Physics papers: I. Foundations of quantum mechanics
This page gives plain-language summaries of physics papers. We draw no link to Dost or to any other text; reading side by side is the reader's own work.
Experiment: measured. Theory: calculated or proved. Interpretation: its meaning is disputed. The labels are our own classification; a paper may have more than one nature.
Main question: Is Ψ merely information, or physical reality?
In quantum mechanics the state of a system is written with a mathematical object called the wave function, denoted Ψ (psi), which changes over time according to the Schrödinger equation. In experiments, however, what we get are always single results: a particle was found here, a counter clicked. The question is whether Ψ is a computational tool for these results (summarizing the information we have) or describes something real that belongs to the system itself. The three papers on this subject range from the first interpretation (Schrödinger, Born) to the tying of this question to a precise theoretical theorem (Pusey-Barrett-Rudolph).
On this page1. What Is the Wave Function? · Schrödinger 1926 — An Undulatory Theory · Born 1926 — Zur Quantenmechanik der Stoß …
Schrödinger summarized, in English, his wave equation and how the energy levels of the hydrogen atom are found with it (a summary of the four-part series 'Quantisierung als Eigenwertproblem' in Annalen der Physik).
This paper is an English summary of the work Schrödinger published in German in 1926 in four parts. The starting point was this: why the energy of atoms can take only certain values was being explained at the time by rules called 'quantum conditions' that were added from outside. Instead of adding these rules from outside, Schrödinger proposed to show that the solutions of a wave equation are, by themselves, well-behaved (finite and single-valued) only for certain values. Much as a taut string can vibrate only at certain frequencies.
The paper first sets up an equation for a wave function called ψ, starting from the energy relation that classical mechanics gives for a particle. Then this equation is applied to the hydrogen atom: under the electric attraction of the nucleus, the well-behaved solutions of the equation exist only at certain energy values, and these values agree with the measured spectral lines of hydrogen. The same method is applied to other systems such as the oscillator.
On interpretation, Schrödinger himself was largely inclined toward a classical reading: he thought of ψ as a real wave spread out in space, and of the particle as a region where this wave is concentrated. He also notes that this wave form gives the same results as the first matrix form of quantum mechanics (Heisenberg); the mathematical equivalence of the two forms was shown in this period.
Another feature of the paper is that it also contains a time-dependent form of the wave equation: this equation gives how ψ changes over time. This is what is known today as the Schrödinger equation, and it has remained the rule for the smooth change of a system in quantum mechanics.
The open point the paper leaves is what exactly ψ describes. While the wave spreads through space, the electron was always seen at a single point in experiments; within a year this mismatch led to a different interpretation.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: The same year Born proposed the probability interpretation in place of reading ψ as a charge density; this interpretation is the standard reading in textbooks today.
Born treated the problem of a particle scattering off an atom with wave mechanics and proposed that the square of the wave function gives a probability (Z. Phys. 37, 863).
Born's paper is about scattering, that is, a beam of particles striking an atom and being dispersed in various directions. Until then Schrödinger's wave mechanics had been used for stationary states inside the atom; Born asked whether the same equation could also describe particles flying freely and striking an obstacle.
In the calculation, the incoming particle is represented by a plane wave, and the effect of the atom enters the equation as a kind of obstacle. In the solution, after the collision the wave acquires a component spreading in all directions. How large the wave is in a given direction turns out to be a quantity that determines how often the particle is scattered in that direction.
What Born leaves here is the stance he takes on what the wave is. The wave is not a substance spread everywhere; in the experiment a single particle goes in a single direction. The wave function gives how probable it is that it goes in a given direction. In a footnote added to the published text, Born says that the probability is proportional to the square of the magnitude of the wave function. In this paper the concept of probability enters quantum mechanics for the first time in this way; what the theory computes is not individual results but the distribution of results.
The practical outcome of the calculation was also important: it made it possible to compute theoretically the scattering frequencies measured in experiments. The name 'Born approximation', used today in the analysis of scattering experiments, is tied to this work.
Its limit is this: the paper does not explain why the theory itself is probabilistic; it only shows that probability arises naturally in the scattering calculation. Whether this says something about nature or about our knowledge remains an open question.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: This interpretation is known as the Born rule and appears today in all quantum mechanics textbooks; what Ψ represents, however, has remained contested (see Pusey-Barrett-Rudolph).
Pusey, Barrett and Rudolph showed that a class of models saying the quantum state is merely information makes predictions that conflict with quantum theory.
