Bose – Einstein Condensation
Abstract
The prevalent interpretation of Quantum Mechanics is that the world consists of two distinct domains:
- a “classical domain” comprising essentially macroscopic objects which behave according to deterministic laws
- a “quantum domain” populated by microscopic objects whose behavior is unpredictable, obeying probabilistic laws .
However, if the theory of Quantum Mechanics is to be a fundamental model of reality, it must be able to explain the emergence of the classically observed phenomena. In particular it must be able to explain and justify the fundamental incompatibility between the world as manifested by the quantum mechanical model and that of perceived reality as reflected in the classical models.
The theoretical relevance of the BE Condensation (Figure 1) resides in that it describes a quantum state of matter in which the classical to quantum transition (and the reverse, quantum to classical transition) can be readily observed, modelled and investigated, which could help to elucidate some of the physical processes underlying the quantum-classical transition.
Outline
- The quantum-classical dichotomy
- The Correspondence principle
- The classical limit of Quantum Mechanics
- The quantum-classical interface
- Persistence of the quantum wave properties in the classical domain Interference/Diffraction with quantum particles
- Macroscopic quantum wave phenomena
- BEC and the quantum-classical transition
- Indiscernibility of identical particles
- Quantum Statistical Distributions
- Summary
- The quantum-classical dichotomy
Quantum mechanics describes the world as if it consisted of two separate realms:
- the first, labeled “classical”, comprising essentially macroscopic objects such as cats, scientists and laboratory instruments…
- the second, labeled “quantum”, populated by microscopic objects such as atoms, nuclei, sub nuclear particles…
In the “classical” domain, on the one hand, things behave “decently” as we would expect them to do from our daily experience. An object remains in one place at a time, separated spatially from other objects, with the occasional interaction during which it exchanges energy, linear momentum, angular momentum (or even particles) according to well-defined conservation laws.
The conservation laws insure that the evolution of the material system is governed by a strict deterministic causality, i.e. equal cause giving rise to an equal effect.
This “classical” object is expected to exist, move and evolve in space and in time (or space-time) independent of the opinion of the curious scientist, and more importantly it continues to do so even when no observer is there to watch it, or a measuring device to record its actions.
Sentient observers in this “classical realm” call their world “objective reality” despite lingering doubt from some of their brighter minds about the unlikelihood of “knowing” the essence of this reality or its Noumena. Nonetheless, the existence of this objective reality is taken for granted.1
In addition, the classical theory of Mechanics assumes that “particles” are the fundamental bricks of this reality, and that the dynamical properties of these particles determine the state of the system of particles under observation and therefore define the observed phenomena.
These particles according to Newton’s model are (or appear) hard, impenetrable, separate, countable, distinguishable and permanent “chunks of matter” moving in well defined trajectories, that are mapped out by deterministic laws.2
Sure enough, the Quantum theory assumes also, like classical Mechanics, that “particles” are the fundamental bricks of reality, and that the dynamical properties of these particles provide a complete description of the state of the system and therefore define the observed phenomena. 3
However, in quantum Mechanics, the Heisenberg Indeterminacy Principle HIP prevents a simultaneous determination of all the dynamical variables of the system. The quantum mechanical description provides only partial statistical information about the behavior of the particles and of the system.
In the quantum domain, the use of the term “particle” to refer to “quantum entities” conveys a misleading picture. The term “particle”, and the picture associated with it, are borrowed from our coarse perception of the macroscopic world around us, a perception which is embedded in the fabric of the classical models of Reality.
It is hopelessly inadequate for communicating the meaning of the quantum particles. even when associated with the term quantum.
In fact, the “quantum” particles are not particles in any normally accepted sense.
They possess a fundamental property usually associated with wave characteristics:
the property of phase .
This phase property allows the superposition, interference, overlapping and entanglement of the “particles”, phenomena normally encountered with the mechanical and electromagnetic waves observed in the classical domain. This wave characteristic is at the core of the so-called “weird“ and counter intuitive behavior observed at the microscopic and submicroscopic level .
Accordingly, in the “quantum domain”, microscopic objects manifest themselves as intangible bundles of energy which may pop into being or vanish under appropriate conditions. The quantum objects or “particles” are entangled, interpenetrating, indiscernible and “ephemeral entities with an “identity crisis”.
As a matter of fact, they appear “undecided” whether to manifest as a dot-like bundle in a localized region of space or spread out wavelike over a wide region of space. The only remaining “particle” properties are inertia, charge and the mysterious spin that are detected as “splashes” of energy on phosphorescent screens or as whimsical tracks in a cloud chamber.
The wave-like behavior of quantum particles is encapsulated in the Schrödinger wave equation. The equation describes the evolution of a mathematical wave function (ψ(r, t) which varies continuously in space and time . The wave function ψ(r, t) represents the state of the quantum system.
More importantly, the wave function is a superposition of all the possible outcomes (or eigenfunctions) of the measurement of a given dynamic variable (or observable) A carried out on the system. This wave function is used to calculate the quantity ψ*Aψ which gives the probability distribution of the possible results of the measurement of A.
The actual measurement process “selects” only one definite outcome out of all the predicted spectrum of values (eigenvalues) corresponding to the dynamic variable. This means that, during the measurement, the wave function “suddenly collapses” to only one of the eigenfunctions corresponding to the measured “eigenvalue”. 4
This quantum mechanical description of the measurement process suggests that dynamical variables, such as position, momentum, spin and energy, may possess (each) many different values simultaneously (or may cycle rapidly between these values), until the system is submitted to a “classical” measuring device.
In particular, a quantum object can be in many different places in space (why not in time too?) at once until an observation of the position is carried out?
The interaction of the system with the “classical” apparatus produces an irreversible collapse of the wave function and therefore of the probability distribution. The collapse of the probability distribution “locks” the value of the dynamical variable.
The quantum theory cannot explain this wave function collapse.
The theory does not explain how, how fast, and when this collapse happens, and does not clarify whether this collapse is a real physical happening or just a mathematical artifact.
In addition, no empirical evidence whatsoever has been found to justify this quasi-magical collapse postulate. Otherwise, we would have been able to dissect it, analyse it and extract a model describing it.
What about QED, QFT and QCD?
A clarification is needed at this point. The theory of Quantum Mechanics discussed above is non-relativistic. The non-relativistic theory of Quantum Mechanics is usually compared and contrasted with “classical” theories such as classical mechanics and classical electrodynamics. Therefore, we have restricted our discussion to non relativistic theories (Quantum and classical).
However various versions of Relativistic Quantum field theories such as QED, QFT, QCD etc., have been developed over the past century with the aim of integrating the theories of restricted Relativity and Quantum mechanics into classical field theory. These theories have provided testable models for fundamental particles and their interactions.
They have culminated in the “Standard Model” which unifies three of the four fundamental interactions in the Universe, namely the weak, the strong and the electromagnetic interactions.
All these theories share the same fundamental features of the non-relativistic quantum theory discussed above. They are probabilistic, non-local, lacking a rigid causality and observer-dependent or at least measurement dependent.
