Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0013
I. Kenyon
Space-time symmetries, conservation laws and Nöther’s theorem are discussed. The Poincaré group, generators and Casimir invariants are outlined. Local charge conservation and the corresponding U(1) charge symmetry underlying electromagnetism are presented, showing the roles of minimal electromagnetic coupling and gauge transformations. Experimental demonstrations of the Aharonov–Bohm effect are described and the topological interpretation is recounted. How the Aharonov–Casher effect survives in the classical world is mentioned. Berry’s revelation of geometric phase is presented. The Bitter–Dubbers experiment confirming this analysis is presented. Some comments are given on a Hilbert space with a simple topology.
{"title":"Symmetry and topology","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0013","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0013","url":null,"abstract":"Space-time symmetries, conservation laws and Nöther’s theorem are discussed. The Poincaré group, generators and Casimir invariants are outlined. Local charge conservation and the corresponding U(1) charge symmetry underlying electromagnetism are presented, showing the roles of minimal electromagnetic coupling and gauge transformations. Experimental demonstrations of the Aharonov–Bohm effect are described and the topological interpretation is recounted. How the Aharonov–Casher effect survives in the classical world is mentioned. Berry’s revelation of geometric phase is presented. The Bitter–Dubbers experiment confirming this analysis is presented. Some comments are given on a Hilbert space with a simple topology.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"222 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115736579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0010
I. Kenyon
EPR showed that quantum mechanics is not a local deterministic theory and on this account they argued that it is incomplete. Quantum mechanics predicts correlations over time-like separations. The suggested resolution in terms of local hidden variables is presented. Bell’s analysis leading to experimental tests is described. The experiment of Aspect, Grangier and Roger vindicating quantum mechanics is described. More refined experiments, avoiding conceivable biases, confirm this result. Then computing based on quantum principles is discussed. Bits with two states in a register would be replaced by qubits with values represented by points on the Bloch sphere. Basic gates are presented. Shor’s algorithm to decompose products of primes is described and a gate structure presented to implement it. Implementation would undermine current encryption methods. Quantum cryptography is described using the BB84 protocol. The no-cloning theorem protects this absolutely against attempts to intercept the encryption data.
{"title":"EPR and Bell’s theorem, and quantum algorithms","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0010","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0010","url":null,"abstract":"EPR showed that quantum mechanics is not a local deterministic theory and on this account they argued that it is incomplete. Quantum mechanics predicts correlations over time-like separations. The suggested resolution in terms of local hidden variables is presented. Bell’s analysis leading to experimental tests is described. The experiment of Aspect, Grangier and Roger vindicating quantum mechanics is described. More refined experiments, avoiding conceivable biases, confirm this result. Then computing based on quantum principles is discussed. Bits with two states in a register would be replaced by qubits with values represented by points on the Bloch sphere. Basic gates are presented. Shor’s algorithm to decompose products of primes is described and a gate structure presented to implement it. Implementation would undermine current encryption methods. Quantum cryptography is described using the BB84 protocol. The no-cloning theorem protects this absolutely against attempts to intercept the encryption data.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129203432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0009
I. R. Kenyon
The distiction between classical product states and quantum entangled states is disclosed with examples. Spontaneous parametric down conversion as a source of entangled photons is described. The action of a perfect beam splitter is analysed using creation and annihilation operators. The HOM interferometer is described. Its use in demonstrating the indistinguishability of photons and in measuring bandwidth of sources at the level of femtoseconds is recounted. Two particle entanglement is analysed using the Bloch sphere representation showing how the full knowledge of the entangled state does not fix the state of the individual particles. The four Bell states, eigenstates of two particle entanglement, are introduced. Teleportation of a photon state using entangled photons is described, and an experiment to entangle the quantum states of atoms at space-like separation outlined.
