At the end of this lecture I may express my critical opinion, that a correct theory should neither lead to infinite zero-point energies nor to infinite zero charges, that it should not use mathematical tricks to subtract infinities or singularities, nor should it invent a “hypothetical world” which is only a mathematical fiction before it is able to formulate the correct interpretation of the actual world of physics.
From the point of view of logic, my report on “Exclusion principle and quantum mechanics” has no conclusion. I believe that it will only be possible to write the conclusion if a theory will be established which will determine the value of the fine-structure constant and will thus explain the atomistic structure of electricity, which is such an essential quality of all atomic sources of electric fields actually occurring in Nature.
REFERENCES
1 A. Landé, Z. Physik, 5 (1921) 231 and Z. Physik, 7 (1921) 398, Physik. Z., 22 (1921) 417.
2 W. Pauli, Z. Physik, 16 (1923) 155.
3 W. Pauli, Z. Physik, 31 (1925) 373.
4 E. C. Stoner, Phil. Mag., 48 (1924) 719.
5 W. Pauli, Z. Physik, 31 (1925) 765.
6 S. Goudsmit and G. Uhlenbeck, Naturwiss., 13 (1925) 953, Nature, 117 (1926) 264.
7 L. H. Thomas, Nature, 117 (1926) 514, and Phil. Mag., 3 (1927) 1. Compare also J. Frenkel, Z. Physik, 37 (1926) 243.
8 Compare Rapport du Sixième Conseil Solvay de Physique, Paris, 1932, pp. 217–225.
9 For this earlier stage of the history of the exclusion principle compare also the author’s note in Science, 103 (1946) 213, which partly coincides with the first part of the present lecture.
10 The Nobel Lectures of W. Heisenberg, E. Schrödinger, and P. A. M. Dirac are collected in Die moderne Atomtheorie, Leipzig, 1934.
11 The articles of N. Bohr are collected in Atomic Theory and the Description of Nature, Cambridge University Press, 1934. See also his article “Light and Life”, Nature, 131 (1933) 421, 457.
12 W. Heisenberg, Z. Physik, 38 (1926) 411 and 39 (1926) 499.
13 E. Fermi, Z. Physik, 36 (1926) 902. P. A. M. Dirac, Proc. Roy. Soc. London, A 112 (1926) 661.
14 S. N. Bose, Z. Physik, 26 (1924) 178 and 27 (1924) 384. A. Einstein, Berl. Ber., (1924) 261; (1925) 1, 18.
15 W. Pauli, Naturwiss., 12 (1924) 741.
16 W. Heisenberg, Z. Physik, 41 (1927) 239, F. Hund, Z. Physik, 42 (1927) 39.
17 D. M. Dennison, Proc. Roy. Soc. London, A 115 (1927) 483.
18 R. de L. Kronig, Naturwiss., 16 (1928) 335. W. Heitler und G. Herzberg, Naturwiss., 17 (1929) 673.
19 G. N. Lewis and M. F. Ashley, Phys. Rev., 43 (1933) 837. G. M. Murphy and H. Johnston, Phys. Rev., 45 (1934) 550 and 46 (1934) 95.
20 Compare for the following the author’s report in Rev. Mod. Phys., 13 (1941) 203, in which older literature is given. See also W. Pauli and V. Weisskopf, Helv. Phys. Acta, 7 (1934) 809.
21 N. Bohr and L. Rosenfeld, Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd., 12 [8] (1933).
22 P. Jordan and E. Wigner, Z. Physik, 47 (1928) 631. Compare also V. Fock, Z. Physik, 75 (1932) 622.
23 W. Pauli, Ann. Inst. Poincaré, 6 (1936) 137 and Phys. Rev., 58 (1940) 716.
24 L. Landau and R. Peierls, Z. Physik, 69 (1931) 56. Compare also the author’s article in Handbuch der Physik, 24, Part 1, 1933, Chap. A, §2.
