. The aim of regarding the quantum theory as a rational generalisation of the classical theories led to the formulation of the socalled correspondence principle. The utilisation of this principle for the interpretation of spectroscopic results was based on a symbolical application of classical electrodynamics, in which the individual transition processes were each associated with a harmonic in the motion of the atomic particles to be expected according to ordinary mechanics. Except in the limit mentioned, where the relative difference between adjacent stationary states may be neglected, such a fragmentary application of the classical theories could only in certain cases lead to a strictly quantitative description of the phenomena. Especially the connexion developed by Ladenburg and Kramers between the classical treatment of dispersion and the statistical laws governing the radiative transition processes formulated by Einstein should be mentioned here. Although it was just Kramers' treatment of dispersion that gave important hints for the rational development of correspondence considerations, it is only through the quantum theoretical methods created in the last few years that the general aims laid down in the principle mentioned have obtained an adequate formulation. As is known, the new development was commenced in a fundamental paper by Heisenberg, where he succeeded in emancipating himself completely from the classical concept of motion by replacing from the very start the ordinary kinematical and mechanical quantities by symbols, which refer directly to the individual processes demanded by the quantum postulate.
Although Heisenberg supports Bohr's original invention of stationary states, he declares them to be "unobservable." Schrödinger, by contrast, visualizes them as his wave functions. See §5.
This was
accomplished by substituting for the Fourier
development of a classical mechanical quantity
a matrix scheme, the elements of which symbolise
purely harmonic vibrations and are associated
with the possible transitions between stationary
states. By requiring that the frequencies ascribed
to the elements must always obey the combination
principle for spectral lines, Heisenberg
could introduce simple rules of calculation for the
symbols, which permit a direct quantum theoretical
transcription of the fundamental equations of
classical mechanics. This ingenious attack on the
dynamical problem of atomic theory proved itself
from the beginning to be an exceedingly powerful
and fertile method for interpreting quantitatively
the experimental results. Through the work of
Born and Jordan as well as of Dirac, the theory
was given a formulation which can compete with
classical mechanics as regards generality and
consistency. Especially the element characteristic
of the quantum theory, Planck's constant, appears
explicitly only in the algorithms to which the
symbols, the socalled matrices, are subjected.
In fact, matrices, which represent canonically
conjugated variables in the sense of the Hamiltonian
equations, do not obey the commutative
law of multiplication, but two such quantities, q
and p, have to fulfil the exchange rule
pq  qp = √1 h / 2π, . . . (3)
Indeed, this exchange relation expresses strikingly the symbolical character of the matrix formulation of the quantum theory. The matrix theory has often been called a calculus with directly observable quantities. It must be remembered, however, that the procedure described is limited just to those problems, in which in applying the quantum postulate the spacetime description may largely be disregarded, and the question of observation in the proper sense therefore placed in the background. In pursuing further the correspondence of the quantum laws with classical mechanics, the stress placed on the statistical character of the quantum theoretical description, which is brought in by the quantum postulate, has been of fundamental importance. Here the generalisation of the symbolical method made by Dirac and Jordan represented a great progress by making possible the operation with matrices, which are not arranged according to the stationary states, but where the possible values of any set of variables may appear as indices of the matrix elements. In analogy to the interpretation considered in the original form of the theory of the 'diagonal elements' connected only with a single stationary state, as time averages of the quantity to be represented, the general transformation theory of matrices permits the representation of such averages of a mechanical quantity, in the calculation of which any set of variables characterising the 'state' of the system have given values, while the canonically conjugated variables are allowed to take all possible values. On the basis of the procedure developed by these authors and in close connexion with ideas of Born and Pauli, Heisenberg has in the paper already cited above attempted a closer analysis of the physical content of the quantum theory, especially in view of the apparently paradoxical character of the exchange relation (3). In this connexion he has formulated the relation
Δq Δp ∼ h, . . . (4)
as the general expression for the maximum accuracy with which two canonically conjugated variables can simultaneously be observed. In this way Heisenberg has been able to elucidate many paradoxes appearing in the application of the quantum postulate, and to a large extent to demonstrate the consistency of the symbolic method. In connexion with the complementary nature of the quantum theoretical description, we must, as already mentioned, constantly keep the possibilities of definition as well as of observation before the mind. For the discussion of just this question the method of wave mechanics developed by Schrödinger has, as we shall see, proved of great help. It permits a general application of the principle of superposition also in the problem of interaction, thus offering an immediate connexion with the above considerations concerning radiation and free particles. Below we shall return to the relation of wave mechanics to the general formulation of the quantum laws by means of the transformation theory of matrices. 5. WAVE MECHANICS AND QUANTUM POSTULATE. Already in his first considerations concerning the wave theory of material particles, de Broglie pointed out that the stationary states of an atom may be visualised as an interference effect of the phase wave associated with a bound electron. It is true that this point of view at first did not, as regards quantitative results, lead beyond the earlier methods of quantum theory, to the development of which Sommerfeld has contributed so essentially. Schrödinger, however, succeeded in developing a wave  theoretical method which has opened up new aspects, and has proved to be of decisive importance for the great progress in atomic physics during the last years.
