Physics papers: XI. Basic phenomena and the birth of the theory
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.
At the end of the nineteenth century, how much light a heated body emits at which colors was not consistent with experiment when calculated with classical physics. The three works in this section try to get past this impasse by supposing that energy is taken and given not as a continuous flow but in certain measures, that is, in 'quanta'. The question asked is this: why do light and atoms exchange energy in definite steps rather than in every amount? As terms, 'black-body radiation' is the light emitted by an ideal body that absorbs all light falling on it and, when heated, produces its own radiation; a 'quantum' is the smallest indivisible share of energy.
On this page51. The Birth of Quantum Theory · Planck 1901 — Ueber das Gesetz der Energ · Einstein 1905 — Über einen die Erzeugung …
Planck derives the energy distribution law of black-body radiation by treating energy as divided into shares of a definite size.
The distribution by wavelength of the light emitted by a heated body was being measured at the time. The distribution that came out of classical theory contradicted experiment at short wavelengths, while the laws that fit the measurements lacked a theoretical basis. In this paper Planck tries to ground in a theoretical calculation a law that fits the measured distribution.
The method is to imagine a system of small oscillating charges that emit and absorb radiation, and to count in how many ways the total energy of this system can be distributed. To be able to make this count, Planck treats the total energy not as a continuous quantity but as a quantity made up of equal shares of a definite size. The size of each share is proportional to the frequency of the oscillation; the constant of proportionality is a new constant of nature, later to be called Planck's constant. This count gives the formula that fits the measured distribution, and the values of two new constants are also obtained from the same calculation.
The paper does not separately argue, in the sources this summary rests on, whether the energy shares really exist in nature or are only a way of calculating; this summary rests not on the text of the paper but on its familiar account. That the calculation gives no result without this assumption of distribution, and that the assumption later became the founding principle of the theory, makes up the paper's importance today.
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: In the following years Einstein treated the energy shares as a property of light itself, and Bohr carried the same constant over to atomic structure. Whether the division was only a way of calculating or a property of nature was debated during this time.
Einstein sets out to explain the photoelectric effect and related phenomena by supposing that light in vacuum behaves as if made up of shares of energy of a definite size.
Maxwell's theory describes light as a continuous wave. Matter, on the other hand, consists of separate atoms and electrons. At the start of the paper Einstein recalls the gap between these two descriptions and says that the wave description alone may fall short in phenomena concerning the production and transformation of light. The word 'heuristic' in the title indicates that what is put forward is not a definitive theory but a viewpoint that works.
The proposed viewpoint is that, in sufficiently dilute radiation, the energy of light behaves not like something continuous spread through space but like shares that are pointlike and move without dividing. The energy of each share is proportional to the frequency of the light. Einstein examines several consequences of this assumption. One of them is that, when light falls on a metal surface, the energy of the electrons that are knocked out depends on the frequency of the light; when the intensity of the light increases the number of electrons knocked out should increase, but the energy of each electron should remain tied to the frequency threshold. These results are made comparable with photoelectric observations.
The paper presents the idea that light behaves like a particle not as a definite truth but as a proposal that the energy shares in Planck's radiation law may be a property of light itself. This summary was written without looking at the text of the paper.
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 explanation of the photoelectric effect along this line was tested over the years with detailed experiments. That light behaves both as a wave and as shares has remained, through the double slit and related experiments, a subject still debated today.
Bohr treats the hydrogen atom as a system with definite energy levels and proposes that the emitted light is determined by the differences between these levels.
According to Rutherford's model of the atom, an atom consists of a small nucleus and electrons orbiting around it. Classical electrodynamics requires that an orbiting charge radiate continuously, lose energy and collapse into the nucleus. Yet atoms are stable, and the light emitted by gases comes in sharp lines at definite frequencies.
Bohr proposes to get past this contradiction with two assumptions that go against classical physics. The first is that the atom can exist only in certain 'stationary states' and does not radiate in these states. The second is that, in passing from one state to another, the atom emits or absorbs light at a frequency proportional to the energy difference. To determine which the stationary states are, he uses the condition that the angular momentum of the electron be an integer multiple of certain units. From this condition the energy levels of hydrogen are calculated, and the line frequencies that come out of them are compared with the series of lines measured in the hydrogen spectrum.