Conversely, the theory of Relativity (General and Restricted) shares the same fundamental scientific premises with classical theories. Despite its far reaching effects on our perception of the phenomena, the theory of Relativity kept the classical foundations of the scientific model of reality essentially unchanged.5
To be sure, space and time ceased to be independent and were merged in space-time whose properties are defined by its matter and energy content, and in addition, mass became a form of energy.
However “relativistic” objects still exist in this relativistic space-time the way they did in the classical space and time. They still possess a well-defined location specified by their space-time coordinates. Objects remain distinguishable and evolve separately, their evolution being governed by deterministic laws obeying the principle of causality, and by their initial conditions (including position and velocity).
The relativistic revolution did not affect the core elements of the classical Universe: locality, separateness, causality, determinism and objective reality. Consequently the study of the phenomena as a series of successive events was still valid.
Natural phenomena could still be described in sufficient detail, dissected, analyzed, classified, modeled and explained as before and objective universal laws obeying the principle of Covariance (Relativity) could be extracted.
By contrast, the Quantum Mechanical theories, at least in the “official” interpretation threaten these very principles upon which Science was historically established.
It is evidently clear that we are faced with two deeply incompatible descriptions of the same reality:
- The classical (including the relativistic) description whose basic elements are locality, causality, determinism and objective reality, on the one hand,
- and the quantum description, which is probabilistic, non-local, non causal and observer dependent or at least measurement dependent, on the other hand.
However, we know that both classical and quantum descriptions are validated by incontrovertible empirical evidence each in its own domain of application.
It is also certain that the classical theories cannot be simply discounted on the ground that Quantum Mechanics has shown that the classical worldview is incomplete, as some quantum enthusiasts like to believe.
If Quantum Mechanics claims to be a general fundamental model of reality, it must be able to explain the emergence of the classically observed phenomena despite the quantum-classical dichotomy.
Quantum Mechanics must be able also to explain and justify the fundamental incompatibility between the phenomena as manifested by the quantum mechanical model and that of the perceived phenomena as reflected in the classical models.
And this brings us to the Correspondence principle.
2- The Correspondence Principle
In order to delineate the relationship between the quantum and classical domains, we must look at a similar situation in the history of science were the relationship was clearly elucidated : that of classical Mechanics with its other modern replacement, the theory of Relativity.
It is well-known that the failure of classical Mechanics to account for the invariance of the speed of light in vacuum, lead to its replacement by the theory of Restricted Relativity.5
However we know with certainty as noted above that classical Mechanics is upheld by thousands of independent measurements and experiments. 6
A widely spread misconception among students and science readers is that Einstein has proven classical Mechanics wrong. This not true of course.
It is more accurate to say that classical Mechanics was found to be limited in its scope of application; it is valid only for phenomena in which the speeds v are much smaller than the speed of light c (3×108 m/s), i.e. v << c.
This classical limit is defined in terms of the parameter v/c, termed deformation parameter. The theory of restricted relativity encompasses classical Mechanics as a limiting case for phenomena where v/c << 1 or v/c –> 0.
Another common mistake among students and science readers is to think that the theory of restricted relativity is applicable to high speeds “only”, while conversely classical Mechanics is valid for low speeds.
In fact the theory of Restricted Relativity, being the more “general” theory, applies for all speeds. At low speeds, classical laws are used only for convenience because advanced methods for solving problems of classical mechanics have been developed and perfected over the three last centuries.
Subsequently, in all processes where classical mechanics is valid, the relevant relativistic laws (and phenomena) must revert to the corresponding classical laws (and phenomena) as a first approximation.
In fact, all the laws of classical mechanics can be recovered from their relativistic counterparts simply by considering the limiting case v/c –> 0, i.e. the Lorentz transformations for the coordinates and the velocities reduce to the Galilean transformations. 7
There is also another example which also explicates this limiting case relationship i.e. that of General Relativity vs. classical Mechanics.
General Relativity reduces to Newton’s law of gravitation for weak gravitational fields. The condition of weak gravitational field can also be achieved at large distances. In addition for small velocities w.r.t to that of light, the laws of motion of classical Mechanics apply. 8
This is termed the Principle of Correspondence.
3- The classical limit of Quantum Mechanics
Contrary to the common belief which ascribes the principle of Correspondence to Bohr, it was Einstein who, in 1916, was the first to expound the Principle of Correspondence as we understand it today in his paper on the “Quantum theory of radiation” , and in 1917 in his presentation of the two theories of restricted and general Relativity: 9
Einstein did not use the term Correspondence explicitly. However the gist of the following quote from reference 9 is certainly clear:
“If we confine the application of the theory to the case where the gravitational fields can be regarded as being weak, and in which all masses move with respect to the co-ordinate system with velocities which are small compared with the velocity of light, we then obtain as a first approximation the Newtonian theory”.
He had presented the principle of correspondence as a heuristic argument supplying additional validation for his General Relativity theory, in the absence, at the time yet to come, empirical test of its prediction concerning the anomalous orbit of Mercury.
However the Principle of Correspondence is associated historically with the theory of Quantum Mechanics and with the name of Bohr who was the first to coin the term.
Bohr used it to justify a number of assumptions in his old Quantum Theory of Atomic emission. (for large quantum numbers n >> 1, atomic spectra merge into the classical continuum). However, Bohr’s Correspondence Principle in its early versions is considered nowadays only for its historical and epistemological interest and has lost its direct theoretical pertinence. 10
The first clear statement of the principle of Correspondence, as we understand it today, in the context of Quantum Mechanics is due to Born in 1933 as follows:
“Judged by the test of experience, the laws of classical physics have brilliantly justified themselves in all processes of motion… It must therefore be laid down as an unconditionally necessary postulate, that the new mechanics… must in all these problems reach the same results as the classical mechanics”.11
By “new mechanics”, Born was referring to the new theory of quantum or wave mechanics which was being developed at that time.
In his book on Quantum Mechanics, Dirac explicited this condition as follows:
“…classical mechanics may be regarded as the limiting case of quantum mechanics when h tends to zero.” 12
This means that by virtue of the Correspondence Principle, Quantum Mechanics should reduce to Classical Mechanics and Classical Electrodynamics for actions S (p.r or = E.t) much larger than Planck’s constant h. The deformation parameter is ℏ/S —> 0.
The Correspondence Principle was also introduced in David Bohm’s well-known textbook in a similar way:
“The correspondence principle […] states that the laws of quantum physics must
be so chosen that in the classical limit, where many quanta are involved, the
quantum laws lead to the classical equations as an average”. 13
Any new general theory which claims to provide a complete account of the natural phenomena must abide by the Correspondence principle.
Otherwise, the theory is only a temporary model of some aspects of reality, no matter how successful it turns out to be in modeling and predicting experimental results.
This criterion is definitive and unavoidable as Born expresses it “It must therefore be laid down as an unconditionally necessary postulate”.
Indeed, the application of the Principle of Correspondence to the theory of Relativity is straightforward. As we noted above, it suffices to assume v/c << 1 in any law of Special relativity, in order to obtain the corresponding classical law as a limiting first approximation .