The problem at the start of the paper was this: does the quantum state really show a property of the system, or is it merely a summary of the information we have? Those who hold that the quantum state is merely information liken it to the situation in classical physics: saying 'the probability of heads is one half' before tossing a coin says nothing about the coin itself; it describes our lack of information. Something else determines the real state of the system, and the quantum state summarizes our knowledge about this real state.
The authors expressed this view mathematically. Suppose every system has a real state; let different quantum states sit over these real states as probability distributions. If the 'merely information' view is correct, the distributions of two different quantum states must be able to partially overlap: the same real state can be compatible with two different pieces of information. The authors added a single extra assumption: the real states of two separately prepared systems are independent of each other.
With this assumption, when two systems are prepared together and a joint measurement is made, the existence of an overlap makes possible an outcome that quantum theory does not predict. That is, if there is overlap, the model conflicts with the theory. Conclusion: under this assumption the quantum state cannot be merely information; it is therefore something that corresponds to the real state.
One must attend to the scope of the theorem: the result does not exclude every view that treats the quantum state as information, but the models carrying the independence assumption above. Views that drop this assumption or set it up in another form fall outside the theorem's scope.
The theorem depends on its assumption: models that reject the independent-preparation assumption remain outside this result.
Afterwards: The result became a subject of both debate and experiment: Leifer's review collects the result and the objections raised to it, and Ringbauer and colleagues carried out a similar test in the laboratory.
2. The Measurement Problem
Main question: What happens during measurement?
Quantum mechanics says the state of a system changes smoothly and deterministically by the Schrödinger equation. But when a measurement is made we see a single result, and the state seems to jump to the state belonging to that result. These two rules (smooth change and the jump at measurement) stand side by side within the same theory, and what a measurement is and where it begins is not said in the theory. This is the measurement problem. Three papers describe the approach that first set up the problem in precise form (von Neumann), the one that asks about the role of consciousness (Wigner), and the one that deals with interaction with the environment (Zurek).
Springer, Berlin (1932; yeniden basım 1996) theory
Von Neumann placed quantum mechanics on a sound mathematical foundation over Hilbert space and for the first time analyzed the measurement process in detail within the theory (the book Mathematische Grundlagen der Quantenmechanik).
This work is not a paper but a book. Its aim is to gather quantum mechanics, independently of the various calculation methods used until then (matrix, wave), into a single mathematical framework. The framework is a vector space called Hilbert space: states are vectors in this space, and measurable quantities are certain operators acting on the space.
On measurement, von Neumann distinguishes two separate processes. The first is the smooth, reversible change of the system according to the Schrödinger equation when no measurement is made. The second is the non-smooth change that occurs in measurement: the state passes at once to the state corresponding to one of the possible values of the measured quantity, and which value will come out can be said only with probability.
Von Neumann then describes the measuring apparatus itself with quantum mechanics. When the measured system and the apparatus interact, by the rules of smooth change the system and apparatus come to an entangled state. You can measure the apparatus with another apparatus, and that one with the next; this chain is called the von Neumann chain. At no link of the chain do the smooth rules of the theory produce a single result. Where we apply the jump along the chain is arbitrary, and the results are not affected by this; this arbitrary cut point is important for the internal consistency of the theory.
Another contribution of the book is that it gives an exact mathematical form for the expectation values of measurable quantities and places the calculation of probability in this framework. The presentation of quantum mechanics in later textbooks largely follows this framework.
The same book also contains a proof arguing that no 'hidden variables' can be found beneath quantum mechanics. The assumptions of this proof were later sharply criticized by Bell.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: The question of what the jump in measurement corresponds to has been open since this book; different interpretations (Copenhagen, many worlds, Bohmian mechanics) give different answers to it.
I. J. Good (ed.), The Scientist Speculates, Heinemann, Londra, 1961; yeniden basım: Philosophical Reflections and Syntheses, Springer 1995, s. 247-260 interpretation
Wigner proposed a thought experiment relating measurement to the observer's consciousness ('Wigner's friend') and an interpretation tied to it.
Wigner's essay appeared in a collection titled 'The Scientist Speculates', and its title tells the nature of the piece: this is not an experiment but a piece of thinking. Wigner's question is at what point the measurement process within quantum mechanics must end in a single result.
In the thought experiment, in a closed laboratory a friend measures a system and sees the result. Wigner, outside, without ever seeing inside the laboratory, describes the measured system, the apparatus and the friend as a single quantum system. If the smooth-change rule of quantum mechanics is applied, everything inside the laboratory, including the friend's state of 'I saw the result', enters a probabilistic superposition. Yet the friend, when asked, says that he saw a definite result.
