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Magnetism, Magnetic Properties, Magnetochemistry
Magnetism, Magnetic Properties, Magnetochemistry 1 Magnetism All matter is electronic Positive/negative charges - bound by Coulombic forces Result of electric field E between charges, electric dipole Electric and magnetic fields = the electromagnetic interaction (Oersted, Maxwell) Electric field = electric +/ charges, electric dipole Magnetic field ??No source?? No magnetic charges, N-S No magnetic monopole Magnetic field = motion of electric charges (electric current, atomic motions) Magnetic dipole – magnetic moment = i A [A m2] 2 Electromagnetic Fields 3 Magnetism Magnetic field = motion of electric charges • Macro - electric current • Micro - spin + orbital momentum Ampère 1822 Poisson model Magnetic dipole – magnetic (dipole) moment [A m2] i A 4 Ampere model Magnetism Microscopic explanation of source of magnetism = Fundamental quantum magnets Unpaired electrons = spins (Bohr 1913) Atomic building blocks (protons, neutrons and electrons = fermions) possess an intrinsic magnetic moment Relativistic quantum theory (P. Dirac 1928) SPIN (quantum property ~ rotation of charged particles) Spin (½ for all fermions) gives rise to a magnetic moment 5 Atomic Motions of Electric Charges The origins for the magnetic moment of a free atom Motions of Electric Charges: 1) The spins of the electrons S. Unpaired spins give a paramagnetic contribution. Paired spins give a diamagnetic contribution. 2) The orbital angular momentum L of the electrons about the nucleus, degenerate orbitals, paramagnetic contribution. The change in the orbital moment -
Quantum Spectropolarimetry and the Sun's Hidden
QUANTUM SPECTROPOLARIMETRY AND THE SUN’S HIDDEN MAGNETISM Javier Trujillo Bueno∗ Instituto de Astrof´ısica de Canarias; 38205 La Laguna; Tenerife; Spain ABSTRACT The dynamic jets that we call spicules. • The magnetic fields that confine the plasma of solar • Solar physicists can now proclaim with confidence that prominences. with the Zeeman effect and the available telescopes we The magnetism of the solar transition region and can see only 1% of the complex magnetism of the Sun. • corona. This is indeed∼regrettable because many of the key prob- lems of solar and stellar physics, such as the magnetic coupling to the outer atmosphere and the coronal heat- Unfortunately, our empirical knowledge of the Sun’s hid- ing, will only be solved after deciphering how signifi- den magnetism is still very primitive, especially concern- cant is the small-scale magnetic activity of the remain- ing the outer solar atmosphere (chromosphere, transition ing 99%. The first part of this paper1 presents a gentle ∼ region and corona). This is very regrettable because many introduction to Quantum Spectropolarimetry, emphasiz- of the physical challenges of solar and stellar physics ing the importance of developing reliable diagnostic tools arise precisely from magnetic processes taking place in that take proper account of the Paschen-Back effect, scat- such outer regions. tering polarization and the Hanle effect. The second part of the article highlights how the application of quantum One option to improve the situation is to work hard to spectropolarimetry to solar physics has the potential to achieve a powerful (European?) ground-based solar fa- revolutionize our empirical understanding of the Sun’s cility optimized for high-resolution spectropolarimetric hidden magnetism. -
Dirac Notation Frank Rioux
Elements of Dirac Notation Frank Rioux In the early days of quantum theory, P. A. M. (Paul Adrian Maurice) Dirac created a powerful and concise formalism for it which is now referred to as Dirac notation or bra-ket (bracket ) notation. Two major mathematical traditions emerged in quantum mechanics: Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. These distinctly different computational approaches to quantum theory are formally equivalent, each with its particular strengths in certain applications. Heisenberg’s variation, as its name suggests, is based matrix and vector algebra, while Schrödinger’s approach requires integral and differential calculus. Dirac’s notation can be used in a first step in which the quantum mechanical calculation is described or set up. After this is done, one chooses either matrix or wave mechanics to complete the calculation, depending on which method is computationally the most expedient. Kets, Bras, and Bra-Ket Pairs In Dirac’s notation what is known is put in a ket, . So, for example, p expresses the fact that a particle has momentum p. It could also be more explicit: p = 2 , the particle has momentum equal to 2; x = 1.23 , the particle has position 1.23. Ψ represents a system in the state Q and is therefore called the state vector. The ket can also be interpreted as the initial state in some transition or event. The bra represents the final state or the language in which you wish to express the content of the ket . For example,x =Ψ.25 is the probability amplitude that a particle in state Q will be found at position x = .25. -
Dirac Equation - Wikipedia
Dirac equation - Wikipedia https://en.wikipedia.org/wiki/Dirac_equation Dirac equation From Wikipedia, the free encyclopedia In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it 1 describes all spin-2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity,[1] and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. -
Quantum Trajectories: Real Or Surreal?
