Matter wave

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Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave. In most cases, however, the wavelength is too small to have a practical impact on day-to-day activities. Hence in our day-to-day lives with objects the size of tennis balls and people, matter waves are not relevant.

The concept that matter behaves like a wave was proposed by Louis de Broglie (/dəˈbrɔɪ/) in 1924. It is also referred to as the de Broglie hypothesis. Matter waves are referred to as de Broglie waves.

The de Broglie wavelength is the wavelength, λ, associated with a massive particle (i.e., a particle with mass, as opposed to a massless particle) and is related to its momentum, p, through the Planck constant, h:

${\displaystyle \lambda ={\frac {h}{p}}={\frac {h}{mv}}.}$

Wave-like behavior of matter was first experimentally demonstrated by George Paget Thomson's thin metal diffraction experiment, and independently in the Davisson–Germer experiment both using electrons, and it has also been confirmed for other elementary particles, neutral atoms and even molecules.

Historical context

At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to Maxwell's equations, while matter was thought to consist of localized particles (see history of wave and particle duality). In 1900, this division was exposed to doubt, when, investigating the theory of black-body radiation, Max Planck proposed that light is emitted in discrete quanta of energy. It was thoroughly challenged in 1905. Extending Planck's investigation in several ways, including its connection with the photoelectric effect, Albert Einstein proposed that light is also propagated and absorbed in quanta; now called photons. These quanta would have an energy given by the Planck–Einstein relation:

${\displaystyle E=h\nu }$

and a momentum

${\displaystyle p={\frac {E}{c}}={\frac {h}{\lambda }}}$

where ν (lowercase Greek letter nu) and λ (lowercase Greek letter lambda) denote the frequency and wavelength of the light, c the speed of light, and h the Planck constant.^[3] In the modern convention, frequency is symbolized by f as is done in the rest of this article. Einstein's postulate was confirmed experimentally by Robert Millikan and Arthur Compton over the next two decades.

de Broglie hypothesis

Propagation of de Broglie waves in 1d – real part of the complex amplitude is blue, imaginary part is green. The probability (shown as the color opacity) of finding the particle at a given point x is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the slope decreases, so the amplitude decreases again, and vice versa. The result is an alternating amplitude: a wave. Top: plane wave. Bottom: wave packet.

De Broglie, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties, electrons also have wave-like properties. By rearranging the momentum equation stated in the above section, we find a relationship between the wavelength, λ, associated with an electron and its momentum, p, through the Planck constant, h:^[4]

${\displaystyle \lambda ={\frac {h}{p}}.}$

The relationship is now known to hold for all types of matter: all matter exhibits properties of both particles and waves.

When I conceived the first basic ideas of wave mechanics in 1923–1924, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that Einstein had introduced for photons in his theory of light quanta in 1905.

— de Broglie^[5]

In 1926, Erwin Schrödinger published an equation describing how a matter wave should evolve – the matter wave analogue of Maxwell's equations — and used it to derive the energy spectrum of hydrogen.

Experimental confirmation

Demonstration of a matter wave in diffraction of electrons

Matter waves were first experimentally confirmed to occur in George Paget Thomson's cathode ray diffraction experiment^[2] and the Davisson-Germer experiment for electrons, and the de Broglie hypothesis has been confirmed for other elementary particles. Furthermore, neutral atoms and even molecules have been shown to be wave-like.

Electrons

Further information: Davisson–Germer experiment and Electron diffraction

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The angular dependence of the diffracted electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for x-rays. At the same time George Paget Thomson at the University of Aberdeen was independently firing electrons at very thin metal foils to demonstrate the same effect. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be exhibited only by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.

This was a pivotal result in the development of quantum mechanics. Just as the photoelectric effect demonstrated the particle nature of light, the Davisson–Germer experiment showed the wave-nature of matter, and completed the theory of wave–particle duality. For physicists this idea was important because it meant that not only could any particle exhibit wave characteristics, but that one could use wave equations to describe phenomena in matter if one used the de Broglie wavelength.

Neutral atoms

Experiments with Fresnel diffraction and an atomic mirror for specular reflection of neutral atoms confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and allow quantum reflection by the tails of the attractive potential. Advances in laser cooling have allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the thermal de Broglie wavelengths come into the micrometre range. Using Bragg diffraction of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold sodium atoms was explicitly measured and found to be consistent with the temperature measured by a different method.