25 P. A. M. Dirac, Proc. Roy. Soc. London, A 117 (1928) 610.
Chapter Five
In the papers covered in the previous chapters, much of the mathematical basis of quantum physics was worked out. However, the more philosophical questions about what quantum theory says about reality and what the “right way” to interpret quantum physics is remain. The standard interpretation of quantum mechanics arose from Niels Bohr and his collaborators, so it has been named the “Copenhagen interpretation.” Two fundamental postulates of the Copenhagen interpretation are that we should only be concerned with what is actually observed and that the quantum wave function or state vector of a system contains all possible information for that system. These two postulates seem very reasonable, but they lead to many strange results. Since the wave function contains all possible information about a system, it is crucially important to understand what it means. How do we interpret the wave function? The mostly widely accepted answer comes from Max Born. He argued that the wave function (or more precisely the square of the amplitude of the wave function) represents the probability that an event will occur. In this sense, quantum mechanics is non-deterministic. In a deterministic theory, when a system starts out with a given initial state, its final state can be calculated from the theory at all times. But in quantum mechanics this is not the case. Identical experiments with identical starting conditions can produce different results. All we can do is calculate the probability that a system will end in a certain final state.
The inherently statistical nature of quantum theory troubled many physicists. In fact, some of the theory’s greatest contributors balked at the strange notions of reality that quantum mechanics seemed to indicate. Notable among these are Albert Einstein and Erwin Schrodinger. Schrodinger was troubled by the idea that according to the standard interpretation of quantum mechanics, a system actually exists in all possible states until a measurement is made and collapses the wave function of that system into a single state. In “The Present Situation in Quantum Mechanics,” Schrodinger introduces his famous cat experiment, which was meant to show the absurdity of the Copenhagen interpretation of quantum mechanics. In this thought experiment he imagines a cat which is in a sealed box with a radioactive source and detector. When a radioactive decay is detected cyanide is released killing the cat. According to the Copenhagen interpretation so long as an observation is not made the radioactive source will exist in a superposition of decayed and non-decayed states. Schrodinger maintained this results in an absurd conclusion, because the cyanide will be simultaneously released and not-released and the cat will simultaneously exist in dead and living states, which is, of course, impossible.
Another attack on the standard interpretation of quantum theory came from Albert Einstein, Boris Podolsky, and Nathan Rosen in a paper in which they argued that quantum mechanics cannot be a complete theory of reality. Like Schrodinger, they were also troubled by the statistical nature of quantum theory, and they wondered if perhaps there was a deeper reality hidden beyond that which was represented by the quantum mechanical wave function. They were also very troubled by the idea of what is now called quantum entanglement. That is, if two systems are allowed to interact and then are separated, a measurement on one system causing the collapse of its wave function can instantaneously cause the wave function of the other system to collapse as well. Einstein called this “spooky action at a distance,” and we can imagine why it was especially troubling to him. At first glance it seems to violate his theory of relativity. It appears that some signal must be traveling instantaneously between the two systems carrying the news that a measurement has been made. It has since been shown that no information is actually carried in the collapse, and so it is not actually a violation of relativity. Nevertheless, the idea of instantaneous collapse is still very troubling to many physicists. Einstein, Podolsky, and Rosen concluded that quantum mechanics as it is formulated cannot be a complete description of reality. In other words, they conclude that there must be elements of reality that are “hidden” from quantum mechanics. When these are taken into consideration, the spooky action at a distance that so troubled Einstein vanishes. It seems that it was Einstein’s hope until his death that a more complete theory of reality—a so-called local hidden-variables theory—would someday replace quantum mechanics.
In 1952 David Bohm produced two papers in which he attempted to create a hidden-variables interpretation of quantum physics. But in order to make his theory match the experimental observations he was not able to make his theory completely deterministic, nor was he able to eliminate the “non-local” spooky action at a distance which so troubled Einstein. In fact, unfortunately
for Einstein it is not likely that such a theory can exist. In the extremely creative paper, “On the Einstein-Podolsky-Rosen Paradox,” John Bell showed that any local hidden-variables theory will make predictions about measurable quantities that differ from those of quantum mechanics. Careful experimentation has continually supported the predictions made by quantum mechanics. Despite the objections of Einstein, Podolsky, and Rosen, quantum mechanics seems to be a more accurate representation of reality than deterministic hidden-variable theories.