Where Heisenberg, and especially Pauli, discounted Bohr's visualization of the stationary states, Schrödinger found a "natural" source of Bohr's postulated quantum numbers in the nodes of his wave functions.
Indeed, the proper
vibrations of the Schrödinger wave equation have
been found to furnish a representation of the
stationary states of an atom meeting all requirements.
The energy of each state is connected with
the corresponding period of vibration according to
the general quantum relation (1). Furthermore,
the number of nodes in the various characteristic
vibrations gives a simple interpretation to the
concept of quantum number which was already
known from the older methods, but at first did not
seem to appear in the matrix formulation. In
addition, Schrödinger could associate with the
solutions of the wave equation a continuous distribution
of charge and current, which, if applied
to a characteristic vibration, represents the
electrostatic and magnetic properties of an atom
in the corresponding stationary state. Similarly,
the superposition of two characteristic solutions
corresponds to a continuous vibrating distribution
of electrical charge, which on classical electrodynamics
would give rise to an emission of radiation,
illustrating instructively the consequences of the
quantum postulate and the correspondence requirement
regarding the transition process between two
stationary states formulated in matrix mechanics.
Max Born's statistical interpretation of the particle wave function (based on Einstein's statistical relation between light waves and the probability of photons) was anathema to Schrödinger
Another application of the method of Schrödinger,
important for the further development, has been
made by Born in his investigation of the problem
of collisions between atoms and free electric
particles. In this connexion he succeeded in
obtaining a statistical interpretation of the wave
functions, allowing a calculation of the probability
of the individual transition processes required by
the quantum postulate. This includes a wavemechanical
formulation of the adiabatic principle
of Ehrenfest, the fertility of which appears strikingly
in the promising investigations of Hund
on the problem of formation of molecules.
Schrödinger hoped to eliminate the discontinuous quantum jumps is Bohr's original quantum postulate, but then he would have no explanation for the observation of particles.
In view of these results, Schrödinger has expressed
the hope that the development of the
wave theory will eventually remove the irrational
element expressed by the quantum postulate and
open the way for a complete description of atomic
phenomena along the line of the classical theories.
In support of this view, Schrödinger, in a recent
paper (Ann. d. Phys., 83, p. 956; 1927), emphasises
the fact that the discontinuous exchange of energy
between atoms required by the quantum postulate,
from the point of view of the wave theory, is
replaced by a simple resonance phenomenon. In
particular, the idea of individual stationary states
would be an illusion and its applicability only an
illustration of the resonance mentioned. It must
be kept in mind, however, that just in the resonance
problem mentioned we are concerned with a closed
system which, according to the view presented here,
is not accessible to observation.
Wave mechanics and matrix mechanics might then be complementary in the same sense that wave and particle viewed as a duality are complementary.
In fact, wave
mechanics just as the matrix theory on this view
represents a symbolic transcription of the problem
of motion of classical mechanics adapted to the
requirements of quantum theory and only to be
interpreted by an explicit use of the quantum
postulate. Indeed, the two formulations of the
interaction problem might be said to be complementary
in the same sense as the wave and
particle idea in the description of the free individuals.