However, in the case of quantum mechanics, the situation is more complicated. QM in its non-relativistic version has two limiting classical theories to contend with. For the laws of motion it should reduce to classical Mechanics while for the radiation laws (emission , absorption) and interaction of radiation with matter it has to satisfy the limit of Maxwell’s Electrodynamics, and in some cases both at the same time. 14
Furthermore, while the theory of Relativity was developed solely from its own postulates without recourse to classical Mechanics, the theory of Quantum mechanics is fundamentally rooted , as we have seen, from an analysis of its postulates, in the Hamiltonian formalism of classical Mechanics and is impossible to elaborate without the latter.
The straightforward application of the condition ℏ/S —> 0 leads to the classical limit in most of the cases using semi-classical approximation techniques.
However it fails to transform the indeterministic quantum mechanical phenomenon into its deterministic classical counterpart. 14
4- The quantum-classical “interface”
A number of scientists and science teachers give the impression that quantum phenomena and entities, whether particles, waves or in between, exist and interact in their own distinct “quantum world” separated from the classical world of measurement devices, cats and humans by an “impassable” divide created by the collapse of the wave function.
The prevalent viewpoint among some workers in the field is that there exists a kind of barrier or divide that separates this Quantum world, behind which “lurk” all kind of weird phenomena and entities to be revealed only by applying something called a “classical” measuring device.
While it is true, as shown by experimental evidence, that these quantum phenomena are “safely” trapped, most of the time, in the quantum domain behind this “barrier”, they tend however to overflow quite readily into the “real” (so-called classical) population using many different routes and under various physical conditions.
Theoretically, this barrier or divide is guarded by an irrational quasi-magical mechanism called the collapse of the wave function (or probability function) which insures that only “rational” occurrences are detected in the real world. The mysterious collapse eliminates the superposition of states thus avoiding “Schrödinger’s cat” style observations of mutually exclusive outcomes (e.g. dead and alive, here and there, in and out, up and down, weird and weirder, etc..).
Out of all possible values allowed for the dynamic variables (position, energy, frequency , momentum, wave-vector etc.. ), by the probability distribution function, only one single value succeeds in crossing the divide, the rest being efficiently “stamped out”?
This fact alone should have raised the mental eye brows of physicists and spurred them to question more forcefully the official dogmas of the Copenhagen interpretation.: why only one ? Since the quantum process is random, why not obtain also a random number of actualized outcomes?
Unless there is a physical mechanism that insures this single occurrence, the whole quantum mechanical model remains an empirical “hocus pocus” collection of recipes which are adjusted to fit the data as we go along.
However, and paradoxically, the Correspondence principle implies that there is nothing called a “Quantum World” of things behaving weirdly according to their own somewhat “irrational” laws in contrast with the so-called classical Universe were objects follow rational deterministic laws. If Quantum mechanics is a fundamental theory this quantum “weirdness” must somehow apply to everything and the barrier should not be there at all.
Indeed, the particles populating the quantum microscopic realm e.g. photons, electrons, protons, neutrons, atoms, molecules…, are exactly the same as the ones that make up the objects observed in the “classical” macroscopic realm.
The quantum wave properties of these particles certainly persist even after whatever measurement, interaction and wave function “collapse” have taken place.
We know for certain that the observed quantum phenomena are a direct consequence of a single fact, the phase property of the particles which gives rise to the quantum wave superposition effects. However, similarly to classical wave superposition, phenomena such as interference and diffraction among other things, require the superposition of coherent waves in order to occur.
Therefore, two physical conditions must be fulfilled by the system for the quantum effects to occur: coherence and quantum wave contact.
This means that for the quantum phenomenon to take place, the quantum waves must have a well defined phase relationship and must overlap in space for the duration of the phenomenon.
Conversely, the physical mechanism underlying the condition ℏ/S —> 0 must therefore be related to physical processes at the end of which the quantum behavior disappears and the classical phenomenon emerges i.e. decoherence and/or “quantum wave disengagement”.
These processes must ideally happen at the hypothetical barrier or boundary separating the two domains for them to be able to explain the experimental facts.
On the other hand, we know from the complexity of the experimental set-ups designed to produce quantum entanglement or Bose-Einstein Condensation (BEC) or even two-slit interference, that the quantum wave coherence, which is responsible for these phenomena, can be easily destroyed by interaction with particles in the external environment a phenomenon called “environmental decoherence”. 15
Evidently, the quantum realm is extremely fragile.
Experiments involving entanglement, BEC, Qubits and similar cases require a high maintenance level and great care to keep the particles in “quantum contact” and isolate them long enough from the collective effects of environmental interactions which result in a rapid degradation of the coherent correlation produced by the phase of the quantum wave.
On the other hand, according to the Principle of Correspondence, any peculiar behavior of the objects at the “quantum” level (atomic and subatomic levels), must either be observable at the macroscopic level or blend into the classical behavior according to a well-defined physical mechanism involving measurable physical occurrences and obeying the fundamental conservation laws of Nature.
In conclusion, there is no quantum-classical divide.
There is no demarcation line between classical and quantum behavior, or classical and quantum realms.
There is no discontinuity separating quantum and classical phenomena.
The transition between classical and quantum is gradual and depends on the relative effect of the perturbation introduced by the measurement/interaction process.
In other words, the classical description merges into the quantum description gradually as the perturbation effect of the quantum of minimum action h becomes more pronounced.
This also means that the quantum (wave) properties of material particles e.g. superposition, interpenetration, entanglement, indiscernibility, uncountability and fuzziness must taper off gradually and reduce to the classical attributes of macroscopic material objects such as impenetrability, separateness, countability, distinguishability and permanence, according to physical processes without the need to invoke discontinuous processes such as the “magical-like” collapse of the wave function, or metaphysical rules such as the Principle of Complementarity.
These gradual processes should give rise to observable features intermediate between the pure quantum phenomena of the isolated system, on the one hand, and the classical phenomena i.e. phenomena associated with partial coherence or partial overlap, on the other.
The quantum- classical barrier is in fact an interface at which the interaction of the quantum system with particles of the (classical) environment lead to gradual “quasi-continuous” and irreversible deterioration of the quantum wave coherence, or to gradual “quasi-continuous” and irreversible quantum disengagement.
Most of the new exciting research is in fact taking place at this interface. This explains the reason that the measurement problem/paradox has occupied a central position in the theory of quantum Mechanics. It has served to reveal the inconsistencies of the official Copenhagen interpretation and point out the incompleteness of the quantum mechanical theory itself.
The elucidation of these mechanisms could lead to the development of a more rational and realistic phenomenological model of the measurement process. The current empirical model of the sudden collapse of the wave function during measurement provides too simplistic an account of what is really happening. This irrational model has contributed to the introduction of contradictions, inconsistencies and metaphysical interpretations such as Complementary, observer-dependent outcomes, many-world quandary and so on.
5– Persistence of quantum wave properties in the “classical” macroscopic domain
For a start, all empirical observations suggest that the “collapse” of the wave-function during the measurement (or any interaction according to Landau 16) does not annihilate the wave nature of the particle which persists in all subsequent interactions right through the “classical domain”.
What is affected by the measurement/interaction is, the particular wave structure (wave function) resulting from the coherent superposition of two or more waves under the boundary conditions of the system.