THE STATISTICAL INTERPRETATION OF QUANTUM MECHANICS
BY
MAX BORN
Nobel Lecture, December 11, 1954
The work, for which I have had the honour to be awarded the Nobel Prize for 1954, contains no discovery of a fresh natural phenomenon, but rather the basis for a new mode of thought in regard to natural phenomena. This way of thinking has permeated both experimental and theoretical physics to such a degree that it hardly seems possible to say anything more about it that has not been already so often said. However, there are some particular aspects which I should like to discuss on what is, for me, such a festive occasion. The first point is this: the work at the Göttingen school, which I directed at that time (1926–1927), contributed to the solution of an intellectual crisis into which our science had fallen as a result of Planck’s discovery of the quantum of action in 1900. Today, physics finds itself in a similar crisis - I do not mean here its entanglement in politics and economics as a result of the mastery of a new and frightful force of Nature, but I am considering more the logical and epistemological problems posed by nuclear physics. Perhaps it is well at such a time to recall what took place earlier in a similar situation, especially as these events are not without a definite dramatic flavour.
The second point I wish to make is that when I say that the physicists had accepted the concepts and mode of thought developed by us at the time, I am not quite correct. There are some very noteworthy exceptions, particularly among the very workers who have contributed most to building up the quantum theory. Planck, himself, belonged to the sceptics until he died. Einstein, De Broglie, and Schrödinger have unceasingly stressed the unsatisfactory features of quantum mechanics and called for a return to the concepts of classical, Newtonian physics while proposing ways in which this could be done without contradicting experimental facts. Such weighty views cannot be ignored. Niels Bohr has gone to a great deal of trouble to refute the objections. I, too, have ruminated upon them and believe I can make some contribution to the clarification of the position. The matter concerns the borderland between physics and philosophy, and so my physics lecture will partake of both history and philosophy, for which I must crave your indulgence.
First of all, I will explain how quantum mechanics and its statistical interpretation arose. At the beginning of the twenties, every physicist, I think, was convinced that Planck’s quantum hypothesis was correct. According to this theory energy appears in finite quanta of magnitude hv in oscillatory processes having a specific frequency v (e.g. in light waves). Countless experiments could be explained in this way and always gave the same value of Planck’s constant h. Again, Einstein’s assertion that light quanta have momentum hv/c (where c is the speed of light) was well supported by experiment (e.g. through the Compton effect). This implied a revival of the corpuscular theory of light for a certain complex of phenomena. The wave theory still held good for other processes. Physicists grew accustomed to this duality and learned how to cope with it to a certain extent.
In 1913 Niels Bohr had solved the riddle of line spectra by means of the quantum theory and had thereby explained broadly the amazing stability of the atoms, the structure of their electronic shells, and the Periodic System of the elements. For what was to come later, the most important assumption of his teaching was this: an atomic system cannot exist in all mechanically possible states, forming a continuum, but in a series of discrete “stationary” states. In a transition from one to another, the difference in energy Em − En is emitted or absorbed as a light quantum hvmn (according to whether Em is greater or less than En). This is an interpretation in terms of energy of the fundamental law of spectroscopy discovered some years before by W. Ritz. The situation can be taken in at a glance by writing the energy levels of the stationary states twice over, horizontally and vertically. This produces a square array
in which positions on a diagonal correspond to states, and non-diagonal positions correspond to transitions.
It was completely clear to Bohr that the law thus formulated is in conflict with mechanics, and that therefore the use of the energy concept in this connection is problematical. He based this daring fusion of old and new on his principle of correspondence. This consists in the obvious requirement that ordinary classical mechanics must hold to a high degree of approximation in the limiting case where the numbers of the stationary states, the so-called quantum numbers, are very large (that is to say, far to the right and to the lower part in the above array) and the energy changes relatively little from place to place, in fact practically continuously.
