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: How do identical, indistinguishable particles distribute over energy levels?
There are two ways of counting particles of the same kind in a box: giving the particles separate identities, or counting them as indistinguishable. Bose and Einstein showed that, first for light and then for matter, counting as indistinguishable gives different results. This counting defines the family of "bosons," in which more than one particle can be in the same quantum state; the coherence of laser light and the frictionless flow of superfluid helium are associated with this behavior.
On this page24. Bose-Einstein Statistics · Bose 1924 — Plancks Gesetz und Lichtquan · Einstein 1925 — Quantentheorie des einat …
Bose derived Planck's radiation law by counting photon states, without appealing to classical electrodynamics.
In 1900 Planck had proposed a law for the spectrum of thermal radiation that fitted experiment, but the derivations up to that time also appealed in part to classical electrodynamics. Bose's question was: can we obtain the law with an argument that only counts light quanta?
Bose distributes the light quanta among cells in space and momentum in which they can be placed, and counts the quanta as indistinguishable from one another. That is, no distinction is made of the form "this quantum is in this cell, that quantum in the other"; only how many quanta are in each cell is counted. This counting comes out different from the classical counting of distinguishable particles and gives Planck's law directly.
The paper reached Zeitschrift für Physik through Einstein: Einstein translated the text into German and sent it for publication. This counting rule is the starting point of the distribution of indistinguishable particles later called "Bose-Einstein statistics."
The counting Bose used deliberately departed from the classical statistics known at the time, and its justification was used in the paper without separate discussion. The nature of this counting, that is, why light quanta must be counted as indistinguishable, was understood more clearly after quantum mechanics was established. In this respect the paper is a work that began with an intuition, whose justification settled afterwards.
Afterwards: Einstein applied Bose's idea in 1924-25 to a massive, ideal gas (see einstein-1925). The statistics of particles that can be in the same state was defined in the following decades as the general property of bosons.
Sitzungsberichte der Preussischen Akademie der Wissenschaften (1925); yeniden basım: Albert Einstein: Akademie-Vorträge, Wiley-VCH 2005, s. 245-257 theory
Einstein applied Bose's counting to a massive ideal gas and predicted that at low temperature the particles could gather in the lowest level.
Einstein realized that the counting Bose used for light was not peculiar to light and could also be applied to a massive gas. In the works he presented at the Prussian Academy of Sciences in 1924-25, he rederived the properties of an ideal gas by counting the particles as indistinguishable. These works were published in German; the Crossref record we have is of the reprint of this paper by Wiley-VCH in 2005.
An important result is the breakdown of the classical expectation when the gas's density or temperature reaches a certain limit. In classical gases, as the temperature falls the particles slow down but remain dispersed. In Einstein's calculation, however, below a certain temperature a significant fraction of the particles "condense" into the same lowest energy state, and for this no attractive force between the particles is needed; the counting rule alone suffices.
At that time this result was received as an abstract calculation; since interactions in real gases alter this picture, direct testing by experiment did not seem possible in the first years.
A limit of the work is the ideal-gas assumption: interactions between the particles are ignored. In real gases interactions change the details of the condensation; in dilute gases, however, the ideal picture is used as a good first approximation. At that time the relation of the condensation to the fluidity in liquid helium was also discussed.
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 condensation was observed in dilute atomic vapors in 1995 (see anderson-1995, topic 26). The later association of superfluid helium with it is a separate and more complicated debate.
25. Fermi-Dirac Statistics
Main question: How do particles that cannot share the same quantum state distribute?
The Pauli exclusion principle says that two electrons in the same atom cannot be in exactly the same quantum state. Fermi-Dirac statistics turns this principle into a counting for a large number of particles: at most one particle can be placed in each state. What is asked is how, with this rule, the temperature-dependent behavior of a gas, in particular the behavior of electrons in a metal, changes.
Fermi, Rendiconti della Accademia dei Lincei (1926), 'Sulla quantizzazione del gas perfetto monoatomico'; İngilizce çeviri: A. Zannoni, arXiv:cond-mat/9912229 theory
Fermi treated an ideal gas as a quantum gas by using the Pauli principle.
Fermi's 1926 paper was published in Italian; it was translated into English by Zannoni in 1999 and placed on arXiv, and the account below rests on the abstract of that translation. According to the translation's abstract, in this paper Fermi for the first time sets out the statistical law obeyed by most of the particles in nature.