The apparent contrast in the utilisation
of the energy concept in the two theories is just
connected with this difference in the startingpoint.
The fundamental difficulties opposing a spacetime description of a system of particles in interaction appear at once from the inevitability of the superposition principle in the description of the behaviour of individual particles. Already for a free particle the knowledge of energy and momentum excludes, as we have seen, the exact knowledge of its spacetime coordinates. This implies that an immediate utilisation of the concept of energy in connexion with the classical idea of the potential energy of the system is excluded. In the Schrödinger wave equation these difficulties are avoided by replacing the classical expression of the Hamiltonian by a differential operator by means of the relation
p = √1 ( h / 2π ) δ / δq, . . . (5)
where p denotes a generalised component of momentum and q the canonically conjugated variable. Hereby the negative value of the energy is regarded as conjugated to the time. So far, in the wave equation, time and space as well as energy and momentum are utilised in a purely formal way. The symbolical character of Schrödinger's method appears not only from the circumstance that its simplicity, similarly to that of the matrix theory, depends essentially upon the use of imaginary arithmetic quantities. But above all there can be no question of an immediate connexion with our ordinary conceptions because the 'geometrical' problem represented by the wave equation is associated with the socalled coordinate space, the number of dimensions of which is equal to the number of degrees of freedom of the system, and hence in general greater than the number of dimensions of ordinary space. Further, Schrödinger's formulation of the interaction problem, just as the formulation offered by matrix theory, involves a neglect of the finite velocity of propagation of the forces claimed by relativity theory. On the whole, it would scarcely seem justifiable, in the case of the interaction problem, to demand a visualisation by means of ordinary spacetime pictures. In fact, all our knowledge concerning the internal properties of atoms is derived from experiments on their radiation or collision reactions, such that the interpretation of experimental facts ultimately depends on the abstractions of radiation in free space, and free material particles. Hence, our whole spacetime view of physical phenomena, as well as the definition of energy and momentum, depends ultimately upon these abstractions. In judging the applications of these auxiliary ideas we should only demand inner consistency, in which connexion special regard has to be paid to the possibilities of definition and observation.
Bohr likes Schrödinger's visualization of the stationary states in his quantum postulate. But we must replace spacetime descriptions  sharply defined mass points  with wavepacket superpositions of wave functions.
In the characteristic vibrations of Schrödinger's
wave equation we have, as mentioned, an adequate
representation of the stationary states of an atom
allowing an unambiguous definition of the energy
of the system by means of the general quantum
relation (1). This entails, however, that in the
interpretation of observations, a fundamental
renunciation regarding the spacetime description
is unavoidable. In fact, the consistent application
of the concept of stationary states excludes, as we
shall see, any specification regarding the behaviour
of the separate particles in the atom. In problems
where a description of this behaviour is essential,
we are bound to use the general solution of the
wave equation which is obtained by superposition
of characteristic solutions. We meet here
with a complementarity of the possibilities of
definition quite analogous to that which we have
considered earlier in connexion with the properties
of light and free material particles. Thus,
while the definition of energy and momentum of
individuals is attached to the idea of a harmonic
elementary wave, every spacetime feature of the
description of phenomena is, as we have seen, based
on a consideration of the interferences taking place
inside a group of such elementary waves. Also in
the present case the agreement between the possibilities
of observation and those of definition can
be directly shown.
According to the quantum postulate any observation regarding the behaviour of the electron in the atom will be accompanied by a change in the state of the atom. As stressed by Heisenberg, this change will, in the case of atoms in stationary states of low quantum number, consist in general in the ejection of the electron from the atom. A description of the 'orbit' of the electron in the atom with the aid of subsequent observations is hence impossible in such a case. This is connected with the circumstance that from characteristic vibrations with only a few nodes no wave packages can be built up which would even approximately represent the 'motion' of a particle. The complementary nature of the description, however, appears particularly in that the use of observations concerning the behaviour of particles in the atom rests on the possibility of neglecting, during the process of observation, the interaction between the particles, thus regarding them as free. This requires, however, that the duration of the process is short compared with the natural periods of the atom, which again means that the uncertainty in the knowledge of the energy transferred in the process is large compared to the energy differences between neighbouring stationary states. In judging the possibilities of observation it must, on the whole, be kept in mind that the wave mechanical solutions can be visualised only in so far as they can be described with the aid of the concept of free particles. Here the difference between classical mechanics and the quantum theoretical treatment of the problem of interaction appears most strikingly. In the former such a restriction is unnecessary, because the 'particles ' are here endowed with an immediate 'reality,' independently of their being free or bound. This situation is particularly important in connexion with the consistent utilisation of Schrödinger's electric density as a measure of the probability for electrons being present within given space regions of the atom. Probabilites for particles in a matterwave field (Born) are analogous to the probabilities for photons in a radiation field (Einstein).