entropy Article Quantum Trajectories: Real or Surreal? Basil J. Hiley * and Peter Van Reeth * Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK * Correspondence: [email protected] (B.J.H.); [email protected] (P.V.R.) Received: 8 April 2018; Accepted: 2 May 2018; Published: 8 May 2018 Abstract: The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls for a re-examination of the meaning of “quantum trajectories”. We will review the arguments that have been assumed to have established that a trajectory has no meaning in the context of quantum mechanics. We show that the conclusion that the Bohm trajectories should be called “surreal” because they are at “variance with the actual observed track” of a particle is wrong as it is based on a false argument. We also present the results of a numerical investigation of a double Stern-Gerlach experiment which shows clearly the role of the spin within the Bohm formalism and discuss situations where the appearance of the quantum potential is open to direct experimental exploration. Keywords: Stern-Gerlach; trajectories; spin 1. Introduction The recent claims to have observed “photon trajectories” [1–3] calls for a re-examination of what we precisely mean by a “particle trajectory” in the quantum domain. Mahler et al. [2] applied the Bohm approach [4] based on the non-relativistic Schrödinger equation to interpret their results, claiming their empirical evidence supported this approach producing “trajectories” remarkably similar to those presented in Philippidis, Dewdney and Hiley [5]. However, the Schrödinger equation cannot be applied to photons because photons have zero rest mass and are relativistic “particles” which must be treated differently. -
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. -
Perturbation Theory and Exact Solutions
PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?. -
Chapter 5 Angular Momentum and Spin
Chapter 5 Angular Momentum and Spin I think you and Uhlenbeck have been very lucky to get your spinning electron published and talked about before Pauli heard of it. It appears that more than a year ago Kronig believed in the spinning electron and worked out something; the first person he showed it to was Pauli. Pauli rediculed the whole thing so much that the first person became also the last ... – Thompson (in a letter to Goudsmit) The first experiment that is often mentioned in the context of the electron’s spin and magnetic moment is the Einstein–de Haas experiment. It was designed to test Amp`ere’s idea that magnetism is caused by “molecular currents”. Such circular currents, while generating a magnetic field, would also contribute to the angular momentum of a ferromagnet. Therefore a change in the direction of the magnetization induced by an external field has to lead to a small rotation of the material in order to preserve the total angular momentum. For a quantitative understanding of the effect we consider a charged particle of mass m and charge q rotating with velocity v on a circle of radius r. Since the particle passes through its orbit v/(2πr) times per second the resulting current I = qv/(2πr), which encircles an area A = r2π, generates a magnetic dipole moment µ = IA/c, qv IA qv r2π qvr q q I = µ = = = = L = γL, γ = , (5.1) 2πr ⇒ c 2πr c 2c 2mc 2mc where L~ = m~r ~v is the angular momentum. Now the essential observation is that the × gyromagnetic ratio γ = µ/L is independent of the radius of the motion. -
The Zeeman Effect
THE ZEEMAN EFFECT OBJECT: To measure how an applied magnetic field affects the optical emission spectra of mercury vapor and neon. The results are compared with the expectations derived from the vector model for the addition of atomic angular momenta. A value of the electron charge to mass ratio, e/m, is derived from the data. Background: Electrons in atoms can be characterized by a unique set of discrete energy states. When excited through heating or electron bombardment in a discharge tube, the atom is excited to a level above the ground state. When returning to a lower energy state, it emits the extra energy as a photon whose energy corresponds to the difference in energy between the two states. The emitted light forms a discrete spectrum, reflecting the quantized nature of the energy levels. In the presence of a magnetic field, these energy levels can shift. This effect is known as the Zeeman effect. Zeeman discovered the