This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution. The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.

The effect has also been used to explain the spatial version of the quantum Zeno effect, in which an otherwise unstable object may be stabilised by rapidly repeated observations.

Molecules

Recent experiments even confirm the relations for molecules and even macromolecules that otherwise might be supposed too large to undergo quantum mechanical effects. In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes. The researchers calculated a De Broglie wavelength of the most probable C₆₀ velocity as 2.5 pm. More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of 10,123 amu. As of 2019, this has been pushed to molecules of 25,000 amu.

Still one step further than Louis de Broglie go theories which in quantum mechanics eliminate the concept of a pointlike classical particle and explain the observed facts by means of wavepackets of matter waves alone.

de Broglie relations

The de Broglie equations relate the wavelength λ to the momentum p, and frequency f to the total energy E of a free particle:

${\displaystyle {\begin{aligned}&\lambda =h/p\\&f=E/h\end{aligned}}}$

where h is the Planck constant. The equations can also be written as

${\displaystyle {\begin{aligned}&\mathbf {p} =\hbar \mathbf {k} \\&E=\hbar \omega \\\end{aligned}}}$

or 

${\displaystyle {\begin{aligned}&\mathbf {p} =\hbar \mathbf {\beta } \\&E=\hbar \omega \\\end{aligned}}}$

where ħ = h/2π is the reduced Planck constant, k is the wave vector, β is the phase constant, and ω is the angular frequency.

In each pair, the second equation is also referred to as the Planck–Einstein relation, since it was also proposed by Planck and Einstein.

Special relativity

Using two formulas from special relativity, one for the relativistic mass energy and one for the relativistic momentum

${\displaystyle E=mc^{2}=\gamma m_{0}c^{2}}$

${\displaystyle {\vec {p}}=m{\vec {v}}=\gamma m_{0}{\vec {v}}}$

allows the equations to be written as

${\displaystyle {\begin{aligned}&\lambda =\,\,{\frac {h}{\gamma m_{0}v}}\,=\,{\frac {h}{m_{0}v}}\,\,\,\,{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\\&f={\frac {\gamma \,m_{0}c^{2}}{h}}={\frac {m_{0}c^{2}}{h}}{\bigg /}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}}$

where ${\displaystyle m_{0}}$ denotes the particle's rest mass, ${\displaystyle v}$ its velocity, ${\displaystyle \gamma }$ the Lorentz factor, and ${\displaystyle c}$ the speed of light in a vacuum. See below for details of the derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with phase velocity (equal to the product of the particle's frequency and its wavelength). In the case of a non-dispersive medium, they happen to be equal, but otherwise they are not.

Group velocity

Albert Einstein first explained the wave–particle duality of light in 1905. Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded, should always equal the group velocity of the corresponding wave. The magnitude of the group velocity is equal to the particle's speed.

Both in relativistic and non-relativistic quantum physics, we can identify the group velocity of a particle's wave function with the particle velocity. Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules.

De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that

${\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}={\frac {\partial (E/\hbar )}{\partial (p/\hbar )}}={\frac {\partial E}{\partial p}}}$

where E is the total energy of the particle, p is its momentum, ħ is the reduced Planck constant. For a free non-relativistic particle it follows that

${\displaystyle {\begin{aligned}v_{g}&={\frac {\partial E}{\partial p}}={\frac {\partial }{\partial p}}\left({\frac {1}{2}}{\frac {p^{2}}{m}}\right)\\&={\frac {p}{m}}\\&=v\end{aligned}}}$

where m is the mass of the particle and v its velocity.

Also in special relativity we find that

${\displaystyle {\begin{aligned}v_{g}&={\frac {\partial E}{\partial p}}={\frac {\partial }{\partial p}}\left({\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4}}}\right)\\&={\frac {pc^{2}}{\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4}}}}\\&={\frac {pc^{2}}{E}}\end{aligned}}}$

where m₀ is the rest mass of the particle and c is the speed of light in a vacuum. But (see below), using that the phase velocity is v_p = E/p = c²/v, therefore

${\displaystyle {\begin{aligned}v_{g}&={\frac {pc^{2}}{E}}\\&={\frac {c^{2}}{v_{p}}}\\&=v\end{aligned}}}$

where v is the velocity of the particle regardless of wave behavior.