Remembering the restriction
mentioned, this interpretation is seen to be a
simple consequence of the assumption that the
probability of the presence of a free electron is
expressed by the electric density associated with
the wavefield in a similar way to that by which the
probability of the presence of a light quantum is
given by the energy density of the radiation.
As already mentioned, the means for a general consistent utilisation of the classical concepts in the quantum theory have been created through the transformation theory of Dirac and Jordan, by the aid of which Heisenberg has formulated his general uncertainty relation (4). In this theory also the Schrödinger wave equation has obtained an instructive application. In fact, the characteristic solutions of this equation appear as auxiliary functions which define a transformation from matrices with indices representing the energy values of the system to other matrices, the indices of which are the possible values of the space coordinates. It is also of interest in this connexion to mention that Jordan and Klein (Zeitsch. f. Phys., 45, 751 ; 1927) have recently arrived at the formulation of the problem of interaction expressed by the Schrödinger wave equation, taking as startingpoint the wave representation of individual particles and applying a symbolic method closely related to the deepgoing treatment of the radiation problem developed by Dirac from the point of view of the matrix theory, to which we shall return below. 6. REALITY OF STATIONARY STATES. In the conception of stationary states we are, as mentioned, concerned with a characteristic application of the quantum postulate. By its very nature this conception means a complete renunciation as regards a time description. From the point of view taken here, just this renunciation forms the necessary condition for an unambiguous definition of the energy of the atom. Moreover, the conception of a stationary state involves, strictly speaking, the exclusion of all interactions with individuals not belonging to the system. The fact that such a closed system is associated with a particular energy value may be considered as an immediate expression for the claim of causality contained in the theorem of conservation of energy. This circumstance justifies the assumption of the supramechanical stability of the stationary states, according to which the atom, before as well as after an external influence, always will be found in a welldefined state, and which forms the basis for the use of the quantum postulate in problems concerning atomic structure. In a judgment of the wellknown paradoxes which this assumption entails for the description of collision and radiation reactions, it is essential to consider the limitations of the possibilities of definition of the reacting free individuals, which is expressed by relation (2). In fact, if the definition of the energy of the reacting individuals is to be accurate to such a degree as to entitle us to speak of conservation of energy during the reaction, it is necessary, according to this relation, to coordinate to the reaction a time interval long compared to the vibration period associated with the transition process, and connected with the energy difference between the stationary states according to relation (1). This is particularly to be remembered when considering the passage of swiftly moving particles through an atom. According to the ordinary kinematics, the effective duration of such a passage would be very small as compared with the natural periods of the atom, and it seemed impossible to reconcile the principle of conservation of energy with the assumption of the stability of stationary states (cf. Zeits. f. Phys., 34, 142 ; 1925). In the wave representation, however, the time of reaction is immediately connected with the accuracy of the knowledge of the energy of the colliding particle, and hence there can never be the possibility of a contradiction with the law of conservation. In connexion with the discussion of paradoxes of the kind mentioned, Campbell (Phil. Mag., i. 1106; 1926) suggested the view that the conception of time itself may be essentially statistical in nature. From the view advanced here, according to which the foundation of spacetime description is offered by the abstraction of free individuals, a fundamental distinction between time and space, however, would seem to be excluded by the relativity requirement. The singular position of the time in problems concerned with stationary states is, as we have seen, due to the special nature of such problems.