The paper sets out to treat not only hydrogen but also heavier atoms and molecules; however, this summary rests not on the full text of the paper but on the familiar account of the theory. The limit of the model is that the calculation fails for atoms with many electrons and that it does not explain why the stationary states satisfy this condition.
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 Bohr model gave way in the following years to wave mechanics and matrix mechanics; the existence of energy levels was kept, and the picture of orbits was abandoned.
52. The Double Slit and Wave-Particle Duality
Main question: Does a particle behave like a wave?
Waves passing through a screen with slits form, behind the screen, light and dark bands that arise from the superposition of wave crests and troughs; this is called interference. Whether particles that can be counted one by one, such as electrons, also form such bands in the same setups is one of the central questions of the theory. The four works in this section ask this question experimentally, starting from electrons and moving up to ever larger molecules. The result is that particles sent one at a time give an interference pattern as they accumulate.
Davisson and Germer scatter an electron beam from a nickel crystal and observe that the electrons are concentrated at certain angles.
In the early 1920s de Broglie had suggested that a wavelength might exist for the particles of matter as well. Davisson and Germer's work is one of the first examples of an experimental test of this proposal. The aim of the work is to measure whether, when an electron beam is scattered from a crystal surface, the electrons come back in all directions or only in certain directions.
In the setup, a single nickel crystal is held in an electron beam, and the number of scattered electrons is measured according to direction and the energy of the electron. The atoms of the crystal form a regular lattice; when waves scatter from such a lattice they reinforce one another at certain angles and cancel at others. The authors report that the electron count also shows definite peaks depending on direction and energy, and they relate the positions of these peaks to the spacings of the crystal lattice and to the de Broglie wavelength for electrons. The result is read as a pattern similar to the diffraction of X-rays from crystals.
This summary rests, without seeing the text of the paper, on the familiar account of the work in Phys. Rev. 30, 705.
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: In the same years another group observed electron diffraction from thin metal films. The wavelike behavior of electrons was also shown in the following decades with multi-slit experiments.
Jönsson sends an electron beam through narrow slits made artificially on a micrometer scale and shows the electron counterpart of the double slit for light.
The double-slit experiment is an interference setup that has been known for light since the early 1800s. Setting up the same arrangement for electrons required making the slits very narrow, appropriate to the electron wavelength, so for a long time it was mentioned only as a thought experiment. In this paper Jönsson finds a way of producing very thin slits artificially and sends the electron beam through them.
The pattern the electrons form on a screen placed behind the slits is recorded. The pattern is observed separately with one slit, two slits and a larger number of slits, and in every case it corresponds to the interference bands predicted by wave theory. The author compares the spacing of these bands with the wavelength computed from the speed of the electron.
This summary was written without seeing the paper directly; the publication is at Zeitschrift für Physik 161, 454. This work does not distinguish when the pattern forms, that is, whether the electrons are sent one by one or as a crowd; this question has been the subject of later experiments.
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: How the pattern builds up with electrons sent one by one was recorded later in the experiment of Tonomura and colleagues.
Tonomura and colleagues record on film how the interference pattern builds up while electrons are sent one at a time.
It had been shown before that electrons behave like waves, but one could ask whether, in a crowded electron beam, the interference arose from interaction among the electrons. The authors of the paper test this question by sending the electrons very sparsely: at most one electron is inside the setup at any one time.
The setup consists of an electron biprism added to an electron microscope and a position-sensitive electron counting system. The biprism, by passing the electron wave along two paths and bringing it together again, plays the role of the double slit. The counting system records the point where each electron reaches the screen. According to the paper's abstract, these records are kept as a film, and it is shown how the interference pattern builds up from electrons arriving one by one.
On the screen each electron leaves a single dot, that is, it looks like a particle. As many electrons accumulate, the dots, instead of being randomly distributed, form light and dark bands. In the language of the theory this observation is described as each electron carrying a wave function of its own and being found at a single point in the measurement. The paper does not separately discuss this description as an interpretation; the aim is the actual recording of an observation that had previously been regarded only as a thought experiment.
