Relativistic Quantum Mechanics Theoretical and Mathematical Physics
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases. -
Symmetry Conditions on Dirac Observables
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 2, 671–676 Symmetry Conditions on Dirac Observables Heinz Otto CORDES Department of Mathematics, University of California, Berkeley, CA 94720 USA E-mail: [email protected] Using our Dirac invariant algebra we attempt a mathematically and philosophically clean theory of Dirac observables, within the environment of the 4 books [10,9,11,4]. All classical objections to one-particle Dirac theory seem removed, while also some principal objections to von Neumann’s observable theory may be cured. Heisenberg’s uncertainty principle appears improved. There is a clean and mathematically precise pseudodifferential Foldy–Wouthuysen transform, not only for the supersymmeytric case, but also for general (C∞-) potentials. 1 Introduction The free Dirac Hamiltonian H0 = αD+β is a self-adjoint square root of 1−∆ , with the Laplace HS − 1 operator ∆. The free Schr¨odinger√ Hamiltonian 0 =1 2 ∆ (where “1” is the “rest energy” 2 1 mc ) seems related to H0 like 1+x its approximation 1+ 2 x – second partial sum of its Taylor expansion. From that aspect the early discoverers of Quantum Mechanics were lucky that the energies of the bound states of hydrogen (a few eV) are small, compared to the mass energy of an electron (approx. 500 000 eV), because this should make “Schr¨odinger” a good approximation of “Dirac”. But what about the continuous spectrum of both operators – governing scattering theory. Note that today big machines scatter with energies of about 10,000 – compared to the above 1 = mc2. So, from that aspect the precise scattering theory of both Hamiltonians (with potentials added) should give completely different results, and one may have to put ones trust into one of them, because at most one of them can be applicable. -
Relational Quantum Mechanics and Probability M
Relational Quantum Mechanics and Probability M. Trassinelli To cite this version: M. Trassinelli. Relational Quantum Mechanics and Probability. Foundations of Physics, Springer Verlag, 2018, 10.1007/s10701-018-0207-7. hal-01723999v3 HAL Id: hal-01723999 https://hal.archives-ouvertes.fr/hal-01723999v3 Submitted on 29 Aug 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Noname manuscript No. (will be inserted by the editor) Relational Quantum Mechanics and Probability M. Trassinelli the date of receipt and acceptance should be inserted later Abstract We present a derivation of the third postulate of Relational Quan- tum Mechanics (RQM) from the properties of conditional probabilities. The first two RQM postulates are based on the information that can be extracted from interaction of different systems, and the third postulate defines the prop- erties of the probability function. Here we demonstrate that from a rigorous definition of the conditional probability for the possible outcomes of different measurements, the third postulate is unnecessary and the Born's rule naturally emerges from the first two postulates by applying the Gleason's theorem. We demonstrate in addition that the probability function is uniquely defined for classical and quantum phenomena. -
Identical Particles
8.06 Spring 2016 Lecture Notes 4. Identical particles Aram Harrow Last updated: May 19, 2016 Contents 1 Fermions and Bosons 1 1.1 Introduction and two-particle systems .......................... 1 1.2 N particles ......................................... 3 1.3 Non-interacting particles .................................. 5 1.4 Non-zero temperature ................................... 7 1.5 Composite particles .................................... 7 1.6 Emergence of distinguishability .............................. 9 2 Degenerate Fermi gas 10 2.1 Electrons in a box ..................................... 10 2.2 White dwarves ....................................... 12 2.3 Electrons in a periodic potential ............................. 16 3 Charged particles in a magnetic field 21 3.1 The Pauli Hamiltonian ................................... 21 3.2 Landau levels ........................................ 23 3.3 The de Haas-van Alphen effect .............................. 24 3.4 Integer Quantum Hall Effect ............................... 27 3.5 Aharonov-Bohm Effect ................................... 33 1 Fermions and Bosons 1.1 Introduction and two-particle systems Previously we have discussed multiple-particle systems using the tensor-product formalism (cf. Section 1.2 of Chapter 3 of these notes). But this applies only to distinguishable particles. In reality, all known particles are indistinguishable. In the coming lectures, we will explore the mathematical and physical consequences of this. First, consider classical many-particle systems. If a single particle has state described by position and momentum (~r; p~), then the state of N distinguishable particles can be written as (~r1; p~1; ~r2; p~2;:::; ~rN ; p~N ). The notation (·; ·;:::; ·) denotes an ordered list, in which different posi tions have different meanings; e.g. in general (~r1; p~1; ~r2; p~2)6 = (~r2; p~2; ~r1; p~1). 1 To describe indistinguishable particles, we can use set notation. -