Unlike complex emergent biological systems, elementary physical particles have no "history." Information about the previous states of all atoms is needed for deterministic theories that reduce biology to physics.
The application of the conception of stationary
states demands that in any observation, say by
means of collision or radiation reactions, permitting
a distinction between different stationary states, we
are entitled to disregard the previous history of the
atom. The fact that the symbolical quantum theory
methods ascribe a particular phase to each stationary
state the value of which depends upon the
previous history of the atom, would for the first
moment seem to contradict the very idea of
stationary states. As soon as we are really concerned
with a time problem, however, the consideration
of a strictly closed system is excluded. The
use of simply harmonic proper vibrations in the
interpretation of observations means, therefore,
only a suitable idealisation which in a more rigorous
discussion must always be replaced by a group
of harmonic vibrations, distributed over a finite
frequency interval. Now, as already mentioned,
it is a general consequence of the superposition
principle that it has no sense to coordinate a
phase value to the group as a whole, in the same
manner as may be done for each elementary wave
constituting the group.
This inobservability of the phase, well known from the theory of optical instruments, is brought out in a particularly simple manner in a discussion of the SternGerlach experiment, so important for the investigation of the properties of single atoms. As pointed out by Heisenberg, atoms with different orientation in the field may only be separated if the deviation of the beam is larger than the diffraction at the slit of the de Broglie waves representing the translational motion of the atoms. This condition means, as a simple calculation shows, that the product of the time of passage of the atom through the field, and the uncertainty due to the finite width of the beam of its energy in the field, is at least equal to the quantum of action. This result was considered by Heisenberg as a support of relation (2) as regards the reciprocal uncertainties of energy and time values. It would seem, however, that here we are not simply dealing with a measurement of the energy of the atom at a given time. But since the period of the proper vibrations of the atom in the field is connected with the total energy by relation (1), we realise that the condition for separability mentioned just means the loss of the phase. This circumstance removes also the apparent contradictions, arising in certain problems concerning the coherence of resonance radiation, which have been discussed frequently, and were also considered by Heisenberg. To consider an atom as a closed system, as we have done above, means to neglect the spontaneous emission of radiation which even in the absence of external influences puts an upper limit to the lifetime of the stationary states. The fact that this neglect is justified in many applications is connected with the circumstance that the coupling between the atom and the radiation field, which is to be expected on classical electrodynamics, is in general very small compared to the coupling between the particles in the atom. It is, in fact, possible in a description of the state of an atom to a considerable extent to neglect the reaction of radiation, thus disregarding the unsharpness in the energy values connected with the lifetime of the stationary states according to relation (2) (cf. Proc. Camb. Phil. Soc., 1924 (Supplement), or Zeits. f. Phys., 13, 117; 1923). This is the reason why it is possible to draw conclusions concerning the properties of radiation by using classical electrodynamics. The treatment of the radiation problem by the new quantum theoretical methods meant to begin with just a quantitative formulation of this correspondence consideration. This was the very startingpoint of the original considerations of Heisenberg. It may also be mentioned that an instructive analysis of Schrödinger's treatment of the radiation phenomena from the point of view of the correspondence principle has been recently given by Klein (Zeits. f. Phys., 41,707; 1927). In the more rigorous form of the theory developed by Dirac (Proc. Roy. Soc., A, vol. 114, p. 243 ; 1927) the radiation field itself is included in the closed system under consideration. Thus it became possible in a rational way to take account of the individual character of radiation demanded by the quantum theory and to build up a dispersion theory, in which the final width of the spectral lines is taken into consideration. The renunciation regarding spacetime pictures characterising this