From this contradiction Wigner draws an interpretation holding that the determination of the measurement result occurs only when a conscious being becomes aware of the result. In other words, the jump occurs where consciousness comes into play; consciousness is an influence lying outside the smooth rules of quantum mechanics.
Wigner's aim in this essay is less to defend an interpretation than to question whether the theory is consistent with a description that leaves consciousness outside. The reasoning in the essay ties the subject of measurement to the question 'who counts as an observer'. The language of the essay is also careful: Wigner presents the conclusion not as a certainty but as a suggestion following the logical consequences of the theory.
This interpretation is referred to as a speculative proposal. The thought experiment, on the other hand, was later taken up again and, independently of the consciousness claim, became the subject of studies questioning what different observers can say about the same event.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: Wigner's thought experiment became a source for later work examining the notion of observer-independent events; the experiment by Proietti and colleagues is along this line.
Zurek reviewed decoherence theory, which explains why the interaction of a quantum system with its environment means that the objects of everyday life do not show quantum superpositions.
Zurek's question is this: if quantum mechanics governs the world, why do tables, cats and even grains of sand never appear to be in two places at once? The paper is a review describing the role that the interaction of a quantum system with its environment plays in this question.
The idea is this. No system stands alone: it constantly interacts with air, light, its own internal parts. This interaction causes some properties of the system to be 'read' by the environment. As the environment reads these properties, the quantum interference (the observable trace of superposition) between different values of the system disappears. This process is called decoherence. The properties the environment 'reads' are selected states; Zurek calls these pointer states and calls the selection process environment-induced superselection (einselection).
The paper argues that this process gives a picture of how the classical world emerges: in large objects decoherence is very fast and only certain states survive; what remains resembles a classical observation. In measurements too, the quantum link between the apparatus and the measured system gives way to a classical relation.
The paper also describes how the rate of decoherence depends on the size of the system and on its interaction with its environment: it can be slow in very small systems and extremely fast in large objects. This is a measure that accounts for quantum effects being easy to observe in the microscopic world and hard to observe in the everyday world.
Its limit is that decoherence alone does not explain why a single result is seen: when one looks together with the environment, all the results are still standing together. For this reason it is contested whether decoherence solves the measurement problem.
Afterwards: Today decoherence is both a phenomenon measured in experiments and one of the main obstacles to quantum computers; whether it solves the measurement problem by itself is a debate that varies with interpretations.
3. The Copenhagen Interpretation
Main question: Is there physical reality before observation?
The Copenhagen interpretation, which took shape around Bohr and Heisenberg in the late 1920s, is the first widespread interpretation of quantum mechanics. It is not a single text; it denotes the close but not identical views of different authors. The common point is the difficulty of speaking about quantum objects apart from the experimental arrangement. Three texts: Heisenberg's uncertainty principle, Bohr's idea of complementarity, and Bohr's reply to the EPR paper.
Heisenberg questioned what words such as 'position' and 'velocity' mean in quantum theory and for the first time set out the uncertainty relation.
The title of Heisenberg's 1927 paper can be translated as 'On the perceptual content of quantum theoretical kinematics and mechanics'. His question is this: the mathematics of quantum mechanics works, but what does it mean in this theory to say 'an electron is here and moving with this velocity'?
Heisenberg's first premise is an operational criterion: an expression like 'the position of a particle' carries meaning only if we can describe an experiment that would measure it. As an example he takes measuring the position of an electron with a microscope. To find the position very precisely, light of very short wavelength, for example gamma rays, is needed. But when a photon of short-wavelength light strikes the electron, it greatly changes the electron's velocity and hence its momentum. The better we know the position, the worse we know the momentum, and vice versa. Heisenberg calls this 'uncertainty' and estimates that their product is of the order of Planck's constant.
The paper places this idea within the mathematical theory: it is the natural consequence of position and momentum being represented in the theory by quantities that do not commute (whose order matters).
Toward the end of the paper, Heisenberg also discusses the effect of this principle on the notion of 'causality': if we cannot know the present exactly, we cannot predict the future exactly either. This claim is also cited as one of the disputed parts of the paper.
The paper has been read in two ways. One reading takes it as a limit concerning the disturbance of the system by measurement. Another takes it as a theoretical limit concerning the absence of simultaneously definite values for the quantities. Heisenberg's own microscope example is close to the first reading, while the exact mathematical form of the relation comes from the later work of Kennard and Robertson.
Afterwards: The limit concerning disturbance during measurement is not the same as Heisenberg's first form: the works of Ozawa and Busch examined this distinction, and Rozema and colleagues tested it experimentally.
