Dynamics of a quasiparticle in the α-T₃ model: Role of pseudospin polarization and transverse magnetic field on zitterbewegung.

PubMed

Biswas, Tutul; Ghosh, Tarun Kanti

2018-01-09

We consider the $\\alpha$-$T_3$ model which provides a smooth crossover between the honeycomb lattice with pseudospin $1/2$ and the dice lattice with pseudospin $1$ through the variation of a parameter $\\alpha$. We study the dynamics of a wave packet representing a quasiparticle in the $\\alpha$-T$_3$ model with zero and finite transverse magnetic field. For zero field, it is shown that the wave packet undergoes a transient $zitterbewegung$ (ZB). Various features of ZB depending on the initial pseudospin polarization of the wave packet have been revealed. For an intermediate value of the parameter $\\alpha$ i.e. for $0<\\alpha<1$ the resulting ZB consists of two distinct frequencies when the wave packet was located initially in $rim$ site. However, the wave packet exhibits single frequency ZB for $\\alpha=0$ and $\\alpha=1$. It is also unveiled that the frequency of ZB corresponding to $\\alpha=1$ gets exactly half of that corresponding to the $\\alpha=0$ case. On the other hand, when the initial wave packet was in $hub$ site, the ZB consists of only one frequency for all values of $\\alpha$. Using stationary phase approximation we find analytical expression of velocity average which can be used to extract the associated timescale over which the transient nature of ZB persists. On the contrary the wave packet undergoes permanent ZB in presence of a transverse magnetic field. Due to the presence of large number of Landau energy levels the oscillations in ZB appear to be much more complicated. The oscillation pattern depends significantly on the initial pseudospin polarization of the wave packet. Furthermore, it is revealed that the number of the frequency components involved in ZB depends on the parameter $\\alpha$. © 2018 IOP Publishing Ltd.

Digital quantum simulation of Dirac equation with a trapped ion

NASA Astrophysics Data System (ADS)

Shen, Yangchao; Zhang, Xiang; Zhang, Junhua; Casanova, Jorge; Lamata, Lucas; Solano, Enrique; Yung, Man-Hong; Zhang, Jingning; Kim, Kihwan; Department Of Physical Chemistry Collaboration

2014-05-01

Recently there has been growing interest in simulating relativistic effects in controllable physical system. We digitally simulate the Dirac equation in 3 +1 dimensions with a single trapped ion. We map four internal levels of 171Yb+ ion to the Dirac bispinor. The time evolution of the Dirac equation is implemented by trotter expansion. In the 3 +1 dimension, we can observe a helicoidal motion of a free Dirac particle which reduces to Zitterbewegung in 1 +1 dimension. This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540. KK acknowledge the support from the recruitment program of global youth experts.

Dual Vector Spaces and Physical Singularities

NASA Astrophysics Data System (ADS)

Rowlands, Peter

Though we often refer to 3-D vector space as constructed from points, there is no mechanism from within its definition for doing this. In particular, space, on its own, cannot accommodate the singularities that we call fundamental particles. This requires a commutative combination of space as we know it with another 3-D vector space, which is dual to the first (in a physical sense). The combination of the two spaces generates a nilpotent quantum mechanics/quantum field theory, which incorporates exact supersymmetry and ultimately removes the anomalies due to self-interaction. Among the many natural consequences of the dual space formalism are half-integral spin for fermions, zitterbewegung, Berry phase and a zero norm Berwald-Moor metric for fermionic states.

Dirac Equation in (1 +1 )-Dimensional Curved Spacetime and the Multiphoton Quantum Rabi Model

NASA Astrophysics Data System (ADS)

Pedernales, J. S.; Beau, M.; Pittman, S. M.; Egusquiza, I. L.; Lamata, L.; Solano, E.; del Campo, A.

2018-04-01

We introduce an exact mapping between the Dirac equation in (1 +1 )-dimensional curved spacetime (DCS) and a multiphoton quantum Rabi model (QRM). A background of a (1 +1 )-dimensional black hole requires a QRM with one- and two-photon terms that can be implemented in a trapped ion for the quantum simulation of Dirac particles in curved spacetime. We illustrate our proposal with a numerical analysis of the free fall of a Dirac particle into a (1 +1 )-dimensional black hole, and find that the Zitterbewegung effect, measurable via the oscillatory trajectory of the Dirac particle, persists in the presence of gravity. From the duality between the squeezing term in the multiphoton QRM and the metric coupling in the DCS, we show that gravity generates squeezing of the Dirac particle wave function.

Simple and accurate sum rules for highly relativistic systems

NASA Astrophysics Data System (ADS)

Cohen, Scott M.