Generalized Lorentz Symmetry and Nonlinear Spinor Fields in a Flat Finslerian Space-Time
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 2, 637–644 Generalized Lorentz Symmetry and Nonlinear Spinor Fields in a Flat Finslerian Space-Time George BOGOSLOVSKY † and Hubert GOENNER ‡ † Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia E-mail: [email protected] ‡ Institute for Theoretical Physics, University of G¨ottingen, G¨ottingen, Germany E-mail: [email protected] The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e. Finslerian space-time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented. 1 Introduction In spite of the impressive successes of the unified gauge theory of strong, weak and electromag- netic interactions, known as the Standard Model, one cannot a priori rule out the possibility that Lorentz symmetry underlying the theory is an approximate symmetry of nature. This implies that at the energies already attainable today empirical evidence may be obtained in favor of violation of Lorentz symmetry. At the same time it is obvious that such effects might manifest themselves only as strongly suppressed effects of Planck-scale physics. Theoretical speculations about a possible violation of Lorentz symmetry continue for more than forty years and they are briefly outlined in [1]. -
Solving the Quantum Scattering Problem for Systems of Two and Three Charged Particles
Solving the quantum scattering problem for systems of two and three charged particles Solving the quantum scattering problem for systems of two and three charged particles Mikhail Volkov c Mikhail Volkov, Stockholm 2011 ISBN 978-91-7447-213-4 Printed in Sweden by Universitetsservice US-AB, Stockholm 2011 Distributor: Department of Physics, Stockholm University In memory of Professor Valentin Ostrovsky Abstract A rigorous formalism for solving the Coulomb scattering problem is presented in this thesis. The approach is based on splitting the interaction potential into a finite-range part and a long-range tail part. In this representation the scattering problem can be reformulated to one which is suitable for applying exterior complex scaling. The scaled problem has zero boundary conditions at infinity and can be implemented numerically for finding scattering amplitudes. The systems under consideration may consist of two or three charged particles. The technique presented in this thesis is first developed for the case of a two body single channel Coulomb scattering problem. The method is mathe- matically validated for the partial wave formulation of the scattering problem. Integral and local representations for the partial wave scattering amplitudes have been derived. The partial wave results are summed up to obtain the scat- tering amplitude for the three dimensional scattering problem. The approach is generalized to allow the two body multichannel scattering problem to be solved. The theoretical results are illustrated with numerical calculations for a number of models. Finally, the potential splitting technique is further developed and validated for the three body Coulomb scattering problem. It is shown that only a part of the total interaction potential should be split to obtain the inhomogeneous equation required such that the method of exterior complex scaling can be applied. -
Statistics of a Free Single Quantum Particle at a Finite
STATISTICS OF A FREE SINGLE QUANTUM PARTICLE AT A FINITE TEMPERATURE JIAN-PING PENG Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China Abstract We present a model to study the statistics of a single structureless quantum particle freely moving in a space at a finite temperature. It is shown that the quantum particle feels the temperature and can exchange energy with its environment in the form of heat transfer. The underlying mechanism is diffraction at the edge of the wave front of its matter wave. Expressions of energy and entropy of the particle are obtained for the irreversible process. Keywords: Quantum particle at a finite temperature, Thermodynamics of a single quantum particle PACS: 05.30.