treatment would seem to offer a striking indication of the complementary character of the quantum theory. This is particularly to be borne in mind in judging the radical departure from the causal description of Nature met with in radiation phenomena, to which we have referred above in connexion with the excitation of spectra. In view of the asymptotic connexion of atomic properties with classical electrodynamics, demanded by the correspondence principle, the reciprocal exclusion of the conception of stationary states and the description of the behaviour of individual particles in the atom might be regarded as a difficulty. In fact, the connexion in question means that in the limit of large quantum numbers where the relative difference between adjacent stationary states vanishes asymptotically, mechanical pictures of electronic motion may be rationally utilised. It must be emphasised, however, that this connexion cannot be regarded as a gradual transition towards classical theory in the sense that the quantum postulate would lose its significance for high quantum numbers. On the contrary, the conclusions obtained from the correspondence principle with the aid of classical pictures depend just upon the assumptions that the conception of stationary states and of individual transition processes are maintained even in this limit. This question offers a particularly instructive example for the application of the new methods. As shown by Schrödinger (Naturwiss., 14, 664 ; 1926), it is possible, in the limit mentioned, by superposition of proper vibrations to construct wave groups small in comparison to the ' size ' of the atom, the propagation of which indefinitely approaches the classical picture of moving material particles, if the quantum numbers are chosen sufficiently large. In the special ease of a simple harmonic vibrator, he was able to show that such wave groups will keep together even for any length of time, and will oscillate to and fro in a manner corresponding to the classical picture of the motion. This circumstance Schrödinger has regarded as a support of his hope of constructing a pure wave theory without referring to the quantum postulate. As emphasised by Heisenberg, the simplicity of the case of the oscillator, however, is exceptional and intimately connected with the harmonic nature of the corresponding classical motion. Nor is there in this example any possibility for an asymptotical approach towards the problem of free particles. In general, the wave group will gradually spread over the whole region of the atom, and the 'motion' of a bound electron can only be followed during a number of periods, which is of the order of magnitude of the quantum numbers associated with the proper vibrations. This question has been more closely investigated in a recent paper by Darwin (Proc. Roy. Soc, A, vol. 117, 258; 1927), which contains a number of instructive examples of the behaviour of wave groups. From the viewpoint of the matrix theory a treatment of analogous problems has been carried out by Kennard (Zeits. f. Phys., 47, 326 ; 1927). Here again we meet with the contrast between the wave theory superposition principle and the assumption of the individuality of particles with which we have been concerned already in the case of free particles. At the same time the asymptotical connexion with the classical theory, to which a distinction between free and bound particles is unknown, offers the possibility of a particularly simple illustration of the above considerations regarding the consistent utilisation of the concept of stationary states. As we have seen, the identification of a stationary state by means of collision or radiation reactions implies a gap in the time description, which is at least of the order of magnitude of the periods associated with transitions between stationary states. Now, in the limit of high quantum numbers these periods may be interpreted as periods of revolution. Thus we see at once that no causal connexion can be obtained between observations leading to the fixation of a stationary state and earlier observations on the behaviour of the separate particles in the atom.
Today quantum mechanics can not only observe "individual particles," but control their transitions between stationary states with exquisite accuracy.
Summarising, it might be said that the concepts
of stationary states and individual transition processes
within their proper field of application
possess just as much or as little 'reality' as the
very idea of individual particles. In both cases
we are concerned with a demand of causality
complementary to the spacetime description, the
adequate application of which is limited only by
the restricted possibilities of definition and of
observation.