Phase velocity

In quantum mechanics, particles also behave as waves with complex phases. The phase velocity is equal to the product of the frequency multiplied by the wavelength.

By the de Broglie hypothesis, we see that

${\displaystyle v_{\mathrm {p} }={\frac {\omega }{k}}={\frac {E/\hbar }{p/\hbar }}={\frac {E}{p}}.}$

Using relativistic relations for energy and momentum, we have

${\displaystyle v_{\mathrm {p} }={\frac {E}{p}}={\frac {mc^{2}}{mv}}={\frac {\gamma m_{0}c^{2}}{\gamma m_{0}v}}={\frac {c^{2}}{v}}={\frac {c}{\beta }}}$

where E is the total energy of the particle (i.e. rest energy plus kinetic energy in the kinematic sense), p the momentum, ${\displaystyle \gamma }$ the Lorentz factor, c the speed of light, and β the speed as a fraction of c. The variable v can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave. Since the particle speed ${\displaystyle v<c}$ for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.

${\displaystyle v_{\mathrm {p} }>c,\,}$

and as we can see, it approaches c when the particle speed is in the relativistic range. The superluminal phase velocity does not violate special relativity, because phase propagation carries no energy. See the article on Dispersion (optics) for details.

Four-vectors

Using four-vectors, the De Broglie relations form a single equation:

${\displaystyle \mathbf {P} =\hbar \mathbf {K} }$

which is frame-independent.

Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by:

${\displaystyle \mathbf {K} =\left({\frac {\omega _{o}}{c^{2}}}\right)\mathbf {U} }$

where

Four-momentum ${\displaystyle \mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)}$

Four-wavevector ${\displaystyle \mathbf {K} =\left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)=\left({\frac {\omega }{c}},{\frac {\omega }{v_{p}}}\mathbf {\hat {n}} \right)}$

Four-velocity ${\displaystyle \mathbf {U} =\gamma (c,{\vec {\mathbf {u} }})=\gamma (c,v_{g}{\hat {\mathbf {n} }})}$

Interpretations

The physical reality underlying de Broglie waves is a subject of ongoing debate. Some theories treat either the particle or the wave aspect as its fundamental nature, seeking to explain the other as an emergent property. Some, such as the hidden variable theory, treat the wave and the particle as distinct entities. Yet others propose some intermediate entity that is neither quite wave nor quite particle but only appears as such when we measure one or the other property. The Copenhagen interpretation states that the nature of the underlying reality is unknowable and beyond the bounds of scientific inquiry.

Schrödinger's quantum mechanical waves are conceptually different from ordinary physical waves such as water or sound. Ordinary physical waves are characterized by undulating real-number 'displacements' of dimensioned physical variables at each point of ordinary physical space at each instant of time. Schrödinger's "waves" are characterized by the undulating value of a dimensionless complex number at each point of an abstract multi-dimensional space, for example of configuration space.

At the Fifth Solvay Conference in 1927, Max Born and Werner Heisenberg reported as follows:

If one wishes to calculate the probabilities of excitation and ionization of atoms [M. Born, Zur Quantenmechanik der Stossvorgange, Z. f. Phys., 37 (1926), 863; [Quantenmechanik der Stossvorgange], ibid., 38 (1926), 803] then one must introduce the coordinates of the atomic electrons as variables on an equal footing with those of the colliding electron. The waves then propagate no longer in three-dimensional space but in multi-dimensional configuration space. From this one sees that the quantum mechanical waves are indeed something quite different from the light waves of the classical theory.

At the same conference, Erwin Schrödinger reported likewise.

Under [the name 'wave mechanics',] at present two theories are being carried on, which are indeed closely related but not identical. The first, which follows on directly from the famous doctoral thesis by L. de Broglie, concerns waves in three-dimensional space. Because of the strictly relativistic treatment that is adopted in this version from the outset, we shall refer to it as the four-dimensional wave mechanics. The other theory is more remote from Mr de Broglie's original ideas, insofar as it is based on a wave-like process in the space of position coordinates (q-space) of an arbitrary mechanical system.[Long footnote about manuscript not copied here.] We shall therefore call it the multi-dimensional wave mechanics. Of course this use of the q-space is to be seen only as a mathematical tool, as it is often applied also in the old mechanics; ultimately, in this version also, the process to be described is one in space and time. In truth, however, a complete unification of the two conceptions has not yet been achieved. Anything over and above the motion of a single electron could be treated so far only in the multi-dimensional version; also, this is the one that provides the mathematical solution to the problems posed by the Heisenberg-Born matrix mechanics.