We must distinguish therefore between the quantum phenomena produced by coherent quantum wave superposition , on the one hand, and the quantum wave characteristic of the particles, on the other hand. The wave characteristic is an intrinsic property of the particle at par with its mass, charge and spin and is not removed by the measurement/interaction processes, .17
Interference/diffraction of quantum particles
A most striking example of the persistence of the quantum properties after the measurement (collapse of the wave function) is the “welcher veg” (or which way) double-slit diffraction experiments with single particles.
The principle of the “which-way” experiment was proposed by Feynman as follows: 18
Let us consider a stream particles with a well- defined kinetic energy (momentum) fired one by one at a screen containing two slits. After passing through the slits, the particles impact a second screen where a detecting apparatus records the arrival of each particle (time and position). As the number of particle impacts increases, a double slit interference pattern typical of (classical) wave behavior is observed.
When one tries to detect through which slit the particles had passed (hence “welcher-weg”), the double slit interference pattern disappears.
The main conclusion is as follows: the act of pinpointing the trajectory of the particle i.e. through which slit the “particle” has passed, destroys the interference pattern produced by the superposition of the waves.19
Figure 2: Single slit interference/diffraction with photons b
This experiment is normally used to verify (test) the dogmatic assertion of the Complementarity principle, in which Bohr postulated that the wave and particle models of the quantum entity are mutually exclusive and cannot be observed simultaneously with the same experimental arrangement.
Practically, the principle of Complementarity affirms that an experimental set-up allowing the particle properties to be observed prevents automatically any observation of the wave properties and vice-versa.20
However, in all experiments of this type, a single slit diffraction dot pattern appears invariably on the the screen even after the particle path had been sharply identified by the path detection apparatus, (Figure 2). 21
The appearance of the single-slit diffraction pattern demonstrates that the wave property of the particles persists even when we know with complete certainty through which slit the electron has passed. The single slit diffraction pattern is an incontrovertible proof that the wave character of the quantum particle is intertwined with its particle characteristic, and is not eliminated by the measurement process.
This is in complete contradiction to the assertion of Bohr’s Complementarity Principle and to the results implied by Feynman’s for the “welcher weg” double slit “thought” interference experiment: if the slit through which the electron passes is identified, the interference term disappears and the pattern is replaced by a Gaussian distributions reflecting the classical probability distribution.18
Furthermore, the disappearance of the double- slit interference pattern can be accounted for, in a simple way, via the destruction of the coherence between the interfering waves due to perturbation of the electron’s motion by the path detection apparatus, a phenomenon termed Decoherence.
The phenomenon of Decoherence has amply been validated experimentally and has been investigated assiduously in the past three decades. 22
Decoherence is understood as quantum entanglement (or correlations ) of the system with the particles in its environment which weakens the phase relations between the components of the system. This entanglement with the environment is manifested as diminution of the interference term of the system’s wave function.
Also, a number of “weak” welcher-weg experiments have been carried out demonstrating partial decoherence: an “unsharp” determination of the particle’s path was obtained without destroying the interference pattern. 23
These experiments were designed to test the “either/or” interpretation of the PIH advocated by Bohr in the Principle of complementarity. This interpretation was found to be too oversimplified to represent what is really taking place experimentally.
Instead, the results of these experiments revealed the occurrence of intermediate situations with partial coherence, unsharp trajectory and low contrast interference fringes.
It is experimentally established that it is always possible to obtain an approximate measure for the particle’s (unsharp) trajectory, using a Wilson or smoke chamber within a finite uncertainty given by the HIP : Δx.Δpx ≥ ℏ/2, without completely destroying the superposition pattern.24
As a matter of fact, these trajectories can be observed and modified using external electric and magnetic fields, and their motion can be modeled mathematically, traced and predicted with sufficient precision. The reason for this is that the energy perturbation due the weak interaction of the electrons with the (detecting) particles in the chamber is still much smaller than the kinetic energy of the electrons.
This is another example which indicates that the transition between classical and quantum is gradual and depends on the relative effect of the perturbation introduced by the measurement/interaction process.
6- Macroscopic quantum wave phenomena
Diffraction/interference phenomena with quantum particles are examples of macroscopic quantum wave effects overflowing through the quantum-classical “divide”.
They are not the only ones, albeit the most frequently used by science teachers and, usually alluded to by science bloggers to illustrate the counter-intuitive aspects of the microscopic world.
In fact, there are many other macroscopic quantum phenomena, some of which too mundane to be noticed or appreciated by seekers of the strange and weird .
Nonetheless, their occurrence constitutes the very foundation of all the achievements of modern engineering and of today’s technological prowess.
We don’t need to dig deep into the sub atomic and sub nuclear particle domain in order to encounter these quantum phenomena.
To begin with, the properties of macroscopic materials whether they be, solids, liquids and even gases among other things can not be understood without invoking the quantum wave properties of their constituents such as, wave superposition, indiscernibility, symmetric/anti-symmetric nature of the wave function or the quantification of linear and angular momenta and energies whenever the particles are confined into the quantum wells of atoms and molecules .
Indeed, the electrical, magnetic, optical , spectral, mechanical and thermodynamic properties of the solid state among other things are a direct result of the quantum wave properties of their constituent electrons.
The electrons are fermions that obey Pauli’s exclusion principle (and the Fermi-Dirac statistics). The wave nature of the electrons is responsible for the periodic potential and energy band structure of crystals and for the existence of the Fermi energy which plays a central role in defining their semi-conducting behaviour.
Semi-conductors are the essential components of many modern technologies (electronics, optoelectronics, photovoltaics, energy storage, communication..). Their peculiar properties are due to this fermionic nature of the electrons.
Moreover, the mere physical “existence” and the properties of matter structures (and of all ordinary objects) is of course due to this fundamental fermionic characteristic of the electrons and other particles.
On the other hand, the phenomena of Superfluidity25 and Superconductivity26 were discovered early on last century (1911). The discovery of low temperature superconductivity is associated with the name of Karmelingh Onnes. He recognized readily the sudden decrease of the resistivity (increase in the conductivity) of the mercury wire at T = 4.2 K, as a new unusual phenomenon and reported it, being the first to coin the term Supraconductivity.
However, Onnes failed to recognize the sudden increase of pressure in his sample chamber as an indicator of a superfluid phase transition in liquid 4He used as coolant. He attributed it to a leak in the sample chamber which reduced the ultra high vacuum.
In fact the superfluid Helium particles behave like a single “bosonic” entity with zero viscosity which seeps unhindered through the container’s walls. The phenomenon of superfluidity was correctly identified and characterized a quarter of a century later (1937) by two groups of researchers studying the viscosity of liquid helium at very low temperatures. 27
Superconductivity and superfluidity are two macroscopic quantum phenomena rooted in the “bosonic” nature of the particles which undergo Bose-Einstein condensation (BEC) at low temperature. The Bose-Einstein condensation (BEC) was first predicted by Einstein in his seminal paper on quantum statistics in 1924. 28
BEC is a peculiar quantum phase change underwent by quantum entities possessing a symmetric wave function (bosons) at low temperature, in which all the particles”condense” in the same quantum state giving rise to some amazing macroscopic quantum mechanical properties such as, coherence , interference, zero viscosity, zero resistivity and tunneling through walls and barriers. 29
7- BEC and the quantum-classical transition
As noted, Bosons are particles which possess a symmetric wave function and obey the Bose-Einstein statistics. This means that, unlike Fermions, they can under certain conditions aggregate in the same quantum state (the ground state of the system) and thus behave like a single entity, sharing the exact same dynamical properties (linear and angular momenta, kinetic energy, velocity, phase …) associated with this quantum state, a phenomenon termed Bose Einstein condensation.