Afterwards: Single-particle experiments of the same kind were done in the following decades with neutrons, atoms and molecules as well.
Arndt and colleagues report that a large molecule made of 60 carbon atoms (C60) shows interference as it passes through a diffraction grating.
Whether the wave behavior shown with electrons and atoms continues in much larger objects with complex internal structure is a question that probes the limit of the theory. In this work, C60 molecules, made of 60 carbon atoms and shaped like a football, are used. Such a molecule has many internal vibrational and rotational states; so it is hard to think of it as a single point.
In the setup, C60 molecules coming out of a source are passed through a grating with very fine spacing and counted by position at a detector behind it. If wave behavior exists, the distribution behind the grating is expected to give not a single narrow beam but a pattern containing interference peaks. The result the paper reports is that this pattern was observed.
This summary was written without looking at the paper itself, relying on its familiar account; numerical details have been deliberately left out. The distinction between the measured distribution and the classical particle expectation is made through the authors' own comparison. It is also known that in the experiment heat and collisions between the molecule and gas can spoil the interference; these effects have been studied later in separate experiments.
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: In the following years interference was shown with even larger molecules; one of them is the work of Fein and colleagues, which is the subject of this section (topic 54).
53. The Uncertainty Principle
Main question: Can position and momentum not both be exact at once, and does measurement disturb?
In quantum mechanics, when position and momentum of a particle are measured, the spreads of the results cannot both be made as small as one likes. Two separate claims are involved here, and they need to be kept apart. The first is that, across many systems prepared in the same way, there is a lower bound on the spreads of the position and momentum results. The second is whether, in a single measurement, there is a lower bound on how much the measurement disturbs the system and on the measurement error. The three papers in this section give the historical and theoretical sides of this distinction.
Kennard shows mathematically that the product of the spreads (standard deviations) of the position and momentum results has a lower bound of half the Planck constant.
Heisenberg had put forward the idea of “uncertainty” in the same year, based on a measurement thought experiment. Kennard’s paper derives the same result inside wave mechanics as an exact inequality, without tying it to a story about measurement. More broadly, the paper deals with the quantum mechanics of simple kinds of motion; the uncertainty inequality is one section of it.
The inequality says this: in many trials in which a particle is prepared in the same way, measuring position gives one spread of results and measuring momentum gives another. The product of these two spreads cannot be smaller than half the reduced Planck constant. Spread here means standard deviation.
This statement is a property of the preparation: it says nothing about whether the measurement disturbs the system in each individual trial, or about the inaccuracy of a measuring instrument. The same system is prepared in separate trials, measured now for position and now for momentum, and each measurement gives its own spread. This summary was written without looking at the paper itself; the paper is in Zeitschrift für Physik 44, 326.
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 inequality was later generalized by Robertson to any two quantities.
Robertson derives, for any two measurable quantities, a lower bound on the product of their spreads that depends on how much the result changes when the order of the quantities is swapped.
Kennard had given an inequality for position and momentum. Robertson’s short note states this in a more general form: let the two quantities be A and B. The product of their spreads is bounded from below to the extent that measuring the two quantities one after the other depends on the order. In equation form it is written ΔA·ΔB ≥ ½|⟨[A,B]⟩|; the square bracket shows how much difference arises when the order of multiplying the two quantities is changed.
For position and momentum this difference is a constant and the bound comes down to half the Planck constant. For two quantities whose order does not matter, the difference is zero and the principle sets no bound. The value of the bound can also depend on the state that is prepared.
Like Kennard’s, the inequality concerns the spread of results over many trials prepared in the same way. It says nothing about the error of the measuring instrument or the disturbance created by the measurement in a single measurement. This summary was written without looking at the text of the paper; the paper is two pages long, in Phys. Rev. 34, 163.
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: In the following decades the inequality was often given in textbooks together with a separate claim about measurement error and disturbance; this second claim is taken up in Ozawa’s paper.
Ozawa shows that the expectation, drawn from Heisenberg’s thought experiment, that the product of measurement error and disturbance cannot be smaller than half the Planck constant does not hold in general.