Bohr first explained the quantum postulate and the idea of complementarity in his 1927 lecture at Como, Italy (Nature 121, 580).
This piece is the printed version of the lecture Bohr gave at a congress in Como in September 1927. Bohr's question is how we are to use the concepts of classical physics (position, momentum, energy, wave, particle) when describing events at the atomic level.
Bohr takes as his starting point the quantum postulate: the action within an atomic process (a quantity in terms of energy times time) is not infinitely divisible but has a definite smallest unit. The consequence of this is that observing an atomic event exerts on that event an influence that cannot be prevented. The observed object and the observing apparatus cannot be treated separately.
From here Bohr moves on to the concept of complementarity. A description in space and time and causality (a description based on the conservation of energy and momentum) cannot be applied precisely at the same time; the two are complementary to each other. Likewise, the wave and particle aspects of light or matter appear in different experimental arrangements and cannot be seen together in the same arrangement. These aspects exclude each other, but both are necessary to describe the atomic event completely.
Bohr also stresses that these descriptions can be made only in classical terms, because we report the results of experiments in the classical language of the apparatus. He interprets Heisenberg's relation within this framework.
The piece also takes up the question of at what point the quantum behavior of the measuring apparatus passes into classical description: a classical description is needed for the transition to observation, but no precise definition of the place of this transition is given. In this piece Bohr also treats the 'discontinuous' appearance of transitions within the atom in this framework, that is, as a consequence of the indivisibility of action.
Its limit: Bohr nowhere gave complementarity a precise definition; later authors interpreted it in different ways.
Afterwards: Bohr's view shifted somewhat with his 1935 reply to the EPR paper; in the following decades complementarity was both debated and taken up in experiments (for example double-slit and which-path experiments).
Bohr replied, under the same title, to the claim of Einstein, Podolsky and Rosen that the quantum description is incomplete (Phys. Rev. 48, 696).
Bohr's reply was published in the same journal and under the same title shortly after the EPR paper appeared. EPR's question was this: if, without disturbing a system in any way, we can predict with certainty the value of a physical quantity by measurement, then there is a reality corresponding to that quantity. If this criterion is accepted, a link set up between two particles that have moved apart shows the quantum description to be incomplete.
In his reply Bohr objects to the phrase 'without in any way disturbing the system' within EPR's criterion. There may be no physical influence going to the second particle; but according to Bohr the choice of measurement affects the very conditions that determine what kinds of predictions can be made for the second particle. The 'reality' of a quantity cannot be spoken of independently of the experimental arrangement.
Bohr thus treats the system made up of the two particles as a single whole. An arrangement measuring position and an arrangement measuring momentum exclude each other; we cannot apply both at once, and so the conclusion that reality must be attributed to both the position and the momentum of one particle does not follow from EPR.
In his reply Bohr also analyzes an arrangement similar to Heisenberg's microscope example: the measurement of the momentum of a particle passing through a slit depends on the movement of the slit itself. This analysis aims to show that a measured quantity is defined only together with the experimental arrangement.
For a long time the reply was regarded as the defense of the Copenhagen interpretation against EPR. Its limit: historians and physicists think differently about exactly what the reply says and what its objection to EPR's criterion is; it has also been written that the language of 'disturbance' receded in Bohr's views after 1935.
Afterwards: The reply remained the main text of this debate until Bell's theorem arrived in 1964; Bell made the difference between the two positions experimentally testable.
4. Many Worlds
Main question: Does the universe branch at measurement?
The measurement problem arises from the jump rule added alongside smooth change in quantum mechanics. Everett's proposal is not to add the jump at all: the Schrödinger equation always holds, and after measurement everything, including the observer, enters a superposition in which every result exists together. The states in which the observer 'sees' each result separately are thought of as separate branches. The question is whether this picture explains why a single result is seen in measurement and where the probabilities come from.
Everett proposed a theory that removes the jump rule in measurement, describes the whole universe with a single wave function, and counts observer states as relative components of this function.
Everett's paper is an answer to the question of whether quantum mechanics can describe measurement without a jump rule. The standard form of the theory contains two rules: the system changes smoothly, while in measurement it jumps to a result. Everett argued that the second rule is unnecessary.
The idea works as follows. Suppose that the measured system and the observer making the measurement together are a single quantum system. When the measurement interaction ends, by the rules of smooth change the joint state becomes a sum of components, each corresponding to a measurement result. In each component the observer is in a state of having 'recorded' a separate result, and in that component there is a single result according to his own memory. No component erases the others. Everett calls this relative state: the observer's state is defined relative to the state of the measured system.