In 1955, Heisenberg reiterated this:

An important step forward was made by the work of Born [Z. Phys., 37: 863, 1926 and 38: 803, 1926] in the summer of 1926. In this work, the wave in configuration space was interpreted as a probability wave, in order to explain collision processes on Schrödinger's theory. This hypothesis contained two important new features in comparison with that of Bohr, Kramers and Slater. The first of these was the assertion that, in considering "probability waves", we are concerned with processes not in ordinary three-dimensional space, but in an abstract configuration space (a fact which is, unfortunately, sometimes overlooked even today); the second was the recognition that the probability wave is related to an individual process.

It is mentioned above that the "displaced quantity" of the Schrödinger wave has values that are dimensionless complex numbers. One may ask what is the physical meaning of those numbers. According to Heisenberg, rather than being of some ordinary physical quantity such as, for example, Maxwell's electric field intensity, or mass density, the Schrödinger-wave packet's "displaced quantity" is probability amplitude. He wrote that instead of using the term 'wave packet', it is preferable to speak of a probability packet. The probability amplitude supports calculation of probability of location or momentum of discrete particles. Heisenberg recites Duane's account of particle diffraction by probabilistic quantal translation momentum transfer, which allows, for example in Young's two-slit experiment, each diffracted particle probabilistically to pass discretely through a particular slit. Thus one does not need necessarily think of the matter wave, as it were, as 'composed of smeared matter'.

These ideas may be expressed in ordinary language as follows. In the account of ordinary physical waves, a 'point' refers to a position in ordinary physical space at an instant of time, at which there is specified a 'displacement' of some physical quantity. But in the account of quantum mechanics, a 'point' refers to a configuration of the system at an instant of time, every particle of the system being in a sense present in every 'point' of configuration space, each particle at such a 'point' being located possibly at a different position in ordinary physical space. There is no explicit definite indication that, at an instant, this particle is 'here' and that particle is 'there' in some separate 'location' in configuration space. This conceptual difference entails that, in contrast to de Broglie's pre-quantum mechanical wave description, the quantum mechanical probability packet description does not directly and explicitly express the Aristotelian idea, referred to by Newton, that causal efficacy propagates through ordinary space by contact, nor the Einsteinian idea that such propagation is no faster than light. In contrast, these ideas are so expressed in the classical wave account, through the Green's function, though it is inadequate for the observed quantal phenomena. The physical reasoning for this was first recognized by Einstein.

De Broglie's phase wave and periodic phenomenon

De Broglie's thesis started from the hypothesis, “that to each portion of energy with a proper mass m₀ one may associate a periodic phenomenon of the frequency ν₀, such that one finds: hν₀ = m₀c². The frequency ν₀ is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory.”  (This frequency is also known as Compton frequency.)

De Broglie followed his initial hypothesis of a periodic phenomenon, with frequency ν₀ , associated with the energy packet. He used the special theory of relativity to find, in the frame of the observer of the electron energy packet that is moving with velocity ${\displaystyle v}$, that its frequency was apparently reduced to

${\displaystyle \nu _{1}=\nu _{0}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\,.}$

De Broglie reasoned that to a stationary observer this hypothetical intrinsic particle periodic phenomenon appears to be in phase with a wave of wavelength ${\displaystyle \lambda }$ and frequency ${\displaystyle f}$ that is propagating with phase velocity ${\displaystyle v_{\mathrm {p} }}$. De Broglie called this wave the “phase wave” («onde de phase» in French). This was his basic matter wave conception. He noted, as above, that ${\displaystyle v_{\mathrm {p} }>c}$, and the phase wave does not transfer energy.

While the concept of waves being associated with matter is correct, de Broglie did not leap directly to the final understanding of quantum mechanics with no missteps. There are conceptual problems with the approach that de Broglie took in his thesis that he was not able to resolve, despite trying a number of different fundamental hypotheses in different papers published while working on, and shortly after publishing, his thesis. These difficulties were resolved by Erwin Schrödinger, who developed the wave mechanics approach, starting from a somewhat different basic hypothesis.