Fermi quantizes a monatomic ideal gas in a harmonic trap (that is, within a restoring force like a spring). The gas's particles are placed in the quantum energy levels the trap allows; according to the Pauli principle at most one particle can enter each level. This rule says that even at temperatures near absolute zero the particles cannot all gather in the lowest level; the levels will be filled in order. For a classical gas the energy at zero temperature is zero; in a Fermi gas, however, the particles stand in filled levels up to a certain energy.
This counting rule was also used to explain the behavior of electrons in metals. The translation's abstract also notes that this trap arrangement resembles the traps used in current experiments.
The paper is cited together with the work Dirac did in the same year; it is generally noted that the two worked independently of each other. Fermi's calculation is for the gas inside a trap; its application to the free electron gas in a metal developed later through the work of figures such as Sommerfeld. The summary here rests on the short introduction of the translation; for details one must look at the translation itself.
Afterwards: In the following years the particles that obey this statistics were given the name "fermion." In cold-atom experiments, compressed fermion gases are being studied systematically.
Proceedings of the Royal Society A 112, 661-677 theory
Dirac derived the same result in the matrix language of quantum mechanics and with antisymmetric wave functions.
In 1926 Dirac takes Heisenberg's matrix mechanics as a general framework and asks how a system made of identical particles is to be described. When we exchange two identical particles the physical state must not change; this requires that the wave function either remain the same or change sign under the exchange.
Dirac shows that when the wave function changes sign (antisymmetric), two particles cannot be in the same state. This is a result of Pauli's exclusion principle arising from within quantum mechanics. In the same framework it is also seen that when the wave function remains symmetric a different counting rule emerges; this rule corresponds to the distribution found by Bose and Einstein.
Thus the two statistics stand side by side as two separate possibilities of the same quantum mechanics, and which particle obeys which is determined afterwards by experiment and theory. The rule called "Fermi-Dirac statistics" today rests on the works Fermi and Dirac did independently of each other and in the same year.
The limit of this work is that it is tied to the form quantum mechanics had at that time: cases in which the number of particles changes are not yet covered. Moreover, why the sign change under the exchange of two particles is realized in nature is not explained in this framework; what determines it is the spin-statistics connection shown later in field theory.
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 connection between spin and statistics of particles (that particles with half-integer spin are fermions) was later shown in general, in relativistic quantum field theory, by the spin-statistics theorem.
26. Bose-Einstein Condensation
Main question: What is macroscopic quantum matter?
When a gas is cooled a great deal, a large part of the particles called bosons can gather in the same lowest quantum state. In this case the whole of the gas behaves like a single wave and quantum effects are carried to visible scales. This state, achieved in the laboratory for the first time in 1995 in a dilute atomic gas, is called Bose-Einstein condensation. The question: can quantum behavior, beyond single atoms, be observed in a cloud of matter as a whole?
Anderson and colleagues observed Bose-Einstein condensation in a dilute vapor of rubidium atoms.
The condensation Einstein predicted around 1925 could not be observed directly for decades. In a dilute atomic gas this requires reaching a very low temperature: the temperature must fall so low that the average distance between atoms and the atom's quantum wavelength are about the same.
The group first cooled rubidium atoms with a laser and placed them in a magnetic trap. Then, by removing the most energetic atoms from the trap (evaporative cooling), it cooled the remaining gas further. The velocity distribution of the atoms was measured after the trap was released. Below a certain temperature, a narrow, sharp peak appeared in the distribution above the broad thermal base: part of the atoms had accumulated with very small velocities, that is, in the lowest quantum state. The peak grew as the temperature fell.
This picture is accepted as the observation of the condensation predicted by Einstein's calculation. There is interaction between the atoms, however weak; for this reason the real condensation is not an exact copy of the ideal-gas picture, and its theoretical details are studied separately.
The limit of the observation is the smallness and short life of the condensate: the cloud consists of very few atoms and can be observed in the trap only for a very short time. Moreover the temperature measurement is indirect; it is inferred from the shape of the velocity distribution. For this reason the paper gives the experiment's result with numbers and leaves the detailed theoretical comparison to later work.
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 year Ketterle's group at MIT independently achieved condensation with sodium atoms. Cornell, Wieman and Ketterle shared the 2001 Nobel Prize in Physics. In later work, interference between condensates was observed and cold atomic gases began to be used for simulation purposes (see greiner-2002).