For the question of where the probabilities come from, Everett uses a measure based on the square of the magnitude of the wave function. The weights of the components are given by this measure, and the observer's memory over a long sequence of measurements appears, with this weighting, in most components to agree with the probability predictions of quantum mechanics.
The paper also examines the case where an observer makes multiple measurements: after each measurement the components branch, and the observer's memory becomes a consistent sequence of results in his own component. This gives the reason why successive measurements by the same observer come out consistent with each other.
The words 'worlds' or 'branching' do not appear in the paper's own language; this name and the branching account come from later authors. Its limit: why the weights should be interpreted as probabilities, and what exactly a 'world' is, remain contested today.
Afterwards: Everett's proposal was developed in the following decades as the 'many worlds' interpretation; decoherence is used in this interpretation to describe the branches separating from one another, while the origin of the probabilities remains a subject of open debate.
5. Hidden Variable Theories
Main question: Is there a deeper theory beneath quantum mechanics?
Quantum mechanics says that different results can come out in systems prepared under the same conditions, and that we can compute only probabilities. Hidden variable theories hold that beneath these probabilities there are variables we have not yet seen, and that once these are known the result would be definite. These two papers set up the clearest attainable form of this idea, a 'pilot wave' that guides the particles.
De Broglie proposed a picture in which the particle is guided by a wave; he also presented the same idea at the 1927 Solvay Congress (J. Phys. Radium 8, 225).
De Broglie's idea of matter waves, proposed in 1924, had led to Schrödinger's wave equation. The 1927 paper takes up the relation of 'wave mechanics to atomic structure and radiation' and proposes a picture that holds wave and particle together.
In the picture the particle is really at a point and follows a path; alongside it there is a wave accompanying it. The velocity of the particle is determined by the form of the wave: the particle goes toward where the wave is strong and in the direction it is inclined. Thus in interference experiments such as the double slit, the particle passes through only one slit, while the wave passes through both slits at once and builds the interference pattern and guides the particle to certain places in the pattern.
The advantage of the idea is that the position of the particle is definite at every moment. Its disadvantage is that in many-particle systems the wave is defined not in a single space but in a combined configuration space, and it requires instantaneous interactions between particles.
De Broglie presented the same idea at the 1927 Solvay Congress. His answer to Wolfgang Pauli's objection on scattering remained weak, and de Broglie abandoned the idea for a time. The idea stayed in the background until Bohm's 1952 work.
In this picture the wave function alone is not counted as a complete description; the real position of the particle is also part of the description. This is the point where it departs from the standard interpretation. For this reason the idea is cited as one of the first theoretical attempts to hold wave and particle together without reducing one to the other.
Its limit: the paper does not fully treat the general many-particle case and does not answer Pauli's objection.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: The idea was rediscovered by Bohm in 1952 and is known today as Bohmian mechanics or the de Broglie–Bohm theory.
Bohm built a theory interpreting quantum mechanics as definite trajectories given by particle positions and the wave function, with the same experimental predictions (Phys. Rev. 85, 166).
Bohm's paper was written to offer a counterexample to the widespread belief that quantum mechanics cannot be completed with hidden variables. His proposal is to build a theory that gives all the predictions of standard quantum mechanics but in which particles have definite positions at every moment.
The theory has two elements: the wave function obeying the Schrödinger equation, and the real positions of the particles. The wave function determines the velocity of the particles (this is called the guidance equation); the position of the particle is definite at every moment. Since initial positions are not known, results are predicted only with probability, and this probability comes out the same as the rule in standard quantum mechanics. In his own account Bohm speaks of a 'quantum potential' by which the wave acts on the particle.
Thus in the double-slit experiment each particle passes through a single slit and follows a definite path, while the interference pattern arises from the guidance of the wave function. Measurement is not a special process; the measuring apparatus too is made up of particles defined within the theory.
Bohm stated explicitly that this theory contains instantaneous influences between distant particles, that is, that it is nonlocal. Twenty years later this feature became a rule with Bell's theorem: Bell showed that the predictions of local hidden variable theories do not agree with quantum mechanics.
The paper consists of two parts; the second part separately treats the measurement process and the working of the theory in the many-particle case. Bohm shows that in the measurement process the apparatus too is made up of the theory's particles and that the result is determined by the real positions of the particles.
Its limit: the theory has difficulty being adapted to relativity and requires an assumption about the initial distribution of the hidden variables.
Afterwards: Bell took Bohm's work as an example showing that a nonlocal hidden variable theory is possible and went on to his 1964 theorem; Bohmian mechanics is today discussed as a serious option among the interpretations.