Heisenberg’s gamma-ray microscope thought experiment says that the more precisely position is measured, the more the momentum is disturbed. The common reading is that the product of the measurement error and the momentum disturbance caused by the measurement also cannot be smaller than half the Planck constant. Ozawa shows that this assumption is wrong.
According to the paper’s abstract, there is a universally valid relation between error and disturbance, and this relation contains an additional term. When the measurement’s interference is independent of the object being measured, this term is not needed and Heisenberg’s bound is preserved. When the interference depends on the measured object, the product of error and disturbance can fall far below the Planck constant. The author gives a measurement model showing this case; in the model, Heisenberg’s lower bound is exceeded by a precise position-measuring instrument that hardly disturbs at all.
The paper also proposes an experiment that would carry this out with a measurement of optical quadratures. This does not refute the Kennard–Robertson inequality for spreads, because that inequality is a property of prepared states. What is refuted is the expectation that a bound of the same form exists for measurement error and disturbance.
Afterwards: The proposed measurement was later investigated in experiments; how measurement error and disturbance ought to be defined has remained a topic debated in the theoretical literature.
54. Superposition and Schrödinger’s Cat
Main question: Can a system be in two states at once, and how large can it grow?
In quantum mechanics the state of a system can be written as a superposition of two distinct states; this raises the question of whether, before measurement, not knowing which state the system is in comes from a lack of information or from the state genuinely having two components. In 1935 Schrödinger presented the theory with a thought experiment showing the oddity that appears when it is applied to a large object. Experimenters, for their part, later tried to record states of being in two places at once in objects that were very small and then gradually larger. The three works in this section are three points along this line.
Schrödinger reviews the then-current state of quantum mechanics and, in this paper, introduces the concept of entanglement and the cat thought experiment.
The paper was published in the journal Naturwissenschaften in three parts and reviews the then-current presentation of quantum mechanics with a critical eye. The text is read within a chain of replies to the EPR paper; for there it takes up a situation in which the states belonging to two systems cannot be separated from each other, and Schrödinger gives this situation the name “Verschränkung”, that is, entanglement.
The set-up of the thought experiment is this: in a closed steel box there are a cat, a Geiger counter and a very small amount of radioactive substance. Within an hour the probability that one of the atoms decays is about one half. If decay occurs, the counter triggers a mechanism and the cat is poisoned. When the theory’s wave function describes the box as a whole, it expresses a state in which the living and the dead cat are mixed in equal shares. Schrödinger presents this as an example of how crude the results can be that the theory yields as a description; opening the box and looking turns this expression at once into a definite outcome.
This summary was written without looking at the paper’s own text, based on its well-known presentation. In the paper the cat is only one of many examples.
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 cat thought experiment remained, in the following decades, a frequently invoked example in discussions of the measurement problem. Why superposition is not seen in everyday life for large objects is discussed through separate interpretations such as collapse models and decoherence.
Monroe and colleagues prepare a single trapped beryllium ion in a superposition of two spatially separated states and verify it by interference.
According to the paper’s abstract, the authors have produced a “Schrödinger cat”-like state at the level of a single atom. A 9Be+ ion held in a trap is first laser-cooled to its zero-point energy. The ion is then prepared in a superposition of two spatially separate coherent harmonic oscillator states.
This state is obtained with a series of laser pulses. The pulses entangle the ion’s internal (electronic) state with its motional state. That is, when the ion’s internal state has one value its motion is at one position, and when it has another value its motion is at the other. That the superposition is really there is verified by observing quantum-mechanical interference between the two localized wave packets.
The authors say that this mesoscopic system may help in understanding the blurred boundary between the quantum and classical worlds; they note that measurement and the phenomenon called decoherence can be studied in a controlled way. The paper uses the cat analogy, but the system is not a cat, it is a single ion; and the distance between the two separated states is only a very small length.
Afterwards: Similar superposition states were later prepared with light fields and other systems as well; this line was continued with experiments aimed at increasing size.
Fein and colleagues demonstrate the quantum superposition of molecules with masses beyond 25 kDa by means of an interference experiment.