-d, 02.70.Rr 1 Quantum mechanics is the theoretical framework that describes phenomena on the microscopic level and is exact at zero temperature. The fundamental statistical character in quantum mechanics, due to the Heisenberg uncertainty relation, is unrelated to temperature. On the other hand, temperature is generally believed to have no microscopic meaning and can only be conceived at the macroscopic level. For instance, one can define the energy of a single quantum particle, but one can not ascribe a temperature to it. However, it is physically meaningful to place a single quantum particle in a box or let it move in a space where temperature is well-defined. This raises the well-known question: How a single quantum particle feels the temperature and what is the consequence? The question is particular important and interesting, since experimental techniques in recent years have improved to such an extent that direct measurement of electron dynamics is possible.1,2,3 It should also closely related to the question on the applicability of the thermodynamics to small systems on the nanometer scale.4 We present here a model to study the behavior of a structureless quantum particle moving freely in a space at a nonzero temperature. -
Baryon Parity Doublets and Chiral Spin Symmetry
PHYSICAL REVIEW D 98, 014030 (2018) Baryon parity doublets and chiral spin symmetry M. Catillo and L. Ya. Glozman Institute of Physics, University of Graz, 8010 Graz, Austria (Received 24 April 2018; published 25 July 2018) The chirally symmetric baryon parity-doublet model can be used as an effective description of the baryon-like objects in the chirally symmetric phase of QCD. Recently it has been found that above the critical temperature, higher chiral spin symmetries emerge in QCD. It is demonstrated here that the baryon parity-doublet Lagrangian is manifestly chiral spin invariant. We construct nucleon interpolators with fixed chiral spin transformation properties that can be used in lattice studies at high T. DOI: 10.1103/PhysRevD.98.014030 I. BARYON PARITY DOUBLETS. fermions of opposite parity, parity doublets, that transform INTRODUCTION into each other upon a chiral transformation [1]. Consider a pair of the isodoublet fermion fields A Dirac Lagrangian of a massless fermion field is chirally symmetric since the left- and right-handed com- Ψ ponents of the fermion field are decoupled, Ψ ¼ þ ; ð4Þ Ψ− μ μ μ L iψγ¯ μ∂ ψ iψ¯ Lγμ∂ ψ L iψ¯ Rγμ∂ ψ R; 1 ¼ ¼ þ ð Þ Ψ Ψ where the Dirac bispinors þ and − have positive and negative parity, respectively. The parity doublet above is a where spinor constructed from two Dirac bispinors and contains eight components. Note that there is, in addition, an isospin 1 1 ψ R ¼ ð1 þ γ5Þψ; ψ L ¼ ð1 − γ5Þψ: ð2Þ index which is suppressed. Given that the right- and left- 2 2 handed fields are directly connected -
Magnetic Moment of a Spin, Its Equation of Motion, and Precession B1.1.6
Magnetic Moment of a Spin, Its Equation of UNIT B1.1 Motion, and Precession OVERVIEW The ability to “see” protons using magnetic resonance imaging is predicated on the proton having a mass, a charge, and a nonzero spin. The spin of a particle is analogous to its intrinsic angular momentum. A simple way to explain angular momentum is that when an object rotates (e.g., an ice skater), that action generates an intrinsic angular momentum. If there were no friction in air or of the skates on the ice, the skater would spin forever. This intrinsic angular momentum is, in fact, a vector, not a scalar, and thus spin is also a vector. This intrinsic spin is always present. The direction of a spin vector is usually chosen by the right-hand rule. For example, if the ice skater is spinning from her right to left, then the spin vector is pointing up; the skater is rotating counterclockwise when viewed from the top. A key property determining the motion of a spin in a magnetic field is its magnetic moment. Once this is known, the motion of the magnetic moment and energy of the moment can be calculated. Actually, the spin of a particle with a charge and a mass leads to a magnetic moment. An intuitive way to understand the magnetic moment is to imagine a current loop lying in a plane (see Figure B1.1.1). If the loop has current I and an enclosed area A, then the magnetic moment is simply the product of the current and area (see Equation B1.1.8 in the Technical Discussion), with the direction n^ parallel to the normal direction of the plane. -
4 Nuclear Magnetic Resonance