2005-03-01

In this paper, I consider the Bethe and Thomas-Reiche-Kuhn sum rules, which together form the foundation of Bethe's theory of energy loss from fast charged particles to matter. For nonrelativistic target systems, the use of closure leads directly to simple expressions for these quantities. In the case of relativistic systems, on the other hand, the calculation of sum rules is fraught with difficulties. Various perturbative approaches have been used over the years to obtain relativistic corrections, but these methods fail badly when the system in question is very strongly bound. Here, I present an approach that leads to relatively simple expressions yielding accurate sums, even for highly relativistic many-electron systems. I also offer an explanation for the difference between relativistic and nonrelativistic sum rules in terms of the Zitterbewegung of the electrons.

Position, spin, and orbital angular momentum of a relativistic electron

NASA Astrophysics Data System (ADS)

Bliokh, Konstantin Y.; Dennis, Mark R.; Nori, Franco

2017-08-01

Motivated by recent interest in relativistic electron vortex states, we revisit the spin and orbital angular momentum properties of Dirac electrons. These are uniquely determined by the choice of the position operator for a relativistic electron. We consider two main approaches discussed in the literature: (i) the projection of operators onto the positive-energy subspace, which removes the Zitterbewegung effects and correctly describes spin-orbit interaction effects, and (ii) the use of Newton-Wigner-Foldy-Wouthuysen operators based on the inverse Foldy-Wouthuysen transformation. We argue that the first approach [previously described in application to Dirac vortex beams in K. Y. Bliokh et al., Phys. Rev. Lett. 107, 174802 (2011), 10.1103/PhysRevLett.107.174802] has a more natural physical interpretation, including spin-orbit interactions and a nonsingular zero-mass limit, than the second one [S. M. Barnett, Phys. Rev. Lett. 118, 114802 (2017), 10.1103/PhysRevLett.118.114802].

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cirilo-Lombardo, Diego Julio; Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna

The central role played by pseudodifferential operators in relativistic dynamics is known very well. In this work, operators like the Schrodinger one (e.g., square root) are treated from the point of view of the non-local pseudodifferential Green functions. Starting from the explicit construction of the Green (semigroup) theoretical kernel, a theorem linking the integrability conditions and their dependence on the spacetime dimensions is given. Relativistic wave equations with arbitrary spin and the causality problem are discussed with the algebraic interpretation of the radical operator and their relation with coherent and squeezed states. Also we perform by means of pure theoreticalmore » procedures (based in physical concepts and symmetry) the relativistic position operator which satisfies the conditions of integrability: it is a non-local, Lorentz invariant and does not have the same problems as the “local”position operator proposed by Newton and Wigner. Physical examples, as zitterbewegung and rogue waves, are presented and deeply analyzed in this theoretical framework.« less

Gouy phase for relativistic quantum particles

NASA Astrophysics Data System (ADS)

Ducharme, R.; da Paz, I. G.

2015-08-01

Exact Hermite-Gaussian solutions to the Klein-Gordon equation for particle beams are obtained here that depend on the 4-position of the beam waist. These are Bateman-Hillion solutions that are shown to include Gouy phase and preserve their forms under Lorentz transformations. As the wave function contains two time coordinates, the particle current must be interpreted in a constraint space to reduce the number of independent coordinates. The form of the constraint space is not certain except in the nonrelativistic limit, but a trial form is proposed, enabling the observable properties of the beam to be calculated for future comparison to experiment. These results can be relevant in the theoretical development of singular electron optics since it was shown that the Gouy phase is crucial in this field as well as to investigate a possible Gouy phase effect in Zitterbewegung phenomenon of spin-zero particles. Additionally, the traditional argument that beam solutions belong to a complex shifted spacetime is shown to necessitate a corresponding Born reciprocal shift in 4-momentum space.

Surface phononic graphene

NASA Astrophysics Data System (ADS)

Yu, Si-Yuan; Sun, Xiao-Chen; Ni, Xu; Wang, Qing; Yan, Xue-Jun; He, Cheng; Liu, Xiao-Ping; Feng, Liang; Lu, Ming-Hui; Chen, Yan-Feng

2016-12-01

Strategic manipulation of wave and particle transport in various media is the key driving force for modern information processing and communication. In a strongly scattering medium, waves and particles exhibit versatile transport characteristics such as localization, tunnelling with exponential decay, ballistic, and diffusion behaviours due to dynamical multiple scattering from strong scatters or impurities. Recent investigations of graphene have offered a unique approach, from a quantum point of view, to design the dispersion of electrons on demand, enabling relativistic massless Dirac quasiparticles, and thus inducing low-loss transport either ballistically or diffusively. Here, we report an experimental demonstration of an artificial phononic graphene tailored for surface phonons on a LiNbO3 integrated platform. The system exhibits Dirac quasiparticle-like transport, that is, pseudo-diffusion at the Dirac point, which gives rise to a thickness-independent temporal beating for transmitted pulses, an analogue of Zitterbewegung effects. The demonstrated fully integrated artificial phononic graphene platform here constitutes a step towards on-chip quantum simulators of graphene and unique monolithic electro-acoustic integrated circuits.