Degenerate matter

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Degenerate_matter 

Degenerate matter is a highly dense state of fermionic matter in which particles must occupy high states of kinetic energy to satisfy the Pauli exclusion principle. The description applies to matter composed of electrons, protons, neutrons or other fermions. The term is mainly used in astrophysics to refer to dense stellar objects where gravitational pressure is so extreme that quantum mechanical effects are significant. This type of matter is naturally found in stars in their final evolutionary states, such as white dwarfs and neutron stars, where thermal pressure alone is not enough to avoid gravitational collapse.

Degenerate matter is usually modelled as an ideal Fermi gas, an ensemble of non-interacting fermions. In a quantum mechanical description, particles limited to a finite volume may take only a discrete set of energies, called quantum states. The Pauli exclusion principle prevents identical fermions from occupying the same quantum state. At lowest total energy (when the thermal energy of the particles is negligible), all the lowest energy quantum states are filled. This state is referred to as full degeneracy. This degeneracy pressure remains non-zero even at absolute zero temperature. Adding particles or reducing the volume forces the particles into higher-energy quantum states. In this situation, a compression force is required, and is made manifest as a resisting pressure. The key feature is that this degeneracy pressure does not depend on the temperature but only on the density of the fermions. Degeneracy pressure keeps dense stars in equilibrium, independent of the thermal structure of the star.

A degenerate mass whose fermions have velocities close to the speed of light (particle energy larger than its rest mass energy) is called relativistic degenerate matter.

The concept of degenerate stars, stellar objects composed of degenerate matter, was originally developed in a joint effort between Arthur Eddington, Ralph Fowler and Arthur Milne. Eddington had suggested that the atoms in Sirius B were almost completely ionised and closely packed. Fowler described white dwarfs as composed of a gas of particles that became degenerate at low temperature. Milne proposed that degenerate matter is found in most of the nuclei of stars, not only in compact stars.

Concept

If a plasma is cooled and under increasing pressure, it will eventually not be possible to compress the plasma any further. This constraint is due to the Pauli exclusion principle, which states that two fermions cannot share the same quantum state. When in this highly compressed state, since there is no extra space for any particles, a particle's location is extremely defined. Since the locations of the particles of a highly compressed plasma have very low uncertainty, their momentum is extremely uncertain. The Heisenberg uncertainty principle states

${\displaystyle \Delta p\Delta x\geq {\frac {\hbar }{2}}}$,

where Δp is the uncertainty in the particle's momentum and Δx is the uncertainty in position (and ħ is the reduced Planck constant). Therefore, even though the plasma is cold, such particles must on average be moving very fast. Large kinetic energies lead to the conclusion that, in order to compress an object into a very small space, tremendous force is required to control its particles' momentum.

Unlike a classical ideal gas, whose pressure is proportional to its temperature

${\displaystyle P=k_{\rm {B}}{\frac {NT}{V}}}$,

where P is pressure, k_B is Boltzmann's constant, N is the number of particles—typically atoms or molecules—, T is temperature, and V is the volume, the pressure exerted by degenerate matter depends only weakly on its temperature. In particular, the pressure remains nonzero even at absolute zero temperature. At relatively low densities, the pressure of a fully degenerate gas can be derived by treating the system as an ideal Fermi gas, in this way

${\displaystyle P={\frac {(3\pi ^{2})^{2/3}\hbar ^{2}}{5m}}\left({\frac {N}{V}}\right)^{5/3}}$,

where m is the mass of the individual particles making up the gas. At very high densities, where most of the particles are forced into quantum states with relativistic energies, the pressure is given by

${\displaystyle P=K\left({\frac {N}{V}}\right)^{4/3}}$,

where K is another proportionality constant depending on the properties of the particles making up the gas.

Pressure vs temperature curves of classical and quantum ideal gases (Fermi gas, Bose gas) in three dimensions.

All matter experiences both normal thermal pressure and degeneracy pressure, but in commonly encountered gases, thermal pressure dominates so much that degeneracy pressure can be ignored. Likewise, degenerate matter still has normal thermal pressure, the degeneracy pressure dominates to the point that temperature has a negligible effect on the total pressure. The adjacent figure shows how the pressure of a Fermi gas saturates as it is cooled down, relative to a classical ideal gas.