To what size superposition persists is a question that has to be probed experimentally. 25 kDa, expressed in atomic mass units, is according to the paper’s title the mass scale reached in this work; that is, a molecular mass has gone beyond 25,000 atomic mass units.
In experiments of this kind, molecules are released from a source, passed through finely spaced structures, and a pattern is looked for at a detector behind them. If wave behavior persists, the pattern gives light and dark bands instead of a single beam; the visibility of the bands measures how coherent the molecule can remain. With large molecules the difficulty is producing the molecule fast and cold enough, protecting it from gas collisions and thermal radiation, and collecting enough molecules for the pattern.
This summary was written without access to the paper’s own abstract; the details of the experimental set-up and of the numbers have deliberately not been given. It would be useful for the reader to look at the paper’s first page.
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 distinction between superposition in larger objects and collapse models is cited as one of the goals of experiments of this kind.
55. Spin and the Pauli Exclusion Principle
Main question: What is spin, and why can two electrons not be in the same state?
Spin is an intrinsic angular momentum carried by a particle; it does not mean that the particle spins like a little top, but it is a property that places it in particular directions in a magnetic field. The electron’s spin can take two values along the field direction. According to the Pauli exclusion principle, two electrons in an atom cannot be the same in all four quantum numbers, spin included; the shell order in the periodic table rests on this. The three papers on this subject include the experiment showing space quantization, the first statement of the exclusion principle, and the proposal of spin.
The 1922 publication of the Stern–Gerlach experiment, which reports that a beam of silver atoms splits into two separate beams as it passes through a non-uniform magnetic field.
In the first twenty years of the twentieth century, the atomic model (the Bohr–Sommerfeld theory) said that an atom’s magnetic moment could stand only in certain directions relative to an external magnetic field; this was called “space quantization” (direction quantization). Classical physics, by contrast, would allow every direction. The question was: can this difference be observed directly?
In the experiment, silver atoms leaving a heated oven formed a narrow beam, the beam passed through a magnetic field in which the force varies with position (a non-uniform field), and finally collected on a glass plate. In such a field the atom is deflected to one side or the other depending on the direction of its magnetic moment. The classical expectation is that atoms whose moments point in random directions would leave a continuous, spread-out trace on the plate. What was observed was two separate traces with the field on and a single trace with the field off.
The result was read as follows: the component of the atom’s magnetic moment along the field direction takes discrete values rather than continuous ones, and for silver the number of these values is two. The theory of the time explained the two traces by the space quantization of the orbital angular momentum.
For the splitting to be visible, the beam had to be very narrow and the non-uniformity of the field very pronounced; the experiment was a measurement made at the limits of atomic beam and high-vacuum techniques of the time. The result has two traces: each atom goes to one of the two traces, and no trace is seen between them. This discreteness is described as one of the first direct demonstrations that a measurement result can take only certain values.
Limit and what was understood later: the orbital angular momentum of silver’s outer electron is zero; the splitting comes not from orbital motion but from the electron’s spin. So the 1922 interpretation differs from today’s explanation, while the observed phenomenon has remained the same.
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: Three years later Uhlenbeck and Goudsmit assigned the electron an intrinsic angular momentum; the two traces in silver are today described as a direct demonstration of the electron’s two spin states. The experiment is also taught as one of the first examples of a situation in which a measurement can give only discrete results.
The paper that proposes an exclusion rule by relating the closing of electron shells in atoms to the complex splittings of spectral lines in a magnetic field.
The paper’s starting point was two separate puzzles. The first was that the shells in the periodic table fill up to 2, 8, 18, 32 electrons and why these numbers close at these values. The second was that spectral lines split in a magnetic field in a more complicated way than expected, that is, the anomalous Zeeman effect. Moreover, it was also unclear why all the electrons in an atom do not gather in the lowest-energy state.
Pauli assigned the electron, in addition to the three quantum numbers known until then, a fourth quantum number carrying “a two-valuedness not describable classically”. He then put forward that in an atom two electrons cannot have the same values in all four quantum numbers. This is the rule known as the exclusion principle.