Dirac Cellular Automaton from Split-step Quantum Walk

PubMed Central

Mallick, Arindam; Chandrashekar, C. M.

2016-01-01

Simulations of one quantum system by an other has an implication in realization of quantum machine that can imitate any quantum system and solve problems that are not accessible to classical computers. One of the approach to engineer quantum simulations is to discretize the space-time degree of freedom in quantum dynamics and define the quantum cellular automata (QCA), a local unitary update rule on a lattice. Different models of QCA are constructed using set of conditions which are not unique and are not always in implementable configuration on any other system. Dirac Cellular Automata (DCA) is one such model constructed for Dirac Hamiltonian (DH) in free quantum field theory. Here, starting from a split-step discrete-time quantum walk (QW) which is uniquely defined for experimental implementation, we recover the DCA along with all the fine oscillations in position space and bridge the missing connection between DH-DCA-QW. We will present the contribution of the parameters resulting in the fine oscillations on the Zitterbewegung frequency and entanglement. The tuneability of the evolution parameters demonstrated in experimental implementation of QW will establish it as an efficient tool to design quantum simulator and approach quantum field theory from principles of quantum information theory. PMID:27184159

Exploring the propagation of relativistic quantum wavepackets in the trajectory-based formulation

NASA Astrophysics Data System (ADS)

Tsai, Hung-Ming; Poirier, Bill

2016-03-01

In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [Phys. Rev. X, 4, 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature—suggesting a “many interacting worlds” interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time—in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.

Does Bohm's Quantum Force Have a Classical Origin?

NASA Astrophysics Data System (ADS)

Lush, David C.

2016-08-01

In the de Broglie-Bohm formulation of quantum mechanics, the electron is stationary in the ground state of hydrogenic atoms, because the quantum force exactly cancels the Coulomb attraction of the electron to the nucleus. In this paper it is shown that classical electrodynamics similarly predicts the Coulomb force can be effectively canceled by part of the magnetic force that occurs between two similar particles each consisting of a point charge moving with circulatory motion at the speed of light. Supposition of such motion is the basis of the Zitterbewegung interpretation of quantum mechanics. The magnetic force between two luminally-circulating charges for separation large compared to their circulatory motions contains a radial inverse square law part with magnitude equal to the Coulomb force, sinusoidally modulated by the phase difference between the circulatory motions. When the particles have equal mass and their circulatory motions are aligned but out of phase, part of the magnetic force is equal but opposite the Coulomb force. This raises a possibility that the quantum force of Bohmian mechanics may be attributable to the magnetic force of classical electrodynamics. It is further shown that relative motion between the particles leads to modulation of the magnetic force with spatial period equal to the de Broglie wavelength.

Hunting for Snarks in Quantum Mechanics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hestenes, David

2009-12-08

A long-standing debate over the interpretation of quantum mechanics has centered on the meaning of Schroedinger's wave function {psi} for an electron. Broadly speaking, there are two major opposing schools. On the one side, the Copenhagen school(led by Bohr, Heisenberg and Pauli) holds that {psi} provides a complete description of a single electron state; hence the probability interpretation of {psi}{psi}* expresses an irreducible uncertainty in electron behavior that is intrinsic in nature. On the other side, the realist school(led by Einstein, de Broglie, Bohm and Jaynes) holds that {psi} represents a statistical ensemble of possible electron states; hence it ismore » an incomplete description of a single electron state. I contend that the debaters have overlooked crucial facts about the electron revealed by Dirac theory. In particular, analysis of electron zitterbewegung(first noticed by Schroedinger) opens a window to particle substructure in quantum mechanics that explains the physical significance of the complex phase factor in {psi}. This led to a testable model for particle substructure with surprising support by recent experimental evidence. If the explanation is upheld by further research, it will resolve the debate in favor of the realist school. I give details. The perils of research on the foundations of quantum mechanics have been foreseen by Lewis Carroll in The Hunting of the Snark{exclamation_point}.« less

Wavefunction Collapse via a Nonlocal Relativistic Variational Principle

DOE Office of Scientific and Technical Information (OSTI.GOV)

Harrison, Alan K.