{"title":"Entanglement","authors":"I. R. Kenyon","doi":"10.1093/oso/9780198808350.003.0009","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0009","url":null,"abstract":"The distiction between classical product states and quantum entangled states is disclosed with examples. Spontaneous parametric down conversion as a source of entangled photons is described. The action of a perfect beam splitter is analysed using creation and annihilation operators. The HOM interferometer is described. Its use in demonstrating the indistinguishability of photons and in measuring bandwidth of sources at the level of femtoseconds is recounted. Two particle entanglement is analysed using the Bloch sphere representation showing how the full knowledge of the entangled state does not fix the state of the individual particles. The four Bell states, eigenstates of two particle entanglement, are introduced. Teleportation of a photon state using entangled photons is described, and an experiment to entangle the quantum states of atoms at space-like separation outlined.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"11 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128460671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0017
I. Kenyon
It is explained how plateaux are seen in the Hall conductance of two dimensional electron gases, at cryogenic temperatures, when the magnetic field is scanned from zero to ~10T. On a Hall plateau σxy = ne 2/h, where n is integral, while the longitudinal conductance vanishes. This is the integral quantum Hall effect. Free electrons in such devices are shown to occupy quantized Landau levels, analogous to classical cyclotron orbits. The stability of the IQHE is shown to be associated with a mobility gap rather than an energy gap. The analysis showing the topological origin of the IQHE is reproduced. Next the fractional QHE is described: Laughlin’s explanation in terms of an IQHE of quasiparticles is presented. In the absence of any magnetic field, the quantum spin Hall effect is observed, and described here. Time reversal invariance and Kramer pairs are seen to be underlying requirements. It’s topological origin is outlined.
解释了在低温下，当磁场从0到~10T扫描时，二维电子气体的霍尔电导是如何出现平台的。在霍尔平台上σ _ (xy) = ne 2/h，其中n为积分，而纵向电导消失。这就是积分量子霍尔效应。这种装置中的自由电子被证明占据量子化朗道能级，类似于经典回旋加速器轨道。IQHE的稳定性被证明与迁移率间隙而不是能量间隙有关。分析显示的拓扑起源的IQHE是复制。接下来描述分数QHE:劳克林在准粒子的IQHE方面的解释被提出。在没有任何磁场的情况下，量子自旋霍尔效应被观察到，并在这里描述。时间反转不变性和克莱默对被认为是潜在的要求。概述了它的拓扑起源。
{"title":"Quantum Hall effects","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0017","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0017","url":null,"abstract":"It is explained how plateaux are seen in the Hall conductance of two dimensional electron gases, at cryogenic temperatures, when the magnetic field is scanned from zero to ~10T. On a Hall plateau σxy = ne\u0000 2/h, where n is integral, while the longitudinal conductance vanishes. This is the integral quantum Hall effect. Free electrons in such devices are shown to occupy quantized Landau levels, analogous to classical cyclotron orbits. The stability of the IQHE is shown to be associated with a mobility gap rather than an energy gap. The analysis showing the topological origin of the IQHE is reproduced. Next the fractional QHE is described: Laughlin’s explanation in terms of an IQHE of quasiparticles is presented. In the absence of any magnetic field, the quantum spin Hall effect is observed, and described here. Time reversal invariance and Kramer pairs are seen to be underlying requirements. It’s topological origin is outlined.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"81 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131544248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0014
I. Kenyon
The superfluid transition of 4He at 2.17K to He-II and the inference of an underlying condensate are introduced. The fountain effect is interpreted. Andronikashvili’s experiment and the determination of superfluid fraction versus temperature are discussed. Sound and second sound are described. Relationships between the condensate and superfluid fractions, and to off diagonal long-range order (ODLRO) are deduced. The revelation of topological quantization of circulation by Vinen’s experiment is recounted. Spontaneous symmetry breaking by the condensate’s phase coherence is explained. Excitations and their dispersion relations described with Landau’s interpretation, including the explanation of the critical velocity of superflow. Vortices, their interpretation in terms of quantized circulation, and their visualization are described.
{"title":"Superfluid 4He","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0014","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0014","url":null,"abstract":"The superfluid transition of 4He at 2.17K to He-II and the inference of an underlying condensate are introduced. The fountain effect is interpreted. Andronikashvili’s experiment and the determination of superfluid fraction versus temperature are discussed. Sound and second sound are described. Relationships between the condensate and superfluid fractions, and to off diagonal long-range order (ODLRO) are deduced. The revelation of topological quantization of circulation by Vinen’s experiment is recounted. Spontaneous symmetry breaking by the condensate’s phase coherence is explained. Excitations and their dispersion relations described with Landau’s interpretation, including the explanation of the critical velocity of superflow. Vortices, their interpretation in terms of quantized circulation, and their visualization are described.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133476375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0004
I. Kenyon
Phonons are introduced as an example of quasi-particles that can only exist in matter. Debye’s quantum model for heat capacity of solids and comparison with experimentin different temperature ranges is presented. The dispersion relations of lattice vibration (phonons) and quantization for chains of atoms presented, revealing the optical and acoustic modes; anharmonic effects are discussed. Crystal lattice structures and Brillouin zones are introduced. Phonon scattering and the Umklapp process described. The variation of the thermal conductivity of dielectrics with temperature is interpreted. X-ray scattering studies of phonon dispersion relations are described. Coupling between phonons with photons in polaritons is explained: Raman scattering studies of GaN used to exhibit the cross-over of their dispersion relations. The Mössbauer effect, a recoilless process, and its dependence on temperature are explained.