Bose-Einstein condensation has been demonstrated in solids (superconductors), in liquids ( superfluids) , in atomic gases and in nanoparticles 30, 31.
On the other hand, Fermions as such cannot undergo Bose-Einstein condensation, since they obey Pauli’s exclusion principle (and the Fermi-Dirac statistics) .
However, weakly bound pairs of Fermions such as 3He-3He pairs, or Cooper electron pairs in superconductors, may form a bosonic compound which undergoes condensation at sufficiently low temperatures. 32
For example, Helium (4He) has a ground state spin equal to zero and is a boson. When the liquid 4He gas is cooled further down below its liquefaction point (~ 4.2 K), it becomes superfluid, all the atoms being in the same energy state i.e. it undergoes a change of phase via BEC. The critical temperature (or λ –point) for the onset of the BEC phenomenon is Tc = 2.172 K (see figure 5 below).
The 3He isotope, on the other hand, with one less neutron has a ground state spin equal to ½ and is a fermion. It becomes superfluid at a much lower temperature (Tc ~ 3×10-3K) by forming weakly interacting pairs i.e. 3He-3He dimers with integer spin which are therefore bosons. 32
The theoretical relevance of the BE Condensate resides in that it describes a quantum state of matter in which the classical-quantum transition (and the reverse quantum-classical transition) can be readily observed and investigated.
The classical limit criterion (ℏ/S –> 0) proposed by Dirac, while correct, is too general and does not provide a straightforward recipe similar to v/c << 1 for restricted relativity. In fact it is not always evident how to apply Dirac’s criterion to reduce Quantum Mechanics to classical mechanics and/or classical electromagnetism.
In addition, the conditions for the classical limit of quantum Mechanical laws and descriptions depend on the type of situation being investigated. The analysis of each situation provides new insight in the workings of the quantum phenomena. In fact, the same physical parameters that can be adjusted to “switch off” the quantum effects and thus achieve the classical limit, may be used conversely to switch on or enhance the quantum effects.
One way to gain insight in the properties of the quantum-classical interface, consists therefore in analyzing a simple physical situation in which the quantum properties are clearly observable and modellable mathematically and use that to extract the physical parameter allowing the reduction of the system to the classical counterpart.
Most situations are too complex to allow a clear straightforward delineation of the physical mechanisms underlying the transition. The Bose-Einstein Condensate consists of a single bosonic entity. It can be readily realized and easily analyzed, simulated and
modeled theoretically.33
8- Indiscernibility of identical particles
The principle of indiscernibility states that identical quantum particles i.e. particles of the same type i.e. with the same intrinsic properties (mass, charge and spin) do not have a proper individual identity. They are in essence indistinguishable from each other and cannot be regarded as separate entities: they cannot in principle be labeled or counted.
The principle is a direct consequence of the superposition of the wave functions and therefore of the wave nature of the particles.
In the region of space where the deBroglie quantum waves overlap, the two particles form a single quantum entity described by a single wave function. They lose their individual identity and become indistinguishable (Figure 3).
Figure 3: Superposition of the two de Broglie waves for d < λg (group wavelength). c
The indiscernibility of particles of same type was already encountered in classical Physics. It was invoked as a property of identical classical particles to solve Gibbs mixing paradox which appeared in the calculation of the entropy of the ideal gas . 34
However classical indiscernibility is fundamentally different from its quantum counterpart. The classical particles do not lose their identity remaining separate and countable.
Indiscernibility in a Two-particle system
Consider two identical indiscernible quantum particles which we will label temporarily particle 1 and particle 2.
The Hamiltonian of the system H (1, 2) and the probability density |ψ (1, 2) |2 must remain unchanged under permutation of the two particles:
|ψ (1, 2) |2= |ψ (2, 1) |2 [1]
Equation [1] shows that the state functions could differ only by a phase factor:
ψ (1, 2) = eiα ψ (2, 1) [2]
Now if we perform a second permutation we return the system to the initial situation:
ψ (1, 2) = ei2α ψ (1, 2)
or [3]
ei2α = 1 —> eiα = ± 1
Substituting in [3], we have:
ψ (1, 2) = ± ψ (2, 1) [4]
Equation [4] shows that we have two types of state functions:
- Symmetric with : ψ (1, 2) = + ψ (2, 1)
And [5]
2. Anti-symmetric : ψ (1, 2) = – ψ (2, 1)
Now consider the two quantum particles labeled 1 and 2 in the same quantum state described by the wave function ψ (1, 2). A permutation of the two particle will eave the wave function unchanged i.e.:
ψ (1, 2) = ψ (2, 1) [6]
By comparing equation [5] and [6] we can deduce the following:
- Symmetric wave functions described by equation [5a] satisfy equation [6] always that is for any number of particles and permutations. The occupation index n (number of particles per single particle state) may take any value from 0 to N that is the total number of particles in the system : n = 0, 1….N
2. Antisymmetric wave functions described by equation [5b] cannot satisfy equation [6], meaning that no two particles may share the same quantum state (Pauli’s exclusion rule). The occupation index n per single particle state may take only two values 0 or 1: n = 0, 1
Indiscernibility in N-particle systems
It can be shown16 that for a system of N identical indiscernible particles described by the state function ψ (1,…i, j,…N)) only two possible situations arise:
- Completely symmetric state function: any permutation results in a state equal to the initial state
- Completely anti-symmetric state function: the function changes sign under an odd number of permutations and remains equal with an even number of permutations
In summary, the state functions of a system of N identical particles are either completely symmetric (occupation index: n = 0, 1…..N) or, completely anti-symmetric (occupation index: n = 0, 1), depending on the nature of the particles.
This statement is termed “Symmetrization postulate”.
Bosons and Fermions
The quantum behavior of Bosons and Fermions has its root in the quantum property of indiscernibility of identical particles.
In Dirac’s theory of relativistic quantum mechanics, it is demonstrated that the type of statistics governing the collective behavior of the particles is related to their spin.
There are two types of elementary and composite particles:
– Fermions such as electrons and other elementary and composite particles which possess a half integer spin.
– Bosons such as photons and other elementary and composite particles which possess an integer spin.
The theory showed that electrons and other fermions are described by antisymmetric wave functions and obey the Fermi-Dirac statistics (hence fermions) which forbids two particles to be in the same state.
On the other hand Photons and other Bosons obey the Bose-Einstein Statistics which allows an unlimited number of particles of the same kind to be in the same state, i.e. may undergo BEC.
8- Quantum Statistical distributions
Statistical distributions are essential for the study of real systems which comprise a very large number of particles. These are probability distribution functions expressed usually in terms of the probability of finding a particle in a particular state with a particular energy value εr , and used to calculate the mean number of particles in a given single particle state, < n(εr) > , or occupation index.