2012-06-18

Since the origin of quantum theory in the 1920's, some of its practitioners (and founders) have been troubled by some of its features, including indeterminacy, nonlocality and entanglement. The 'collapse' process described in the Copenhagen Interpretation is suspect for several reasons, and the act of 'measurement,' which is supposed to delimit its regime of validity, has never been unambiguously defined. In recent decades, nonlocality and entanglement have been studied energetically, both theoretically and experimentally, and the theory has been reinterpreted in imaginative ways, but many mysteries remain. We propose that it is necessary to replace the theory by one thatmore » is explicitly nonlinear and nonlocal, and does not distinguish between measurement and non-measurement regimes. We have constructed such a theory, for which the phase of the wavefunction plays the role of a hidden variable via the process of zitterbewegung. To capture this effect, the theory must be relativistic, even when describing nonrelativistic phenomena. It is formulated as a variational principle, in which Nature attempts to minimize the sum of two spacetime integrals. The first integral tends to drive the solution toward a solution of the standard quantum mechanical wave equation, and also enforces the Born rule of outcome probabilities. The second integral drives the collapse process. We demonstrate that the new theory correctly predicts the possible outcomes of the electron two-slit experiment, including the infamous 'delayed-choice' variant. We observe that it appears to resolve some long-standing mysteries, but introduces new ones, including possible retrocausality (a cause later than its effect). It is not clear whether the new theory is deterministic.« less

Semirelativity in semiconductors: a review.

PubMed

Zawadzki, Wlodek

2017-09-20

An analogy between behavior of electrons in narrow-gap semiconductors (NGS) and relativistic electrons in vacuum is reviewed. Energy band structures [Formula: see text] are considered for various NGS materials and their correspondence to the energy-momentum relation in special relativity is emphasized. It is indicated that special relativity for vacuum is analogous to a two-band [Formula: see text] description for NGS. The maximum electron velocity in NGS is [Formula: see text], which corresponds to the light velocity in vacuum. An effective mass of charge carriers in semiconductors is introduced, relating their velocity to quasimomentum and it is shown that this mass depends on electron energy (or velocity) in a way similar to the mass of free relativistic electrons. In [Formula: see text] alloys one can reach vanishing energy gap at which electrons and light holes become three-dimensional massless Dirac fermions. A wavelength [Formula: see text] is defined for NGS, in analogy to the Compton wavelength in relativistic quantum mechanics. It is estimated that [Formula: see text] is on the order of tens of Angstroms in typical semiconducting materials which is experimentally confirmed in tunneling experiments on energy dispersion in the forbidden gap. Statistical properties of the electron gas in NGS are calculated and their similarity is demonstrated to those of the Juttner gas of relativistic particles. Interband electron tunneling in NGS is described and shown to be in close analogy to the predicted but unobserved tunneling between negative and positive energies resulting from the Dirac equation for free electrons. It is demonstrated that the relativistic analogy holds for orbital and spin properties of electrons in the presence of an external magnetic field. In particular, it is shown that the spin magnetic moment of both NGS electrons and relativistic electrons approaches zero with increasing energy. This conclusion is confirmed experimentally for NGS. Electrons in crossed electric and magnetic fields are described theoretically and experimentally. It is only the two-band description for NGS, equivalent to the Dirac or Klein-Gordon equations for free particles, that gives a correct account of experimental results in this situation. A transverse Doppler shift in the cyclotron resonance observed in crossed fields in InSb indicates that there exists a time dilatation between an oscillating electron and an observer. The phenomenon of Zitterbewegung (ZB, trembling motion) for electrons in NGS is considered theoretically, following the original proposition of Schrödinger for free relativistic electrons in vacuum. The two descriptions are in close analogy, but the frequency of ZB for electrons in NGS is orders of magnitude lower and its amplitude orders of magnitude higher making possible experimental observations in semiconductors considerably more favorable. Finally, graphene and carbon nanotubes, as well as topological insulators are considered in the framework of relativistic analogy. These systems, with their linear energy-quasimomentum dispersions, illustrate the extreme semirelativistic regime. Experimental results for the energy dispersions and the Landau quantizations in the presence of a magnetic field are quoted and their analogy to the behavior of free relativistic electrons is discussed. Approximations and restrictions of the relativistic analogy are emphasized. On the other hand, it is indicated that in various situations it is considerably easier to observe semirelativistic effects in semiconductors than the relativistic effects in vacuum.

Semirelativity in semiconductors: a review

NASA Astrophysics Data System (ADS)