While degeneracy pressure usually dominates at extremely high densities, it is the ratio between degenerate pressure and thermal pressure which determines degeneracy. Given a sufficiently drastic increase in temperature (such as during a red giant star's helium flash), matter can become non-degenerate without reducing its density.

Degeneracy pressure contributes to the pressure of conventional solids, but these are not usually considered to be degenerate matter because a significant contribution to their pressure is provided by electrical repulsion of atomic nuclei and the screening of nuclei from each other by electrons. The free electron model of metals derives their physical properties by considering the conduction electrons alone as a degenerate gas, while the majority of the electrons are regarded as occupying bound quantum states. This solid state contrasts with degenerate matter that forms the body of a white dwarf, where most of the electrons would be treated as occupying free particle momentum states.

Exotic examples of degenerate matter include neutron degenerate matter, strange matter, metallic hydrogen and white dwarf matter.

Degenerate gases

Degenerate gases are gases composed of fermions such as electrons, protons, and neutrons rather than molecules of ordinary matter. The electron gas in ordinary metals and in the interior of white dwarfs are two examples. Following the Pauli exclusion principle, there can be only one fermion occupying each quantum state. In a degenerate gas, all quantum states are filled up to the Fermi energy. Most stars are supported against their own gravitation by normal thermal gas pressure, while in white dwarf stars the supporting force comes from the degeneracy pressure of the electron gas in their interior. In neutron stars, the degenerate particles are neutrons.

A fermion gas in which all quantum states below a given energy level are filled is called a fully degenerate fermion gas. The difference between this energy level and the lowest energy level is known as the Fermi energy.

Electron degeneracy

In an ordinary fermion gas in which thermal effects dominate, most of the available electron energy levels are unfilled and the electrons are free to move to these states. As particle density is increased, electrons progressively fill the lower energy states and additional electrons are forced to occupy states of higher energy even at low temperatures. Degenerate gases strongly resist further compression because the electrons cannot move to already filled lower energy levels due to the Pauli exclusion principle. Since electrons cannot give up energy by moving to lower energy states, no thermal energy can be extracted. The momentum of the fermions in the fermion gas nevertheless generates pressure, termed "degeneracy pressure".

Under high densities the matter becomes a degenerate gas when the electrons are all stripped from their parent atoms. In the core of a star, once hydrogen burning in nuclear fusion reactions stops, it becomes a collection of positively charged ions, largely helium and carbon nuclei, floating in a sea of electrons, which have been stripped from the nuclei. Degenerate gas is an almost perfect conductor of heat and does not obey the ordinary gas laws. White dwarfs are luminous not because they are generating any energy but rather because they have trapped a large amount of heat which is gradually radiated away. Normal gas exerts higher pressure when it is heated and expands, but the pressure in a degenerate gas does not depend on the temperature. When gas becomes super-compressed, particles position right up against each other to produce degenerate gas that behaves more like a solid. In degenerate gases the kinetic energies of electrons are quite high and the rate of collision between electrons and other particles is quite low, therefore degenerate electrons can travel great distances at velocities that approach the speed of light. Instead of temperature, the pressure in a degenerate gas depends only on the speed of the degenerate particles; however, adding heat does not increase the speed of most of the electrons, because they are stuck in fully occupied quantum states. Pressure is increased only by the mass of the particles, which increases the gravitational force pulling the particles closer together. Therefore, the phenomenon is the opposite of that normally found in matter where if the mass of the matter is increased, the object becomes bigger. In degenerate gas, when the mass is increased, the particles become spaced closer together due to gravity (and the pressure is increased), so the object becomes smaller. Degenerate gas can be compressed to very high densities, typical values being in the range of 10,000 kilograms per cubic centimeter.