effect in 1896 and obtained the charge to mass ratio e/m of the electron (one year before Thompson’s measurement) by measuring the spectral line broadening of a sodium discharge tube in a magnetic field. At the time neither the existence of the electron nor nucleus were known. Quantum mechanics was not to be invented for another decade. Zeeman’s colleague, Lorentz, was able to explain the observation by postulating the existence of a moving “corpuscular charge” that radiates electromagnetic waves. (See supplementary reading on the discovery) Qualitatively the Zeeman effect can be understood as follows. In an atomic energy state, an electron orbits around the nucleus of the atom and has a magnetic dipole moment associated with its angular momentum. -
Type of Presentation: Poster IT-16-P-3287 Electron Vortex Beam
Type of presentation: Poster IT-16-P-3287 Electron vortex beam diffraction via multislice solutions of the Pauli equation Edström A.1, Rusz J.1 1Department of Physics and Astronomy, Uppsala University Email of the presenting author: [email protected] Electron magnetic circular dichroism (EMCD) has gained plenty of attention as a possible route to high resolution measurements of, for example, magnetic properties of matter via electron microscopy. However, certain issues, such as low signal-to-noise ratio, have been problematic to the applicability. In recent years, electron vortex beams\cite{Uchida2010,Verbeeck2010}, i.e. electron beams which carry orbital angular momentum and are described by wavefunctions with a phase winding, have attracted interest as potential alternative way of measuring EMCD signals. Recent work has shown that vortex beams can be produced with a large orbital moment in the order of l = 100 [6, 7]. Huge orbital moments might introduce new effects from magnetic interactions such as spin-orbit coupling. The multislice method[2] provides a powerful computational tool for theoretical studies of electron microscopy. However, the method traditionally relies on the conventional Schrödinger equation which neglects relativistic effects such as spin-orbit coupling. Traditional multislice methods could therefore be inadequate in studying the diffraction of vortex beams with large orbital angular momentum. Relativistic multislice simulations have previously been done with a negligible difference to non-relativistic simulations[4], but vortex beams have not been considered in such work. In this work, we derive a new multislice approach based on the Pauli equation, Eq. 1, where q = −e is the electron charge, m = γm0 is the relativistically corrected mass, p = −i ∇ is the momentum operator, B = ∇ × A is the magnetic flux density while A is the vector potential and σ = (σx , σy , σz ) contains the Pauli matrices. -
Chapter 5 the Dirac Formalism and Hilbert Spaces
Chapter 5 The Dirac Formalism and Hilbert Spaces In the last chapter we introduced quantum mechanics using wave functions defined in position space. We identified the Fourier transform of the wave function in position space as a wave function in the wave vector or momen tum space. Expectation values of operators that represent observables of the system can be computed using either representation of the wavefunc tion. Obviously, the physics must be independent whether represented in position or wave number space. P.A.M. Dirac was the first to introduce a representation-free notation for the quantum mechanical state of the system and operators representing physical observables. He realized that quantum mechanical expectation values could be rewritten. For example the expected value of the Hamiltonian can be expressed as ψ∗ (x) Hop ψ (x) dx = ψ Hop ψ , (5.1) h | | i Z = ψ ϕ , (5.2) h | i with ϕ = Hop ψ . (5.3) | i | i Here, ψ and ϕ are vectors in a Hilbert-Space, which is yet to be defined. For exampl| i e, c|omi plex functions of one variable, ψ(x), that are square inte 241 242 CHAPTER 5. THE DIRAC FORMALISM AND HILBERT SPACES grable, i.e. ψ∗ (x) ψ (x) dx < , (5.4) ∞ Z 2 formt heHilbert-Spaceofsquareintegrablefunctionsdenoteda s L. In Dirac notation this is ψ∗ (x) ψ (x) dx = ψ ψ . (5.5) h | i Z Orthogonality relations can be rewritten as ψ∗ (x) ψ (x) dx = ψ ψ = δmn. (5.6) m n h m| ni Z As see aboveexpressions look likeabrackethecalledthe vector ψn aket vector and ψ a bra-vector. -
Relativistic Quantum Mechanics 1
Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4).