Zawadzki, Wlodek

2017-09-01

An analogy between behavior of electrons in narrow-gap semiconductors (NGS) and relativistic electrons in vacuum is reviewed. Energy band structures \\varepsilon ≤ft(\\mathbf{k}\\right) are considered for various NGS materials and their correspondence to the energy-momentum relation in special relativity is emphasized. It is indicated that special relativity for vacuum is analogous to a two-band \\mathbf{k}\\centerdot \\mathbf{p} description for NGS. The maximum electron velocity in NGS is u≃ 1× {{10}8}~\\text{cm}~{{\\text{s}}-1} , which corresponds to the light velocity in vacuum. An effective mass of charge carriers in semiconductors is introduced, relating their velocity to quasimomentum and it is shown that this mass depends on electron energy (or velocity) in a way similar to the mass of free relativistic electrons. In \\text{H}{{\\text{g}}1-x}\\text{C}{{\\text{d}}x}\\text{Te} alloys one can reach vanishing energy gap at which electrons and light holes become three-dimensional massless Dirac fermions. A wavelength {λz} is defined for NGS, in analogy to the Compton wavelength in relativistic quantum mechanics. It is estimated that {λz} is on the order of tens of Angstroms in typical semiconducting materials which is experimentally confirmed in tunneling experiments on energy dispersion in the forbidden gap. Statistical properties of the electron gas in NGS are calculated and their similarity is demonstrated to those of the Juttner gas of relativistic particles. Interband electron tunneling in NGS is described and shown to be in close analogy to the predicted but unobserved tunneling between negative and positive energies resulting from the Dirac equation for free electrons. It is demonstrated that the relativistic analogy holds for orbital and spin properties of electrons in the presence of an external magnetic field. In particular, it is shown that the spin magnetic moment of both NGS electrons and relativistic electrons approaches zero with increasing energy. This conclusion is confirmed experimentally for NGS. Electrons in crossed electric and magnetic fields are described theoretically and experimentally. It is only the two-band description for NGS, equivalent to the Dirac or Klein-Gordon equations for free particles, that gives a correct account of experimental results in this situation. A transverse Doppler shift in the cyclotron resonance observed in crossed fields in InSb indicates that there exists a time dilatation between an oscillating electron and an observer. The phenomenon of Zitterbewegung (ZB, trembling motion) for electrons in NGS is considered theoretically, following the original proposition of Schrödinger for free relativistic electrons in vacuum. The two descriptions are in close analogy, but the frequency of ZB for electrons in NGS is orders of magnitude lower and its amplitude orders of magnitude higher making possible experimental observations in semiconductors considerably more favorable. Finally, graphene and carbon nanotubes, as well as topological insulators are considered in the framework of relativistic analogy. These systems, with their linear energy-quasimomentum dispersions, illustrate the extreme semirelativistic regime. Experimental results for the energy dispersions and the Landau quantizations in the presence of a magnetic field are quoted and their analogy to the behavior of free relativistic electrons is discussed. Approximations and restrictions of the relativistic analogy are emphasized. On the other hand, it is indicated that in various situations it is considerably easier to observe semirelativistic effects in semiconductors than the relativistic effects in vacuum.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ling, Meng-Chieh

Graphene, a two-dimensional (2D) honeycomb structure allotrope of carbon atoms, has a long history since the invention of the pencil [Petroski (1989)] and the linear dispersion band structure proposed by Wallace [Wal]; however, only after Novoselov et al. successively isolated graphene from graphite [Novoselov et al. (2004)], it has been studied intensively during the recent years. It draws so much attentions not only because of its potential application in future electronic devices but also because of its fundamental properties: its quasiparticles are governed by the two-dimensional Dirac equation, and exhibit a variety of phenomena such as the anomalous integer quantummore » Hall effect (IQHE) [Novoselov et al. (2005)] measured experimentally, a minimal conductivity at vanishing carrier concentration [Neto et al. (2009)], Kondo effect with magnetic element doping [Hentschel and Guinea (2007)], Klein tunneling in p-n junctions [Cheianov and Fal’ko (2006), Beenakker (2008)], Zitterbewegung [Katsnelson (2006)], and Schwinger pair production [Schwinger (1951); Dora and Moessner (2010)]. Although both electron-phonon coupling and photoconductivity in graphene also draws great attention [Yan et al. (2007); Satou et al. (2008); Hwang and Sarma (2008); Vasko and Ryzhii (2008); Mishchenko (2009)], the nonequilibrium behavior based on the combination of electronphonon coupling and Schwinger pair production is an intrinsic graphene property that has not been investigated. Our motivation for studying clean graphene at low temperature is based on the following effect: for a fixed electric field, below a sufficiently low temperature linear eletric transport breaks down and nonlinear transport dominates. The criteria of the strength of this field [Fritz et al. (2008)] is eE = T2/~vF (1.1) For T >√eE~vF the system is in linear transport regime while for T <√eE~vF the system is in nonlinear transport regime. From the scaling’s point of view, at the nonlinear transport regime the temperature T and electric field E are also related. In this thesis we show that the nontrivial electron distribution function can be associated with an effective temperature T which exhibits a dependence on electric field E and electron-phonon coupling g: T ∝ E1/4g(1.2) The anamolous exponent 1/4 may obtained from scaling. Meanwhile, yet we cannot obtain the distribution function, however, argument based on scaling gives us the current dependence on electric field: J ∝√Eg2 (1.3) which is a very different result compared with the results in which electrons do not experience scattering. This result provides us with important insighht into the correct nonequilibrium distribution function because now we know what the electric field dependence of current must be. Due to the applied field, the electronic system produces heat which prevents us from reaching a steady state. In order to remove Joule heat, we imagine that we have a graphene flake attached to a semiconductor substrate. Joule heat either transport to its environment or to the substrate as shown in 1.1. The red lines represent heat current flowing from high temperature sample to the low temperature reservoir. However, for a very large system, the temperature gradient is 0 in the plane so heat cannot be conducted outside in the horizontal direction, while the energy gap in semiconductor also forbids electron current from flowing into the substrate. But for phonon thermal current, the temperature gradient is large in the vertical direction, so heat can be transported into the substrate via phonons. There are two possible channels of phonon degrees of freedom, acoustic phonon and optical phonon. As we can see from Fig. 1.2 [Kusminskiy et al. (2009)], since the optical phonon excitation energy is too large for a low temperature system, it is note likely to be excited by the nonlinear electric field, so the possible way left is by electron-acoustic phonon scattering. Here acoustic phonon acts as a heat bath to absorb the Joule heat created by pair production process. Hence the scattering process is determined by electron-acoustic phonon interaction which will be introduced in section 3.3.« less