{"title":"Phonons","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0004","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0004","url":null,"abstract":"Phonons are introduced as an example of quasi-particles that can only exist in matter. Debye’s quantum model for heat capacity of solids and comparison with experimentin different temperature ranges is presented. The dispersion relations of lattice vibration (phonons) and quantization for chains of atoms presented, revealing the optical and acoustic modes; anharmonic effects are discussed. Crystal lattice structures and Brillouin zones are introduced. Phonon scattering and the Umklapp process described. The variation of the thermal conductivity of dielectrics with temperature is interpreted. X-ray scattering studies of phonon dispersion relations are described. Coupling between phonons with photons in polaritons is explained: Raman scattering studies of GaN used to exhibit the cross-over of their dispersion relations. The Mössbauer effect, a recoilless process, and its dependence on temperature are explained.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130167876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0008
I. Kenyon
Field or second quantization is carried through for electromagnetism, giving creation and annihilation operators for photons. Vacuum energy arises from field fluctuations, which causes the Casimir force and the Lamb shift of spectral lines. The connection between absorption, spontaneous emission and the stimulated emission of radiation is shown to emerge naturally. This yields Einstein’s equations for radiation in thermal equilibrium. The prerequisites for lasing, the operation and the properties of lasers are described. Fully coherent (Laser) states are expressed in terms of Fock states. The first and second order coherence of lasers and thermal sources are worked out. The Hanbury Brown and Twiss experiment is described and the application of the principle to determining stellar sizes and interaction regions in particle collisions from meson correlations are described.
{"title":"Field quantization","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0008","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0008","url":null,"abstract":"Field or second quantization is carried through for electromagnetism, giving creation and annihilation operators for photons. Vacuum energy arises from field fluctuations, which causes the Casimir force and the Lamb shift of spectral lines. The connection between absorption, spontaneous emission and the stimulated emission of radiation is shown to emerge naturally. This yields Einstein’s equations for radiation in thermal equilibrium. The prerequisites for lasing, the operation and the properties of lasers are described. Fully coherent (Laser) states are expressed in terms of Fock states. The first and second order coherence of lasers and thermal sources are worked out. The Hanbury Brown and Twiss experiment is described and the application of the principle to determining stellar sizes and interaction regions in particle collisions from meson correlations are described.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130197118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0019
I. Kenyon
Quantum chromodynamics the quantum gauge theory of strong interactions is presented: SU(3) being the (colour) symmetry group. The colour content of strongly interacting particles is described. Gluons, the field particles, carry colour so that they mutually interact – unlike photons. Renormalization leads to the coupling strength declining at large four momentum transfer squared q 2 and to binding of quarks in hadrons at small q 2. The cutoff in the range of the strong interaction is shown to be due to this low q 2 behaviour, despite the gluon being massless. In high energy interactions, say proton-proton collisions, the initial process is a hard (high q 2) parton+parton to parton+parton process. After which the partons undergo softer interactions leading finally to emergent hardrons. Experiments at DESY probing proton structure with electrons are described. An account of electroweak unification completes the book. The weak interaction symmetry group is SUL(2), L specifying handedness. This makes the electroweak symmetry U(1)⊗SUL(2). The weak force carriers, W± and Z0, are massive, which is at odds with the massless carriers required by quantum gauge theories. How the BEH mechanism resolves this problem is described. It involves spontaneous symmetry breaking of the vacuum with scalar fields. The outcome are massive gauge field particles to match the W± and Z0 trio, a massless photon, and a scalar field with a massive particle, the Higgs boson. The experimental programmes that discovered the vector bosons in 1983 and the Higgs in 2012 are described, including features of generic detectors. Finally puzzles revealed by our current understanding are outlined.