Chapter 4, page 1 4 Nuclear Magnetic Resonance Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet. Since then, the splitting of energy levels proportional to an external magnetic field has been called the "Zeeman effect". The "Zeeman resonance effect" causes magnetic resonances which are classified under radio frequency spectroscopy (rf spectroscopy). In these resonances, the transitions between two branches of a single energy level split in an external magnetic field are measured in the megahertz and gigahertz range. In 1944, Jevgeni Konstantinovitch Savoiski discovered electron paramagnetic resonance. Shortly thereafter in 1945, nuclear magnetic resonance was demonstrated almost simultaneously in Boston by Edward Mills Purcell and in Stanford by Felix Bloch. Nuclear magnetic resonance was sometimes called nuclear induction or paramagnetic nuclear resonance. It is generally abbreviated to NMR. So as not to scare prospective patients in medicine, reference to the "nuclear" character of NMR is dropped and the magnetic resonance based imaging systems (scanner) found in hospitals are simply referred to as "magnetic resonance imaging" (MRI). 4.1 The Nuclear Resonance Effect Many atomic nuclei have spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is L =+h II(1). (4.01) The component in the direction of an applied field is Lz = Iz h ≡ m h. (4.02) The external field is usually defined along the z-direction. The magnetic quantum number is symbolized by Iz or m and can have 2I +1 values: Iz ≡ m = −I, −I+1, ..., I−1, I. -
Zero-Point Energy of Ultracold Atoms
Zero-point energy of ultracold atoms Luca Salasnich1,2 and Flavio Toigo1 1Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Universit`adi Padova, via Marzolo 8, 35131 Padova, Italy 2CNR-INO, via Nello Carrara, 1 - 50019 Sesto Fiorentino, Italy Abstract We analyze the divergent zero-point energy of a dilute and ultracold gas of atoms in D spatial dimensions. For bosonic atoms we explicitly show how to regularize this divergent contribution, which appears in the Gaussian fluctuations of the functional integration, by using three different regular- ization approaches: dimensional regularization, momentum-cutoff regular- ization and convergence-factor regularization. In the case of the ideal Bose gas the divergent zero-point fluctuations are completely removed, while in the case of the interacting Bose gas these zero-point fluctuations give rise to a finite correction to the equation of state. The final convergent equa- tion of state is independent of the regularization procedure but depends on the dimensionality of the system and the two-dimensional case is highly nontrivial. We also discuss very recent theoretical results on the divergent zero-point energy of the D-dimensional superfluid Fermi gas in the BCS- BEC crossover. In this case the zero-point energy is due to both fermionic single-particle excitations and bosonic collective excitations, and its regu- larization gives remarkable analytical results in the BEC regime of compos- ite bosons. We compare the beyond-mean-field equations of state of both bosons and fermions with relevant experimental data on dilute and ultra- cold atoms quantitatively confirming the contribution of zero-point-energy quantum fluctuations to the thermodynamics of ultracold atoms at very low temperatures. -
Discrete Variable Representations and Sudden Models in Quantum
Volume 89. number 6 CHLMICAL PHYSICS LEITERS 9 July 1981 DISCRETEVARlABLE REPRESENTATIONS AND SUDDEN MODELS IN QUANTUM SCATTERING THEORY * J.V. LILL, G.A. PARKER * and J.C. LIGHT l7re JarrresFranc/t insrihrre and The Depwrmcnr ofC7temtsrry. Tire llm~crsrry of Chrcago. Otrcago. l7lrrrors60637. USA Received 26 September 1981;m fin11 form 29 May 1982 An c\act fOrmhSm In which rhe scarrcnng problem may be descnbcd by smsor coupled cqumons hbclcd CIIIW bb bans iuncltons or quadrature pomts ISpresented USCof each frame and the srnrplyculuatcd unitary wmsformatlon which connects them resulis III an cfliclcnt procedure ror pcrrormrnpqu~nrum scxrcrrn~ ca~cubr~ons TWO ~ppro~mac~~~ arc compxcd wrh ihe IOS. 1. Introduction “ergenvalue-like” expressions, rcspcctnely. In each case the potential is represented by the potcntud Quantum-mechamcal scattering calculations are function Itself evahrated at a set of pomts. most often performed in the close-coupled representa- Whjle these models have been shown to be cffcc- tion (CCR) in which the internal degrees of freedom tive III many problems,there are numerousambiguities are expanded in an appropnate set of basis functions in their apphcation,especially wtth regardto the resulting in a set of coupled diiierentral equations UI choice of constants. Further, some models possess the scattering distance R [ 1,2] _The method is exact formal difficulties such as loss of time reversal sym- IO w&in a truncation error and convergence is ob- metry, non-physical coupling, and non-conservation tained by increasing the size of the basis and hence the of energy and momentum [ 1S-191. In fact, it has number of coupled equations (NJ While considerable never been demonstrated that sudden models fit into progress has been madein the developmentof efficient any exactframework for solutionof the scattering algonthmsfor the solunonof theseequations [3-71, problem.