BOOK REVIEWS: Quantum Mechanics: Fundamentals

NASA Astrophysics Data System (ADS)

Whitaker, A.

2004-02-01

This review is of three books, all published by Springer, all on quantum theory at a level above introductory, but very different in content, style and intended audience. That of Gottfried and Yan is of exceptional interest, historical and otherwise. It is a second edition of Gottfried’s well-known book published by Benjamin in 1966. This was written as a text for a graduate quantum mechanics course, and has become one of the most used and respected accounts of quantum theory, at a level mathematically respectable but not rigorous. Quantum mechanics was already solidly established by 1966, but this second edition gives an indication of progress made and changes in perspective over the last thirty-five years, and also recognises the very substantial increase in knowledge of quantum theory obtained at the undergraduate level. Topics absent from the first edition but included in the second include the Feynman path integral, seen in 1966 as an imaginative but not very useful formulation of quantum theory. Feynman methods were given only a cursory mention by Gottfried. Their practical importance has now been fully recognised, and a substantial account of them is provided in the new book. Other new topics include semiclassical quantum mechanics, motion in a magnetic field, the S matrix and inelastic collisions, radiation and scattering of light, identical particle systems and the Dirac equation. A topic that was all but totally neglected in 1966, but which has flourished increasingly since, is that of the foundations of quantum theory. John Bell’s work of the mid-1960s has led to genuine theoretical and experimental achievement, which has facilitated the development of quantum optics and quantum information theory. Gottfried’s 1966 book played a modest part in this development. When Bell became increasingly irritated with the standard theoretical approach to quantum measurement, Viki Weisskopf repeatedly directed him to Gottfried’s book. Gottfried had devoted a chapter of his book to these matters, titled ‘The Measurement Process and the Statistical Interpretation of Quantum Mechanics’. Gottfried considered the von Neumann or Dirac ‘collapse of state-vector’ (or ‘reduction postulate’ or ‘projection postulate’) was unsatisfactory, as he argued that it led inevitably to the requirement to include ‘consciousness’ in the theory. He replaced this by a more mathematically and conceptually sophisticated treatment in which, following measurement, the density matrix of the correlated measured and measuring systems, rho, is replaced by hat rho, in which the interference terms from rho have been removed. rho represents a pure state, and hat rho a mixture, but Gottfried argued that they are ‘indistinguishable’, and that we may make our replacement, ‘safe in the knowledge that the error will never be found’. Now our combined state is represented as a mixture, it is intuitive, Gottfried argued, to interpret it in a probabilistic way, |cm|2 being the probability of obtaining the mth measurement result. Bell liked Gottfried’s treatment little more than the cruder ‘collapse’ idea of von Neumann, and when, shortly before Bell’s death, his polemical article ‘Against measurement’ was published in the August 1990 issue of Physics World (pages 33-40), his targets included, not only Landau and Lifshitz’s classic Quantum Mechanics, pilloried for its advocacy of old-fashioned collapse, and a paper by van Kampen in Physica, but also Gottfried’s approach. Bell regarded his replacement of rho by hat rho as a ‘butchering’ of the density matrix, and considered, in any case, that even the butchered density matrix should represent co-existence of different terms, not a set of probabilities. Gottfried has replied to Bell ( Physics World, October 1991, pages 34-40; Nature 405, 533-36 (2000)). He has also become a major commentator on Bell’s work, for example editing the section on quantum foundations in the World Scientific edition of Bell’s collected works. Thus it is exceedingly interesting to discover how he has responded to Bell’s criticisms in the new edition of the book. To commence with general discussion of the new book, the authors recognise that the graduate student of today almost certainly has substantial experience of wave mechanics, and is probably familiar with the Dirac formalism. The 1966 edition had what seems, at least in retrospect, a relatively soft opening covering the basic ideas of wave mechanics and a substantial number of applications; it did not reach the Dirac formalism in the first two hundred pages, though it then moved on to tackle rather advanced topics, including a very substantial section on symmetries, which tackled a range of sophisticated issues. The new edition has been almost entirely rewritten; even at the level of basic text, it is difficult to trace sentences or paragraphs that have moved unscathed from one edition to the next. As well as the new topics, many of the old ones are discussed in much greater depth, and the general organisation is entirely different. As compared with the steady rise in level of the 1966 edition, the level of this book is fairly consistent throughout, and from the perspective of a beginning graduate student, I would estimate, a little tough. A brief introductory chapter gives a useful, though not particularly straightforward, discussion of complementarity, uncertainty and superposition, and concludes with an informative though very short summary of the discovery of quantum mechanics, together with a few nice photographs of some of its founders. There follow two substantial chapters which