The classical statistical distribution is the Maxwell- Boltzmann (MB) distribution discussed in a previous blog. 35
The MB gives, for a (closed) system, in thermal equilibrium with a heat source at temperature T, the probability P(εr) that a particle (or more) has an energy value εr :
P(εr) = Z-1.exp( – βεr ) [7]
Z is a constant termed partition function Z and β= 1/kT
In this case, the average number of particles in a given particle state εr is given by:
< n(εr) > = exp[-β(εr – μ)] [8]
Where, the parameter μ is the thermodynamic potential.36
The Fermi-Dirac distribution gives:
< n(εr) > = [ exp β(εr – μ)) + 1]-1 [9]
The FD distribution for electrons in solids is expressed usually as follows:
F(ε) = [ exp β(ε – EF) + 1]-1 [10]
Where EF = μ, is termed Fermi energy .
The Fermi energy EF (electrons potential) is a characteristic of the material and can be determined empirically from measurements of CV vs T for a given metal. It represents the highest energy level occupied by electrons at T ~ 0K.
On the other hand, he Bose-Einstein distribution is given by:
< n(εr) > = [exp β(εr – μ) – 1]-1 [11]
In the special cases of photons and phonons (quantized mechanical wave):
< n(εr) > = [ exp(βεr ) – 1]-1 [12]
Where μ = 0.
The three distributions are plotted on the same graph in figure 4, below:
Figure 4: Average occupancy for the 3 statistical distributions at a given temperature d
Bose-Einstein condensation
From Quantum Mechanics we know that the energy of a particle of mass m confined in a given volume (say a cubic box of side a) is quantized and given by :
εn = ℏ2π2. n2 / 2ma2 [13]
With, n ∈ N .
The ground state energy is thus ε0 = 0 J, and the first excited state is ε1 = ℏ2π2 / 2ma2
For a box of macroscopic size (1 cm3), and for 4He gaz, we calculate: ε1 ~ 10-37 J >> kT ( kTroom ~ 25 meV or 4×10-21J)
For classical distinguishable particles, the probability of the particle possessing an energy value εr is given by the Maxwell- Boltzman distribution i.e equation [7]:
P(εr) = Z-1.exp( – βεr )
Consider two energy levels labeled i and j with j > i in a system of N particles.
Using equation [7], we can write:
P(εj) / P(εi) = exp. [-Δε/kT)] [14]
Where k is Boltzmann’s constant and Δε = εj – εi.
For Δε << kT, as is the case here, the ratio Δε/kT → 0, and the exponential →1.
This gives:
P(εj) = P(εi)
Subsequently, the ground state population remains negligible for all practical purposes. Note: for the ground state population to become appreciable the temperature has to become extremely low i.e. kT ~ 10-37, corresponding to T~ 10-16 K a truly unreachable temperature to date.
However, Figure 4 shows that the Bose-Einstein probability, which merges with the MB probability curve for high energy levels ε, separates from it and starts rising much more steeply at a relatively low temperature. This indicates that as ε approaches the ground state value, the particles start aggregating in the ground state much more rapidly than in the MB case, i.e. undergo a BE Condensation.
Let s look more closely at the statistical distribution of bosons given by equation [11]:
< n(εr) > = exp[β(εr – μ) – 1]-1
This equation is valid only for values of μ < ε0, where ε0 is the particle energy in the ground state or the lowest energy. Otherwise the average occupation < n(ε0) > becomes negative which is not possible, ( note that μ <0).
Since e0 = 0, we can write:
N0 = < n(ε0) > = [exp(-βμ) – 1]-1 [15]
Where N0 is the ground state population.
As μ –> ε0 = 0 , the population of the ground state increases appreciably until all the particles condense in this ground state.
The onset of the BE condensation takes place at the critical temperature Tc which is given as a function of the systems parameters as follows:37
Tc ~ 3.31 ( ℏ2/m.k) (N/V)2/3 [16]
Where N is the total number of particles, and V the volume assuming uniform system.
The particles’ deBroglie wavelength at the critical temperature (λc) is defined as follows:
kTc = ℏ2k2/2m = ℏ2π2/2mλc2
Or [17]
λc = (ℏ2π2/2mkTc)1/2
The critical particle density (at Tc) is calculated to be:
nc = (Nc/V) ~ 2.61 (λc )-3 [18]
The average distance between particles is simply d = (V/N)1/3 .
Which means that at Tc, the average distance between particles is comparable to the thermal wavelength of the particles λc.
The fraction of particles in the ground state is then given by:
N0/N = ( 1 – T/Tc )3 [19]
As the temperature decreases below TC, the proportion (N0/N) of particles condensed in the ground state increases.
In summary there are two conditions for the occurrence of BE Condensation:
T < Tc
And [20]
d = (V/N)1/3 < λc
A spectacular experimental validation of the existence of Tc and of the BE Condensation can be found in the measurement of the specific heat capacity of 4He at low temperatures, illustrated in Fig 5 , below:: 38
Fig. 5: Heat capacity of liquid 4He versus temperature, showing the typical λ-shape at the critical temperature ( TC ~ 2.17 K at 1 atm.), and revealing the phase transition between the normal (HE I) and the Superfluid (He II). e
Conditions for the quantum-classical transition
The graph of the statistical distributions, displayed in Figure 4, demonstrates the gradual merging of the quantum properties of the particles into the classical properties as a result of decoherence.
The classical limit is obtained for large values of the exponential (exp [β(εr – μ)] >> 1), In this case the factor (± 1) becomes negligibly small compared to the exponential factor and the two quantum distributions reduce to the Maxwell-Boltzmann distribution (equation [8]).
This is clearly shown mathematically in equations [10] and [1]:
< n(εr) > = [ exp β(er – μ)) ± 1]-1 —> < n(εr) > = [ expβ(εr – μ)]-1 [21]
It is interesting to find the experimental conditions under which the classical situation is obtained, that is when the particles lose their quantum properties and start behave like classical particles.
Consider a gas of N particles at (V, T). The classical limit must satisfy the condition:
exp [β(εr – μ)] >> 1.
This means that the occupation index or the average number of particles per state of energy εr must be very small:
< n(εr) > << 1 [22]
This condition can be achieved via two pathways:
a- Reduce the total number of particle N (or the particle concentration N/V) for a given temperature.
b- Increase the system temperature T for given number of particles N.
Increasing the temperature increases the number of energy levels accessible to the particles which results in the decrease of the average number of particles per energy level εr i.e. the occupation index < n(εr) > .
This means that for sufficiently high temperatures and/or low concentrations the quantum behavior of the particles is suppressed and eventually eliminated giving way to classical behavior. There are different ways of expressing this limiting condition and also justifying it physically.
In the first situation, the mean separation between particles becomes much larger than the deBroglie wavelength λBroglie :
d >> λBroglie
The particles become too far apart for their associated deBroglie waves to overlap spatially. This negates the wave function superposition which is responsible for their quantum properties (indiscernibility, entanglement, interference).
In the second situation and as T (hence kT) increases, the thermal disorder of the particles increases. The increase in thermal collisions scrambles the phases of the particles . This counteracts the quantum ordering (coherence) resulting from the entanglement of the particles due to spatial overlap of the wave functions, and eventually destroys it.