This rule gives the number of electrons that can fit in a shell (2n² in the n-th shell) and explains the shell closings. It also organizes the splitting patterns in the anomalous Zeeman effect through the additional two-valued number.
One consequence of the rule is this: two electrons sharing the same orbital state must differ in the fourth number, that is, at most two electrons can occupy each orbital state. For this reason electrons cannot all gather in the lowest-energy state; the shells fill in order, and the chemical properties of the elements are tied to this filling pattern. The fourth number is not merely a correction term; it enters the shell structure itself.
The paper did not explain with a model what the two-valuedness measures; Pauli deliberately did not tie this number to a mechanical picture. The rule itself was also laid down as a principle expressing a regularity of observation; it was not derived from quantum mechanics.
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: In 1926 Fermi and Dirac placed the same rule, within many-particle quantum mechanics, as the wave function changing sign under the exchange of two electrons (antisymmetry). The link between spin and statistics was taken up separately in the following years in relativistic field theory.
Naturwissenschaften 13, 953-954 (Crossref kaydında yazar P. Ehrenfest olarak görünür; not Uhlenbeck ve Goudsmit'e atfedilir) — citation verified second-hand theory
The short note that proposes, as a physical content for Pauli’s fourth quantum number, that the electron has an intrinsic angular momentum and a magnetic moment.
The note’s title proposes that, in place of Pauli’s “non-mechanical constraint”, a condition on the internal behavior of each electron be put. That is, the question is what Pauli’s fourth quantum number corresponds to.
By examining the fine structure of spectral lines and the anomalous Zeeman effect, Uhlenbeck and Goudsmit proposed that every electron can be given two new degrees of freedom. The proposal is this: the electron has an intrinsic angular momentum of magnitude ħ/2 and a magnetic moment equal to one Bohr magneton; this moment has only two orientations relative to an external field. Thus the fourth quantum number is read as an orientation number taking two values, and the two beams in silver are understood in the same way.
The proposal was not accepted at once. If the electron is regarded as a spinning sphere as small as the classical electron radius, the speed of its surface exceeds the speed of light; this objection came from Lorentz and showed that “rotation” as a picture is problematic. In addition, the fine-structure calculation gave one extra factor.
The notable side of the proposal is that it connects two separate pieces of data with a single assumption: the two-valuedness in Pauli’s rule and the fine-structure splittings in spectral lines. The note is short and does not present a detailed calculation; the calculation was developed in later publications.
Limit: what is proposed in the paper is not that the electron really rotates but the existence of an intrinsic angular momentum and magnetic moment; the word “spin” remains an analogy. Bibliographic note: the Crossref record lists Ehrenfest as the author; the paper is commonly referred to as the note of Uhlenbeck and Goudsmit.
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: In 1926 Thomas published the calculation that removed the extra factor in the fine structure by means of a relativistic correction (today called Thomas precession). In 1928 Dirac’s relativistic electron equation gave spin spontaneously, without adding it separately as an assumption.
56. Quantum Tunneling
Main question: How does a particle pass through a wall without going over it?
In classical physics a particle whose energy does not reach the height of a barrier cannot get over the barrier. In quantum mechanics the particle is described by a wave function; this wave leaks into the barrier, decaying as it goes, and if the barrier is thin enough it emerges on the other side with a non-zero amplitude. That is, the particle has a probability of being found on the far side; this phenomenon is called tunneling. The two papers on this subject contain the first major application of tunneling, alpha decay, and its controlled experimental demonstration.
The paper that calculates the emission of an alpha particle from an atomic nucleus as the tunneling of the particle through the nucleus’s potential barrier.
In radioactive alpha decay, half-lives vary over a very wide range: from fractions of a second to billions of years. There was a regular relation (the Geiger–Nuttall relation) between the energy of the emitted alpha particle and the half-life, and it could not be explained. Moreover, the alpha particle’s energy was far below the top of the repulsive electric barrier outside the nucleus; classically the particle would not be expected to get out.
Gamow modeled the nucleus as a two-region potential for the alpha particle: an attractive well inside and, outside, a barrier coming from the nucleus’s electric repulsion. In the solution of the Schrödinger equation the wave function decays exponentially inside the barrier but reappears on the other side of the barrier with a non-zero amplitude. The ratio of these amplitudes gives the probability of the particle “passing” the barrier.