are preparation for the later study of actual systems. The first, called ‘The Formal Framework’ is a fairly comprehensive survey of the methods of quantum theory---Hilbert space, Dirac notation, mixtures, the density matrix, entanglement, canonical quantization, equations of motion, symmetries, conservation laws, propagators, Green’s functions, semiclassical quantum mechanics. The level of mathematical rigour is stated as ‘typical of the bulk of theoretical physics literature---slovenly’; those unhappy with this are directed to the well-known books of Jordan and Thirring. The next chapter---‘Basic Tools’---explains a set of topics which students will need to use when studying particular systems---angular momentum and its addition, free particles, the two-body system, and the standard approximation techniques. There follow chapters on low-dimensional systems---harmonic oscillator, Aharanov--Bohm effect, one-dimensional scattering, WKB and so on; hydrogenic atoms---the Kepler problem, fine and hyperfine structure, Zeeman and Stark effects; and on two-electron atoms---spin and statistics. As in the first edition, there is a substantial treatment of symmetries, including time reversal, Galileo transformations, the rotation group, the Wigner-Eckart theorem and the Berry phase. There are two long chapters on scattering---elastic and inelastic respectively, including an account of the S matrix. The treatment of electrodynamics is much extended and modernised compared to that in the first edition. There are discussions of the quantization of the free field, causality and uncertainty in electrodynamics, vacuum fluctuations including the Casimir effect and the Lamb shift, and radiative transitions. There is a treatment of quantum optics, but this a only a brief introduction to a rapidly expanding subject, designed to facilitate understanding of the experiments on Bell’s inequalities discussed in the later chapter on interpretation. Other topics are the photoeffect in hydrogen, scattering of photons, resonant scattering and spontaneous decay. Identical particles are discussed, with a treatment of second quantization and an introduction to Bose--Einstein condensation, and the last chapter is a brief introduction to relativistic quantum mechanics, including the Dirac equation, the electromagnetic interaction of a Dirac particle, the scattering of ultra-relativistic electrons and a treatment of bound states in a Coulomb field. Gottfried and Yan’s response both to the growing interest in work on foundational matters in general, and to the specific criticism of Bell on the previous edition is included in the chapter entitled `Interpretation'. This chapter appears to be something of a hybrid. The first four sections broadly discuss hidden variables. An account of the Einstein--Podolsky--Rosen approach is followed by a general study of hidden variables, including a discussion of what the authors call the Bell--Kochen--Specker theorem. Bell’s theorem is analysed in some detail; also included are the Clauser--Horne inequality and the experimental test of the Bell inequality by Aspect. There is an interesting discussion of locality. Granted that both quantum mechanics and experiment (the latter admittedly with a remaining loophole) are in conflict with what the authors call a classical conception of locality as embodied in the Bell inequality, they ask whether quantum mechanics is actually non-local if one uses a definition of locality entailing no ingredients unknown to quantum mechanics. Their answer is that it is a matter of taste. In the statistical distribution of measurement outcomes on separate systems in entangled states, there is no hint of non-locality and no question of superluminal signalling. But quantum mechanics displays perfect correlations between distant outcomes, even though Bell’s theorem demonstrates that pre-existing values cannot be assumed. The second part of this chapter is a discussion of the measurement procedure similar to that in the first edition. The authors aim to show how measurement results are obtained and displayed, and how the appropriate probabilities are determined. The expression of this intention, however, is accompanied by the statement that they are not attempting to derive the statistical interpretation of quantum mechanics, which is assumed, but to examine whether it gives a consistent account of measurement. The conclusion is that after a measurement, interference terms are ‘effectively’ absent; the set of ‘one-to-one correlations between states of the apparatus and the object’ has the same form as that of everyday statistics and is thus a probability distribution. This probability distribution refers to potentialities, only one of which is actually realized in any one trial. Opinions may differ on whether their treatment is any less vulnerable to criticisms such as those of Bell. To sum up, Gottfried and Yan’s book contains a vast amount of knowledge and understanding. As well as explaining the way in which quantum theory works, it attempts to illuminate fundamental aspects of the theory. A typical example is the ‘fable’ elaborated in Gottfried’s article in Nature cited above, that if Newton were shown Maxwell’s equations and the Lorentz force law, he could deduce the meaning of E and B, but if Maxwell were shown Schrödinger’s equation, he could not deduce the meaning of Psi. For use with a well-constructed course (and, of course, this is the avowed purpose of the book; a useful range of problems is provided for each chapter), or for the relative expert getting to grips with particular aspects of the subject or aiming for a deeper understanding, the book is certainly ideal. It might be suggested, though, that, even