Theoretical physics maintained itself on this concept for the next ten years. The problem was this: an harmonic oscillation not only has a frequency, but also an intensity. For each transition in the array there must be a corresponding intensity. The question is how to find this through the considerations of correspondence? It meant guessing the unknown from the available information on a known limiting case. Considerable success was attained by Bohr himself, by Kramers, Sommerfeld, Epstein, and many others. But the decisive step was again taken by Einstein who, by a fresh derivation of Planck’s radiation formula, made it transparently clear that the classical concept of intensity of radiation must be replaced by the statistical concept of transition probability. To each place in our pattern or array there belongs (together with the frequency vmn = (En − Em)/h) a definite probability for the transition coupled with emission or absorption.
In Göttingen we also took part in efforts to distil the unknown mechanics of the atom from the experimental results. The logical difficulty became ever sharper. Investigations into the scattering and dispersion of light showed that Einstein’s conception of transition probability as a measure of the strength of an oscillation did not meet the case, and the idea of an amplitude of oscillation associated with each transition was indispensable. In this connection, work by Ladenburghe, Kramerhf, Heisenberghg, Jordan and mehh should be mentioned. The art of guessing correct formulae, which deviate from the classical formulae, yet contain them as a limiting case according to the correspondence principle, was brought to a high degree of perfection. A paper of mine, which introduced, for the first time I think, the expression quantum mechanics in its title, contains a rather involved formula (still valid today) for the reciprocal disturbance of atomic systems.
Heisenberg, who at that time was my assistant, brought this period to a sudden endhi. He cut the Gordian knot by means of a philosophical principle and replaced guess-work by a mathematical rule. The principle states that concepts and representations that do not correspond to physically observable facts are not to be used in theoretical description. Einstein used the same principle when, in setting up his theory of relativity, he eliminated the concepts of absolute velocity of a body and of absolute simultaneity of two events at different places. Heisenberg banished the picture of electron orbits with definite radii and periods of rotation because these quantities are not observable, and insisted that the theory be built up by means of the square arrays mentioned above. Instead of describing the motion by giving a coordinate as a function of time, x(t), an array of transition amplitudes xmn should be determined. To me the decisive part of his work is the demand to determine a rule by which from a given
can be found (or, more general, the multiplication rule for such arrays).
By observation of known examples solved by guess-work he found this rule and applied it successfully to simple examples such as the harmonic and anharmonic oscillator.
This was in the summer of 1925. Heisenberg, plagued by hay fever took leave for a course of treatment by the sea and gave me his paper for publication if I thought I could do something with it.
The significance of the idea was at once clear to me and I sent the manuscript to the Zeitschrift für Physik. I could not take my mind off Heisenberg’s multiplication rule, and after a week of intensive thought and trial I suddenly remembered an algebraic theory which I had learned from my teacher, Professor Rosanes, in Breslau. Such square arrays are well known to mathematicians and, in conjunction with a specific rule for multiplication, are called matrices. I applied this rule to Heisenberg’s quantum condition and found that this agreed in the diagonal terms. It was easy to guess what the remaining quantities must be, namely, zero; and at once there stood before me the peculiar formula
This meant that coordinates q and momenta p cannot be represented by figure values but by symbols, the product of which depends upon the order of multiplication - they are said to be “non-commuting”.
I was as excited by this result as a sailor would be who, after a long voyage, sees from afar, the longed-for land, and I felt regret that Heisenberg was not there. I was convinced from the start that we had stumbled on the right path. Even so, a great part was only guess-work, in particular, the disappearance of the non-diagonal elements in the above-mentioned expression. For help in this problem I obtained the assistance and collaboration of my pupil Pascual Jordan, and in a few days we were able to demonstrate that I had guessed correctly. The joint paper by Jordan and myselfhj contains the most important principles of quantum mechanics including its extension to electrodynamics. There followed a hectic period of collaboration among the three of us, complicated by Heisenberg’s absence. There was a lively exchange of letters; my contribution to these, unfortunately, have been lost in the political disorders. The result was a three-author paperhk which brought the formal side of the investigation to a definite conclusion. Before this paper appeared, came the first dramatic surprise: Paul Dirac’s paper on the same subjecthl. The inspiration afforded by a lecture of Heisenberg’s in Cambridge had led him to similar results as we had obtained in Göttingen except that he did not resort to the known matrix theory of the mathematicians, but discovered the tool for himself and worked out the theory of such non-commutating symbols.
The Dreams That Stuff is Made of Page 42