For the decay rate, Gamow assumed that the alpha particle strikes the barrier many times per second inside the nucleus and multiplied this number by the passage probability. The result showed that the decay rate depends exponentially on energy; thus the Geiger–Nuttall relation and the wide range of half-lives came out of a calculation. A small difference in energy changes the time greatly.
A practical consequence of the calculation is that the passage probability depends exponentially on the width and the height of the barrier: if the barrier thickens a little, or the particle’s energy drops a little, the probability falls very fast. This is the reason half-lives are spread over such a wide range.
Limit: the model treats the nucleus as a simple potential; details inside the nucleus, such as the assumption that the alpha particle is already formed in the nucleus, are left outside the calculation.
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: Gurney and Condon independently published a similar explanation in Nature in the same year. Tunneling was later observed in other fields as well; the scanning tunneling microscope in the second paper on this subject is an example of this.
The first experiment to measure the tunneling current in a controlled way by making the vacuum gap between two electrodes adjustable from outside (the experimental basis of the scanning tunneling microscope).
The paper’s question is whether electrons cross an empty gap between two conductors, measured while the gap width is controlled. Tunneling had earlier been observed across a solid insulating film; measuring it reproducibly in an adjustable vacuum gap, however, was an experimental difficulty.
The authors built a set-up in which the gap between two electrodes could be adjusted from outside in a reproducible way, and measured the tunnel resistance while the gap width was varied. The set-up also allowed the electrode surfaces to be examined and treated.
As a result the measured tunnel resistance came out depending exponentially on the width of the gap. The paper shows that vacuum tunneling was observed through this exponential dependence. The physical meaning of this is: the wave function decays exponentially in the gap, so as the gap widens the probability of finding the electron at the other electrode, and hence the current, falls rapidly.
This experiment is the basis of the scanning tunneling microscope, in which a tip is scanned over a surface and the surface is mapped at the atomic scale: the exponential dependence of the current on the gap gives a very sensitive measure of the tip’s distance from the surface.
The exponential dependence is treated in the paper as the distinguishing sign of vacuum tunneling. The tunnel current depends not only on the gap but also on the surface condition of the electrodes; for this reason the ability to examine and treat the surfaces in the same set-up is also important for making the measurement on clean surfaces. The paper is a short experimental report and presents the method, the measured curve and the possibility of surface treatment together; it gives no theoretical tunneling calculation, it presents an experimental measurement.
Limit: the paper shows the existence of tunneling and its dependence on the gap; surface imaging was developed in later work.
Afterwards: Binnig and Rohrer shared the 1986 Nobel Prize in Physics with Ruska for the scanning tunneling microscope built on this work.
57. Delayed Choice and the Quantum Eraser
Main question: Does a choice made afterwards affect the past?
In a double-slit experiment, if there is information that determines which path the particle took, the interference pattern disappears; if this information is erased, interference can return. In delayed-choice experiments the decision to read or erase the path is made after the particle has passed the slits. The question is whether this late decision determines the particle’s earlier behavior. The three papers on this subject contain the quantum eraser proposal, its implementation with entangled photon pairs, and Wheeler’s single-photon interferometer experiment.
The theoretical paper that discusses the effect on the interference pattern of the choice of reading or erasing path information afterwards in a double-slit-like arrangement, and proposes the “quantum eraser”.
In a double-slit experiment, if there is information that determines which slit the particle went through, the interference pattern disappears. The question is: what happens if this information is stored in another system (for example a photon or the internal state of an atom) and, instead of being read afterwards, is erased?
The paper considers an arrangement in which the information about which path a particle took is transferred to a second system. As long as the path information stays in that system, no interference is seen. But if that second system is measured in such a way as to erase the path information, that is, if a measurement is made so that the path information is no longer to be found anywhere, the interference pattern of the first particle can be seen again, in subsets separated according to the outcome of the measurement.