compared to the first edition, the isolated learner might find the wide range of topics, and the very large number of mathematical and conceptual techniques, introduced in necessarily limited space, somewhat overwhelming. The second book under consideration, that of Schwabl, contains ‘Advanced’ elements of quantum theory; it is designed for a course following on from one for which Gottfried and Yan, or Schwabl’s own `Quantum Mechanics' might be recommended. It is the second edition in English, and is a translation of the third German edition. It has a restricted range of general topics, and consists of three parts entitled `Nonrelativistic Many-Particle Systems', `Relativistic Wave Equations', and `Relativistic Fields'. Thus it studies in some depth areas of physics which are either dealt with in an introductory fashion, or not reached at all, by Gottfried and Yan. Despite its more advanced level, this book may actually be the more accessible to an isolated learner, because the various aspects are developed in an unhurried fashion; the author remarks that ‘the inclusion of all mathematical steps and full presentation of intermediate calculations ensures ease of understanding’. Many useful student problems are included. The presentation is said to be rigorous, but again this is a book for the physicist rather than the mathematician. The treatment of many-particle systems begins with a rather general introduction to second quantization, and then applies this formalism to spin-1/2 fermions and bosons. The study of fermions includes consideration of the Fermi sphere, the electron gas, and the Hartree--Fock equations for atoms; that of bosons includes Bose--Einstein condensation, Bogoliubov theory of the weakly interacting Bose gas, and a brief account of superfluidity. The last section of this part of the book investigates in detail the dynamics of many-particle systems on a microscopic quantum-mechanical basis using, in particular, the dynamical correlation functions. In the second part which considers relativistic wave equations, the Klein--Gordon and Dirac equations are derived, and the Lorentz covariance of the Dirac equation is established. The role of angular momentum in relativistic quantum mechanics is explained, as a preliminary to the study of the energy levels in a Coulomb potential using both the Klein--Gordon and Dirac equations, the latter being solved exactly for the hydrogen atom. For larger atoms, the Foldy--Wouthuysen transformation is explained, and also relativistic corrections and the Lamb shift. There is an interesting chapter on the physical interpretation of the Dirac equation, including such topics as the negative energy solutions, the Zitterbewegung and the Klein paradox. The last chapter in this part of the book is an extensive treatment of the symmetries and other properties of the Dirac equation, including the behaviour under rotation, translation, reflection, charge conjugation and time reversal. Helicity is explained, and the behaviour of zero-mass fermions is discussed; even though it now seems certain that neutrinos do not have zero-mass, this treatment provides a good approximation to their behaviour if they have high enough momenta. The last section on relativistic fields contains chapters on the quantization of relativistic fields, the free Klein--Gordon and Dirac fields, quantization of the radiation field, interacting fields and quantum electrodynamics, including the S matrix, Wick’s theorem and Feynman diagrams. Schwabl’s book would be excellent for those requiring a detailed presentation of the topics it includes, at a level of rigour appropriate to the physicist. It includes a substantial number of interesting problems. The third book under consideration, that by Gustafson and Sigal is very different from the others. In academic level, at least the initial sections may actually be slightly lower; the book covers a one-term course taken by senior undergraduates or junior graduate students in mathematics or physics, and the initial chapters are on basic topics, such as the physical background, basic dynamics, observables and the uncertainty principle. However the level of mathematical sophistication is far higher than in the other books. While the mathematical prerequisites are modest---real and complex analysis, elementary differential equations and preferably Lebesgue integration, a third of the book is made up of what are called mathematical supplements---on operator adjoints, the Fourier transform, tensor products, the trace and trace class operators, the Trotter product formula, operator determinants, the calculus of variations (a substantial treatment in a full chapter), spectral projections, and the projecting-out procedure. On the basis of these supplements, the level of mathematical sophistication and difficulty is increased substantially in the middle section of the book, where the topics considered are many-particle systems, density matrices, positive temperatures, the Feynman path integral, and quasi-classical analysis, and there is a final substantial step for the concluding chapters on resonances, an introduction to quantum field theory, and quantum electrodynamics of non-relativistic particles. A supplementary chapter contains an interesting approach to the remormalization group due to Bach, Fröhlich and Sigal himself. This book is well-written, and the topics discussed have been well thought-out. It would provide a useful approach to quantum theory for the mathematician, and would also provide access for the physicist to some mathematically advanced methods and topics, but the physicist would definitely have to be prepared to work hard at the mathematics required.