As predicted (& 4), the mechanism underlying the condition ℏ/S —> 0 involves physical processes at the end of which the quantum behavior disappears and the classical phenomenon emerges i.e. decoherence and/or “quantum wave disengagement”.
Conversely, two physical conditions must be fulfilled by the system for the quantum effects to occur:
– coherence which prevails at T < Tc
– wave overlap or “quantum contact” which occurs at d < λBroglie
These processes occur at the boundary separating the two domains classical and quantum, which manifests itself in the case of BEC, as phase change initiated at the critical temperature Tc.
9- Summary
Any new general theory which claims to provide a complete account of the natural phenomena must abide by the Correspondence principle.
This means that Quantum Mechanics should reduce to Classical Mechanics and Classical Electrodynamics for actions S much larger than Planck’s constant i.e. ℏ/S —> 0.
This criterion is definitive and unavoidable as Born expresses it “It must therefore be laid down as an unconditionally necessary postulate”.
On the other hand, the observed quantum phenomena are a direct consequence of the phase property possessed by the particles. This phase property is the cause of the phenomena of interference, indiscernibility and entanglement of the quantum particles, phenomena normally associated with mechanical and electromagnetic waves observed in the classical domain.
Two physical conditions must be fulfilled by the system for the quantum effects to occur: phase coherence and wave overlap or “quantum contact”.
The physical mechanism underlying the quantum to classical transition must therefore be connected with specific physical processes at the end of which the quantum behavior disappears and the classical phenomenon emerges i.e. decoherence and/or “quantum disengagement”.
The Bose-Einstein Condensation (BEC) provides an ideal situation for the study of the quantum -classical transition. The condensate consists of a single bosonic entity. The analysis of BEC shows that there are two conditions for the occurrence the classical to quantum transition i.e. BEC:
- The temperature has to be below a critical value to preserve the phase coherence:
T < Tc
- The particle density has to be large enough so that the average distance between the particles becomes smaller than the deBroglie wavelength to insure quantum superposition:
d = (V/N)1/3 < λBroglie
These processes occur at the boundary separating the classical and quantum domains, which manifests itself in the case of BEC, as a change of state (phase transition) initiated at the critical temperature Tc.
The quantum-classical divide is an interface in which the two realms merge into each other. All the new exciting research is in fact taking place at this interface. This explains the reason behind the central position occupied by the measurement problem/paradox in the theory of quantum Mechanics.
The controversy over the measurement problem has served to reveal the inconsistencies of the official Copenhagen interpretation and point out the incompleteness of the theory itself . It has instigated new theoretical and experimental activities which enriched and improved our understanding of the quantum world with the discovery of hitherto unknown phenomena and applications.
The development of the quantum Decoherence model, in particular, has introduced tangible physical mechanisms in the measurement/interaction problem . The elucidation of these mechanisms could lead to the development of a more rational and realistic phenomenological model of the measurement process and of Quantum Mechanics.
Images
- “Bose-Einstein condensate: a new quantum phase of matter”.
Credit: NIST/JILA/CU-Boulder
Each image shows a cloud of Rubidium-87 behaving like a single quantum entity , a boson. The temperature drops from left to right : 200 nK, 100 nK, 20 nK. As the temperature decreases, the atoms are seen to accumulate in a single distribution, which becomes more and more peaked. The right most peak shows the condensation of about 2000 Rubidium atoms in the same dynamic state, i.e. with the same velocity as predicted by the Bose-Einstein statistics.
b- Young’s Interference Experiment with a single photon
https://photonterrace.net/en/photon/duality/
c- Entanglement and Indiscernibility of identical particles
Credit: Daniel Suchet, “La quête des températures ultrabasses”
d- Graphs of quantum and classical statistical distributions
https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics
e- λ- point diagram for 4He, showing the typical shape at the critical temperature ( TC ~ 2.17 K at 1 atm.), revealing the phase transition between the normal (HE I) and the Superfluid (He II) phases. The experimental points are for the Heat capacity of liquid 4He at saturated vapor pressure versus temperature
Z. Korichi1 and M. T. Meftah1. Physical Science International Journal
8(1): 1-13, 2015, Article no.PSIJ.20366
References and Notes
1- See for example:
a- Gottfried Wilhelm Leibniz, 2007; 2013
https://plato.stanford.edu/entries/leibniz/#TruReaTruFac
“….one’s body or even a stone is real because it is an object of perception that fits into an account of the world that is both coherent from the point of view of the single perceiver and in harmony with the perceptions of other minds.”
b- Kant. I, (1781). “Critique of pure reason”
“There are things which are given to us, as objects of our senses situated outside of us, but of what they may well be in themselves (noumena), we know nothing, we only know their phenoumena, that is, the representations they produce in us by affecting our senses”.
2- A detailed analysis of the concept of particle and of its origin is found in my blog:
3- This applies to non-relativistic quantum mechanics. Relativistic quantum field theories describe particles as excited vibrational states within a quantum field. In quantum Electrodynamics, the field is electromagnetic and the particles are photons. In Dirac’s relativistic quantum field theory, the particles are electron-positron pairs in an electron-positron quantum field. There is some similarity with the Bohm-de Broglie quantum field guiding the particles and determining their trajectory.
4- See my blog for a brief review of the formalism:
https://particlemysteries.wordpress.com/2023/02/08/the-wave-particle-conundrum-iii/
5- See my article: “The-universe-according-to-the-principle-of-relativity”
6- Classical mechanics still accounts for most of the observable macroscopic phenomena in its field of application. It is still used to construct high precision machines and mechanical systems to predict and account for the motion of planes, rockets, spaceships, comets, asteroids and planets where minor relativistic corrections are sometimes needed.
7- All relativistic relations reduce to their classical counterparts for v/c<< 1 (except E = mc2 , which has no classical counterpart)
A pertinent example is that of relativistic kinetic energy given by:
T = (γ -1 )mc2
γ is the Lorentz factor given by: γ = (1 – v2/c2)-1/2
For v/c –> 0, γ ~ 1 + v2/2c2 + higher order terms.
By substituting in the expression of T we have:
T ~ mv2/2
Therefore the classical expression of the kinetic energy is a first order approximation of the relativistic expression in the limit v/c –> 0.
8-Tolish, A. General Relativity and the Newtonian Limit
9- a- Einstein, A. (1916), Zur Quantentheorie der Strahlung, Deutsche Physikalische Gesellschaft. Verhandlungen 16(18), 47–62.
b- Einstein, A. (1917). Über die spezielle und die allgemeine Relativitätstheorie : Gemeinverständlich.
English version, (1920). Relativity: The Special and the General Theory: Popular Exposition; authorised translation by Robert W. Lawson, D.Sc., University of Sheffield
10- Bohr’s Correspondence principle(2010-2020), & 6:
Interpretations in the current Physics Literature.
Stanford Encyclopedia of Philosophy
https://plato.stanford.edu/entries/bohr-correspondence/
In section 2
“Among current Bohr scholars there is a consensus that Bohr did not intend his correspondence principle to designate some sort of general requirement that quantum mechanics recover the predictions of classical mechanics in the classical limit, despite the prevalence of this interpretation in the physics literature”.