The point the paper emphasizes is that the disappearance of interference comes from the existence of path information and that, for this, it is not necessary to disturb the particle’s position or momentum by measurement. The phrase “delayed choice” in the title refers to the arrangement in which the decision to erase or read can be made even after the particle has been detected.
The theoretical contribution of the proposal is that it shows the principle of complementarity (that wave and particle behavior cannot both be fully seen together in one experiment) can be described not by a mechanical jolt given during measurement but by the connection between the measured system and the recording system. The paper ties the question of when interference will appear to the question “how much path information is there”.
Limit: this is a theoretical proposal, not an experimental report. Interference is seen in subsets separated according to the outcome of the erasing measurement; when all events are taken together, no interference pattern appears on its own.
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 proposal was implemented with entangled photon pairs in the 2000 work of Kim and colleagues.
The paper that carries out, with entangled photon pairs, the delayed-choice quantum eraser experiment proposed by Scully and Drühl.
The paper reports the “delayed-choice quantum eraser” experiment proposed in 1982 by Scully and Drühl. The question asked is whether a quantum object’s behaving both like a wave and like a particle can be observed together by means of its entangled twin.
In the set-up a crystal produces entangled photon pairs. One photon of the pair goes to a detector after a double-slit-like passage; the path of the other photon is divided among detectors that can either erase the information “it came by both paths” or give the information “which path it came by”. Thanks to entanglement, the path information of the second photon is tied to that of the first. The detection of the second photon takes place after the detection of the first.
The result the paper reports is this: path information, even after it has been recorded, can be erased or marked by the entangled twin. Depending on which subset one looks at in the first photon’s detector, that is, on which detector the twin went to, interference fringes are seen or not seen. When all events are taken together, the first photon alone has no interference pattern.
In the arrangement, the place where the first photon is detected is a detector that is moved to scan the fringes. Beam splitters on the second photon’s path direct the photon to one of four detectors: two erase the path information, two give the path information. In the counts coincident with the eraser detectors, interference fringes are seen; the fringes of the two eraser detectors are shifted so as to complement each other, so in total there is no interference.
Limit: interference appears only in subsets paired with (coincidence-counted with) the second photon. For this reason the experiment does not show that a photon’s behavior was changed afterwards in the past; the result should be read within the framework of entanglement and pairing.
Afterwards: For a survey of delayed-choice experiments one may look at the 2016 review by Ma and colleagues. This result is consistent with the calculations of standard quantum mechanics; whether the past changes is a matter of interpretation depending on how the result is told, and it is disputed.
The experiment that carries out Wheeler’s two-path interferometer thought experiment with single photons and with the choice made by a random number generator.
In Wheeler’s delayed-choice thought experiment, a single photon enters a two-path interferometer. After the photon has entered, the configuration at the interferometer’s exit is chosen: either the two paths are recombined and interference is observed (the “closed” configuration), or the paths are not recombined and the path the photon came by is measured (the “open” configuration). The question is: is whether the photon behaved “as a wave or as a particle” at the moment it entered determined by this choice made afterwards?
This paper reports a nearly ideal realization of the experiment, in the authors’ words. The light pulses are genuine single photons; this ensures that the path measurements are not ambiguous. The two paths are spatially separate and interference is observed with high visibility.
The decision of open or closed configuration is made with a quantum random number generator, and this choice is, in the relativistic sense, spacelike separated from the photon’s entry into the interferometer; that is, no signal can travel between the choice and the photon’s entry at the speed of light.
Results: in the closed configuration the interference visibility is 94 percent; in the open configuration the path taken by the photon is determined with an error probability of less than 1 percent.
The single-photon condition is important: if there were more than one photon in the light pulse, the path measurement would become ambiguous; the authors emphasize using genuine single photons for this reason. The spacelike separation of the choice also means that news of the choice cannot reach the photon inside the interferometer even at the speed of light.
Limit: the result is consistent with the prediction of standard quantum mechanics. The experiment provides a test for models that assume the photon, on entering, had “chosen” one path or two paths; but the account does not require saying that the past was changed afterwards.
Afterwards: The 2016 review by Ma and colleagues surveys the different realizations of delayed-choice experiments. By which interpretation the result is to be told has remained disputed.