There is an upper limit to the mass of an electron-degenerate object, the Chandrasekhar limit, beyond which electron degeneracy pressure cannot support the object against collapse. The limit is approximately 1.44 solar masses for objects with typical compositions expected for white dwarf stars (carbon and oxygen with two baryons per electron). This mass cutoff is appropriate only for a star supported by ideal electron degeneracy pressure under Newtonian gravity; in general relativity and with realistic Coulomb corrections, the corresponding mass limit is around 1.38 solar masses. The limit may also change with the chemical composition of the object, as it affects the ratio of mass to number of electrons present. The object's rotation, which counteracts the gravitational force, also changes the limit for any particular object. Celestial objects below this limit are white dwarf stars, formed by the gradual shrinking of the cores of stars that run out of fuel. During this shrinking, an electron-degenerate gas forms in the core, providing sufficient degeneracy pressure as it is compressed to resist further collapse. Above this mass limit, a neutron star (partially supported by neutron degeneracy pressure) or a black hole may be formed instead.

Neutron degeneracy

Main article: neutron star

Neutron degeneracy is analogous to electron degeneracy and is demonstrated in neutron stars, which are partially supported by the pressure from a degenerate neutron gas. The collapse may happen when the core of a white dwarf exceeds approximately 1.4 solar masses, which is the Chandrasekhar limit, above which the collapse is not halted by the pressure of degenerate electrons. As the star collapses, the Fermi energy of the electrons increases to the point where it is energetically favorable for them to combine with protons to produce neutrons (via inverse beta decay, also termed electron capture and "neutronization"). The result is an extremely compact star composed of nuclear matter, which is predominantly a degenerate neutron gas, sometimes called neutronium, with a small admixture of degenerate proton and electron gases (and at higher densities, muons).

Neutrons in a degenerate neutron gas are spaced much more closely than electrons in an electron-degenerate gas because the more massive neutron has a much shorter wavelength at a given energy. Typical separations are comparable with the size of the neutron and the range of the strong nuclear force, and it is actually the repulsive nature of the latter at small separations that primarily supports neutron stars more massive than 0.7 solar masses (which includes all measured neutron stars). In the case of neutron stars and white dwarfs, this phenomenon is compounded by the fact that the pressures within neutron stars are much higher than those in white dwarfs. The pressure increase is caused by the fact that the compactness of a neutron star causes gravitational forces to be much higher than in a less compact body with similar mass. The result is a star with a diameter on the order of a thousandth that of a white dwarf.

There is an upper limit to the mass of a neutron-degenerate object, the Tolman–Oppenheimer–Volkoff limit, which is analogous to the Chandrasekhar limit for electron-degenerate objects. The limit for objects supported by ideal neutron degeneracy pressure is only 0.75 solar masses.^[10] For more realistic models including baryon interaction, the precise limit is unknown, as it depends on the equations of state of nuclear matter, for which a highly accurate model is not yet available. Above this limit, a neutron star may collapse into a black hole or into other, denser forms of degenerate matter.^[a]

Proton degeneracy

Sufficiently dense matter containing protons experiences proton degeneracy pressure, in a manner similar to the electron degeneracy pressure in electron-degenerate matter: protons confined to a sufficiently small volume have a large uncertainty in their momentum due to the Heisenberg uncertainty principle. However, because protons are much more massive than electrons, the same momentum represents a much smaller velocity for protons than for electrons. As a result, in matter with approximately equal numbers of protons and electrons, proton degeneracy pressure is much smaller than electron degeneracy pressure, and proton degeneracy is usually modeled as a correction to the equations of state of electron-degenerate matter.

Quark degeneracy

At densities greater than those supported by neutron degeneracy, quark matter is expected to occur. Several variations of this hypothesis have been proposed that represent quark-degenerate states. Strange matter is a degenerate gas of quarks that is often assumed to contain strange quarks in addition to the usual up and down quarks. Color superconductor materials are degenerate gases of quarks in which quarks pair up in a manner similar to Cooper pairing in electrical superconductors. The equations of state for the various proposed forms of quark-degenerate matter vary widely, and are usually also poorly defined, due to the difficulty of modeling strong force interactions.

Quark-degenerate matter may occur in the cores of neutron stars, depending on the equations of state of neutron-degenerate matter. It may also occur in hypothetical quark stars, formed by the collapse of objects above the Tolman–Oppenheimer–Volkoff mass limit for neutron-degenerate objects. Whether quark-degenerate matter forms at all in these situations depends on the equations of state of both neutron-degenerate matter and quark-degenerate matter, both of which are poorly known. Quark stars are considered to be an intermediate category between neutron stars and black holes.