11- Full quote of Born’s statement from reference 10:
“The leading idea (Bohr’s correspondence principle, 1923) may be stated broadly as follows. Judged by the test of experience, the laws of classical physics have brilliantly justified themselves in all processes of motion… It must therefore be laid down as an unconditionally necessary postulate, that the new mechanics … must in all these problems reach the same results as the classical mechanics. In other words, it must be demanded that, for the limiting cases of large masses and of orbits of large dimensions, the new mechanics passes over into classical mechanics”.
12- P. A. M. Dirac, (1958). The Principles of Quantum Mechanics, 4th ed. (Oxford University Press, Oxford, 1958), p. 88.
13- D. Bohm, (1951). Quantum Theory. New York: Prentice Hall.
14- U. Klein, (2012). “What is the limit h → 0 of quantum theory? “
Am. .J. Phys. vol. 80, 1009
15- See for example: Tebbenjohans et al. (2021). “Quantum control of a nanoparticle optically levitated in cryogenic free space”. Nature, volume 595, pages378–382 (2021).
16- Landau and Lifshitz, Mecanique Quantique (1967, French translation, Mir)
“By measurement, we mean in quantum mechanics any process of interaction of a classical object and a quantum entity taking place, moreover, independently of any observer. ”
17- “We will not speculate on the way these intrinsic properties are wrapped up into the particle-bundle. One could consider the wave as an additional degree of freedom for the particle. It goes without doubt that the construction of a satisfactory model of the way the wave and the particle characteristics are intertwined is required in order for the theory of quantum mechanics to move forward towards a rational scientific model of Nature”.
18- Richard Feynman, The Feynman Lectures on Physics, (Addison wesley 1963), Volume III, Chapter I” Quantum Behavior”.
19- See for example:
Tonomura A., Endo J., Matsuda T., Kawasaki T., and Ezawa H. “Demonstration of single electron build-up of an interference pattern”, Am. J. Phys. 57: 117 (1989)
« Double-slit experiment: Quantum measurement »
https://www.hitachi.com/rd/research/materials/quantum/doubleslit/index.html
20- Quoted in: https://mappingignorance.org/2016/02/11/the-complementarity-principle
“The wave and particle models are both required for a complete description of matter and of electromagnetic radiation. Since these two models are mutually exclusive, they cannot be used simultaneously. Each experiment, or the experimenter who designs the experiment, selects one or the other description as the proper description for that experiment”.
21- “Young’s Interference Experiment with a single photon” (https://photonterrace.net/en/photon/duality/)
22- a- Paz, J.P. and W.H. Zurek (2000). “Environment-Induced Decoherence and the Transition from Quantum to classical”.
b- Erich Joos, (1999). “Elements of Environmental Decoherence”
c- W. H. Zurek (2002). “Decoherence and the Transition from Quantum to Classical—Revisited”
Los Alamos Science: November 2002, Number 27.
23- Here is a sample:
a- Shahriar S. Afshar (2005), “Violation of the principle of Complementarity, and its implications” Proceedings of SPIE Vol. 5866 SPIE, Bellingham, WA, 2005.
https://arxiv.org/ftp/quant-ph/papers/0701/0701027.pdf.
b- Greenberger DM, Yasin A. (1988) “Simultaneous wave and particle knowledge in a neutron interferometer”. Phys. Lett. A.;128: 391–394.
c- Mittelstaedt P, Prieur A, Schieder R. (1987) “Unsharp particle-wave duality in a photon split-beam experiment”. Found. Phys. 1987;17:891–903.
d- Kocsis, Sacha et al. (2011) « Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer [archive] ». Science 332, nᵒ 6034 (2011): 11791173.
24- R. Auccaise et.al (2012). ”Experimental analysis of the quantum complementarity principle ”. Phys. Rev. A 85, 032121
25- Superconductivity
https://en.wikipedia.org/wiki/Superconductivity
26- Superfluidity
https://en.wikipedia.org/wiki/Superfluidity
27- a- Kapitza, P. (1938). “Viscosity of liquid helium below the lamda-point.
Nature, 141 (3558).
b- Allen, J.F.; Misener, A.D. (1938). “Flow of liquid Helium II. Nature, 142 (3597).
28- A. Einstein, (1924). “Quantum Theory of a Monoatomic Ideal Gas”
English translation of “Quantentheorie des einatomigen idealen Gases”, published in
Königliche Preußische Akademie der Wissenschaften. Sitzungsberichte: (1924) 261–267
http://www.fisica.uns.edu.ar/albert/archivos/46/156/1787030330_apuntes.pdf
29- Bose-Einstein Condensate
https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate
30- Anderson, M. H. et al.. (1995). ”Observation of BEC in dilute atomic vapor” . Science. 269 (5221): 198–201.
31- F. Tebbenjohans et al. (2021). “Quantum control of a nanoparticle optically levitated in cryogenic free space”. Nature, volume 595, pages378–382 (2021).
32- Fermionic condensate – Wikipedia
https://simple.wikipedia.org/wiki/Fermionic_condensate
33- Richard Feynman, « Simulating physics with computers», International Journal of Theoretical Physics, vol. 21, 1982.
Feynman proposes to use a quantum elements obeying quantum mechanical laws capable of being adjusted to simulate any physical system you wish to study i.e. a universal quantum simulator. To be successful the system must as simple as possible.
34- Gibbs mixing paradox
When two gases of “different” nature are mixed, the (theoretical) total entropy increases (by ΔS = Rln2, for equal N and V). The increase in entropy is termed mixing entropy and is independent of the nature of the two gases. Gibbs noted that the same discontinuous increase is obtained even when two “identical” gases are mixed, contrary to experimental facts, in which case ΔS should be 0.
Gibbs suggested that for identical gases the particles are indistinguishable and the state of the gas remains unaffected by exchange of positions (permutations) between any two particles. For a gas of N identical particles the number of permutation is N!
He reduced the partition function Z by a factor of (N!), and injected it in the classical expression of the entropy, obtaining the corrected expression Scor:
Zred=Z/N!
Scor = kln(Z/N!) + E/T
This correction solved the mixing paradox, or at least this aspect of it. The discontinuous increase of S independently of the nature of the gases is of course “paradoxical” on its own. But this is another story.
35- https://particlemysteries.wordpress.com/2023/05/21/the-wave-particle-conundrum-iii/ the Quantum Mechanical model of the Universe
36- The thermodynamic potential μ is a parameter that characterizes thermodynamic equilibrium in open system (variable number of particles N) and chemical reactions.
It determines the contribution of Gibbs energy dG to the internal energy variation dU of the system, resulting from the exchange of particles, where dG = μdN.
37- For the detailed mathematical model, see for example:
a- Matthew Schwartz, “Lecture 12: Bose-Einstein Condensation”. Spring 2019.
Statistical Mechanics
b- “Bose-Einstein Condensation of An Ideal Gas” & 3.2.
38- Z. Korichi and M. T. Meftah, (2015). “Application of Fractional Classical and Quantum Mechanics to Statistical Physics System”
Physical Science International Journal, 8(1): 1-13, 2015, Article no.PSIJ.20366
The God particle and other mysteries 