{"title":"Particle physics II","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0019","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0019","url":null,"abstract":"Quantum chromodynamics the quantum gauge theory of strong interactions is presented: SU(3) being the (colour) symmetry group. The colour content of strongly interacting particles is described. Gluons, the field particles, carry colour so that they mutually interact – unlike photons. Renormalization leads to the coupling strength declining at large four momentum transfer squared q\u0000 2 and to binding of quarks in hadrons at small q\u0000 2. The cutoff in the range of the strong interaction is shown to be due to this low q\u0000 2 behaviour, despite the gluon being massless. In high energy interactions, say proton-proton collisions, the initial process is a hard (high q\u0000 2) parton+parton to parton+parton process. After which the partons undergo softer interactions leading finally to emergent hardrons. Experiments at DESY probing proton structure with electrons are described. An account of electroweak unification completes the book. The weak interaction symmetry group is SUL(2), L specifying handedness. This makes the electroweak symmetry U(1)⊗SUL(2). The weak force carriers, W± and Z0, are massive, which is at odds with the massless carriers required by quantum gauge theories. How the BEH mechanism resolves this problem is described. It involves spontaneous symmetry breaking of the vacuum with scalar fields. The outcome are massive gauge field particles to match the W± and Z0 trio, a massless photon, and a scalar field with a massive particle, the Higgs boson. The experimental programmes that discovered the vector bosons in 1983 and the Higgs in 2012 are described, including features of generic detectors. Finally puzzles revealed by our current understanding are outlined.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"3 Dermatol Sect 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134430109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0007
I. Kenyon
A derivation of Fermi’s golden rule is given: this is the interface into which matrix elements from theory can be slotted to provide a prediction testable by experiment. The example of the prediction of the 2p→1s decay in hydrogen is worked through in detail. Selection rules, spectral line shapes (Breit–Wigner and Gaussian) and broadening processes are explained. The formula for the experimental cross-section in terms of the matrix element is produced. The Born approximation is presented and applied to Rutherford scattering. Then the decay rate for allowed β-decays is calculated in Fermi’s model and fitted to the observed rates. Low energy s-wave scattering is analysed in terms of phase shift and scattering length. The example of cold alkali metal atom scattering (≤10−6eV) is treated in preparation for use later with gaseous Bose–Einstein condensates. Ramsauer–Townsend effect explained.
{"title":"Transitions","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0007","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0007","url":null,"abstract":"A derivation of Fermi’s golden rule is given: this is the interface into which matrix elements from theory can be slotted to provide a prediction testable by experiment. The example of the prediction of the 2p→1s decay in hydrogen is worked through in detail. Selection rules, spectral line shapes (Breit–Wigner and Gaussian) and broadening processes are explained. The formula for the experimental cross-section in terms of the matrix element is produced. The Born approximation is presented and applied to Rutherford scattering. Then the decay rate for allowed β-decays is calculated in Fermi’s model and fitted to the observed rates. Low energy s-wave scattering is analysed in terms of phase shift and scattering length. The example of cold alkali metal atom scattering (≤10−6eV) is treated in preparation for use later with gaseous Bose–Einstein condensates. Ramsauer–Townsend effect explained.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133736715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-10DOI: 10.1093/oso/9780198808350.003.0003
I. Kenyon
Indistinguishability of like particles, and the fermion and boson exchange symmetries discussed.Pauli exclusion principle and features of multi-electron atoms, including selection rules are discussed. Degeneracy pressure and the formation of compact stellar objects is analysed. Quantum exchange force between electrons and its contribution to ferromagnetism is outlined. Fermi-Dirac and Bose-Einstein statistics, includng the chemical potential are derived. The conditions for Bose-Einstein condensation are deduced; condensates and their stability are considered.
{"title":"Quantum statistics","authors":"I. Kenyon","doi":"10.1093/oso/9780198808350.003.0003","DOIUrl":"https://doi.org/10.1093/oso/9780198808350.003.0003","url":null,"abstract":"Indistinguishability of like particles, and the fermion and boson exchange symmetries discussed.Pauli exclusion principle and features of multi-electron atoms, including selection rules are discussed. Degeneracy pressure and the formation of compact stellar objects is analysed. Quantum exchange force between electrons and its contribution to ferromagnetism is outlined. Fermi-Dirac and Bose-Einstein statistics, includng the chemical potential are derived. The conditions for Bose-Einstein condensation are deduced; condensates and their stability are considered.","PeriodicalId":165376,"journal":{"name":"Quantum 20/20","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129861941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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