7. THE PROBLEM OF THE ELEMENTARY PARTICLES. When due regard is taken of the complementary feature required by the quantum postulate, it seems, in fact, possible with the aid of the symbolic methods to build up a consistent theory of atomic phenomena, which may be considered as a rational generalisation of the causal spacetime description of classical physics. This view does not mean, however, that classical electron theory may be regarded simply as the limiting case of a vanishing quantum of action. Indeed, the connexion of the latter theory with experience is based on assumptions which can scarcely be separated from the group of problems of the quantum theory. A hint in this direction was already given by the wellknown difficulties met with in the attempts to account for the individuality of ultimate electrical particles on general mechanical and electrodynamical principles. In this respect also the general relativity theory of gravitation has not fulfilled expectations. A satisfactory solution of the problems touched upon would seem to be possible only by means of a rational quantumtheoretical transcription of the general field theory, in which the ultimate quantum of electricity has found its natural position as an expression of the feature of individuality characterising the quantum theory. Recently Klein (Zeits. f. Phys., 46, 188; 1927) has directed attention to the possibility of connecting this problem with the fivedimensional unified representation of electromagnetism and gravitation proposed by Kaluza. In fact, the conservation of electricity appears in this theory as an analogue to the conservation theorems for energy and momentum. Just as these concepts are complementary to the spacetime description, the appropriateness of the ordinary fourdimensional description as well as its symbolical utilisation in the quantum theory would, as Klein emphasises, seem to depend essentially on the circumstance that in this description electricity always appears in welldefined units, the conjugated fifth dimension being as a consequence not open to observation. Quite apart from these unsolved deepgoing problems, the classical electron theory up to the present time has been the guide for a further development of the correspondence description in connexion with the idea first advanced by Compton that the ultimate electrical particles, besides their mass and charge, are endowed with a magnetic moment due to an angular momentum determined by the quantum of action. This assumption, introduced with striking success by Goudsmit and Uhlenbeck into the discussion of the origin of the anomalous Zeeman effect, has proved most fruitful in connexion with the new methods, as shown especially by Heisenberg and Jordan. One might say, indeed, that the hypothesis of the magnetic electron, together with the resonance problem elucidated by Heisenberg (Zeits. f. Phys., 41, 239; 1927), which occurs in the quantumtheoretical description of the behaviour of atoms with several electrons, have brought the correspondence interpretation of the spectral laws and the periodic system to a certain degree of completion. The principles underlying this attack have even made it possible to draw conclusions regarding the properties of atomic nuclei. Thus Dennison (Proc. Roy. Soc., A, vol. 115, 483; 1927), in connexion with ideas of Heisenberg and Hund, has succeeded recently in a very interesting way in showing how the explanation of the specific heat of hydrogen, hitherto beset with difficulties, can be harmonised with the assumption that the proton is endowed with a moment of momentum of the same magnitude as that of the electron. Due to its larger mass, however, a magnetic moment much smaller than that of the electron must be associated with the proton. The insufficiency of the methods hitherto developed as concerns the problem of the elementary particles appears in the questions just mentioned from the fact that they do not allow of an unambiguous explanation of the difference in the behaviour of the electric elementary particles and the 'individuals' symbolised through the conception of fight quanta expressed in the socalled exclusion principle formulated by Pauli. In fact, we meet in this principle, so important for the problem of atomic structure as well as for the recent development of statistical theories, with one among several possibilities, each of which fulfils the correspondence requirement. Moreover, the difficulty of satisfying the relativity requirement in quantum theory appears in a particularly striking fight in connexion with the problem of the magnetic electron. Indeed, it seemed not possible to bring the promising attempts made by Darwin and Pauli in generalising the new methods to cover this problem naturally, in connexion with the relativity kinematical consideration of Thomas so fundamental for the interpretation of experimental results. Quite recently, however, Dirac (Proc. of the Roy. Soc., A, 117, 610; 1928) has been able successfully to attack the problem of the magnetic electron through a new ingenious extension of the symbolical method and so to satisfy the relativity requirement without abandoning the agreement with spectral evidence. In this attack not only the imaginary complex quantities appearing in the earlier procedures are involved, but his fundamental equations themselves contain quantities of a still higher degree of complexity, that are represented by matrices. Already the formulation of the relativity argument implies essentially the union of the spacetime coordination and the demand of causality characterising the classical theories. In the adaptation of the relativity requirement to the quantum postulate we must therefore be prepared to meet with a renunciation as to visualisation in the ordinary sense going still further than in the formulation of the quantum laws considered here.
Bohr is disingenuous. He has for decades denied Einstein's extraordinary insights into light quanta and the waveparticle duality that is the core of complementarity.
Indeed, we find ourselves here on the very path
taken by Einstein of adapting our modes of perception
borrowed from the sensations to the gradually
deepening knowledge of the laws of Nature. The
hindrances met with on this path originate above
all in the fact that, so to say, every word in the
language refers to our ordinary perception. In
the quantum theory we meet this difficulty at once
in the question of the inevitability of the feature
of irrationality characterising the quantum postulate.
I hope, however, that the idea of complementarity
is suited to characterise the situation,
which bears a deepgoing analogy to the general
difficulty in the formation of human ideas, inherent
in the distinction between subject and object.
For Teachers
For Scholars
