DOCSLIB.ORG

Understanding Electron Spin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Understanding Electron Spin Understanding Electron Spin Jan C. A. Boeyens University of the Witwatersrand, P.O. Wits, 2050 Johannesburg, South Africa Electron spin is of obvious importance in chemistry. It is and in the same manner therefore surprising that even textbooks with "Quantum" titles often make no attempt to provide an intelligible ac- [iYi*]= LG. count thereof. It is more fashionable to introduce the con- A A cept as a mystery with its origin in either relativity or deep [L,L,] = i*iy quantum theory. The operator for the square angular momentum is con- This state of affairs is only partially explained by the fact structed from that the Schrodinger equation was formulated (1)before nnnn the recognition of electron spin (2, 3) because several pro- L~=L:+L;+L,~ posals outlining a physical basis of spin were published Now, the angular-dependent part of the central-field soon after (4, 5).The overriding consideration seems to be Schrodinger wave function, in spherical polar coordinates, that a consistent mathematical description of the two-level is an eigenfunct*n of 2'. spin system emerged for the first time in Dirac's relativis- tic wave equation (6).Thereby spin could be described as a ~~y.a(e,v)= e(e + i)t~~~(e,,~) relativistic effect without further elucidation. e=o,i, ..(n-1) S~inand Angular- Momentum In classical mechanics angular momentum is calculated as the vector ~roductof generalized coordinates and mo- A - The eigenvalues ofL, are (2t 1)-fold degenerate. States menta. + with e = 0 should have zero angular momentum, contrary e=qxp to experiment. Hydrogen s states (e = 0) show fine struc- which in component form reduces to ture in the form of doublet levels that represent an addi- tional degeneracy of 2s + 1= 2. This remarkable state of affairs implies that, whereas classical angular momentum is a con erved quantity, the observable quantity associated with 1 is not. The quan- tum-mechanical conserved quantity, implied by the rota- To convert this into a quantum-mechanical statement tional symmetry of space, is therefore associated with an- other operator (71, the momentum is represented by an operator A n n Si= -iRV J=L+S where the commutators are defined by operating on some f,for example, A A stc., forL ,but nowS cannot be expressed in terms of$ and Operating on this with L, gives q operators. However, the following set of 2 x 2 matrices (Pauli matrices) has the required property. Likewise ~. Using A The quantity corresponding to the eigenvalues of S,, s = *l/zh is known as the spin of the system. etc., by subtraction one has Quantum Mechanics of Spin It follows from the foregoing that quantum theory in the traditional Schrodinger formulation is not complete. On the other hand, spin appears in Dirac's equation not be- This defines the commutator cause it is relativistic, but because the wave function has the correct transformation properties. Instead of deriving a relativistic equation from the en- ergy-momentum expression, 412 Journal of Chemical Education obeys the Galilean transformation x'+Rx+ut+a by substituting a E + iti- t'+t+b at The correct Galilean invariance can he ensured by fol- p -, itiV lowing the same linearization procedure as Dirac. This means finding the linear classical form that yields the cor- to yield rect quantum formulation, that is, -ti2 -a2w = ti2c2~2v+ m2c4~ at2 (AE + B.p+ c)'~I= 0 Dirac wrote (A'(E)' + 2WEp)+ 2AC(E)+ 2BCY.p) + B'$) + c2)= 0 iti-a~ = Hv at To match eq 1this requires A2 = AB = BC = C2 = 0 and To ensure that space and time dimensions are treated in AC + CA= 2m with B2 = 1. Because B is a vector, we can the same wav. as reauired bv-. s~ecial relativity, it would not clearly formulate the required condition in terms of Pauli he acceptabcto sudstitute and other matrices as Instead, use a linearized square mot, which formally has the form1 H = -ea.p - pmc 2 and solve for a and p. In component form one has 2 2 2 24 (a#, + a& +a& + be2)=P, +py+p2 + m c and the wave function as a column vector, This condition cannot be satisfied unless the coefficients a and p are treated as anticommuting matrices rather than complex numbers. This means that Thus. ((AC + CA)E + B~~~)= 0 The linearized Schrodinger equation in this form is The simplest solution is possible in terms of 4 x 4 matri- (AE+B.p+C)ry=O ces, that is, that is, where 0;s are the same Pauli matrices introduced above. This requires the wave function to be formulated as a four-dimensional vector, usually written as E'P + up)^ = 0 Because o2= 1, it follows that The two-component vectors q and x are called spinors. that is, Exactly the same procedures can be followed with the Schrodinger equation (8).In its nonrelativistic form it de- rives from the classical expression and likewise for q. 2 Both q and therefore satisfy the Schrodinger equation E-~=o x 2m with the same eigenvalue E = p2/2m. by introducing the quantum operators To describe an electron in an electromagnetic field repre- sented by a scalar potential V, and vector potential A, the quantum operators become without demonstrating that the form into 'This ensures that both time and space derivatives appear in first order. Volume 72 Number 5 May 1995 413 An Absolute Direction This conservation law can be stated in another way, in terms of a nonobservable (111, which in this case is an ab- solute spatial direction. Whenever an absolute direction is observed, the conservation no longer holds. This could mean that in the quantum domain, space cannot be as- sumed to be rotationally symmetrical and the special di- rection corresponds to the observed spin alignment. Simple Electrodynamic Effect Alternatively it could be viewed as a simple electrody- namic effect like the instantaneous magnetic interaction where between two like charges approaching at right angles. By the right-hand rule the forces between the charges are, as V(iwA)=VxA=eurlA=B shown in the figure, equal but not opposite as required to where B is the magnetic field. conserve momentum. The missing momentum has been The equation reduces to transferred to the field, and only when the field momen- tum is added to the mechanical momentum of the charges does conservation of momentum prevail (12). Similar considerations apply to angular momentum (13). In this instance conservation requires that the field ac- which corresponds to an equation ascribed to Pauli (9)and quires angular momentum, in addition to the mechanical obtained by empirically adding an operator to the Schrod- angular momentum that a particle may have. In the case inger Hamiltonian to account for electron spin. of a quantum-mechanical particle this statement is The importance of this equation lies in the term that re- equivalent to the suggestion (5, 14) that spin should be re- flects an intrinsic magnetic moment garded as a circulating flow of momentum density in the electron field, of the same kind as carried by the fields of a fie circularly polarized electromagnetic wave. This means p=- 2mc that spin is not associated with the internal structure of the electron, but rather with the structure of its wave field. which is exactly as required to explain the fine structure of The HamiltonJacobi Equation electronic soectra in terms of a s~inaneular - momentum of 012 and a ~Lndhfactor of 2. Both of the foregoing interpretations imply that an elec- tron wave function has more than mathematical signifi- The Physical Basis of Spin cance and refers to a physically real wave as proposed by Conservation of Angular Momentum De Broglie (15) and later more fully developed by Bohm (16)and others (17)in terms of the quantum potential (16) The two-valuedness of electron spin reflected here by the and the quantum torque (17). To understand these con- matrix nature of o appears to be a strict quantum property, cepts the spinor wave function is written in terms of the but this has never been demonstrated conclusively. This Euler angles (18), question probably cannot be addressed without examining the conservation of angular momentum in more detail. It occurs as a consequence of the isotropy or spherical rota- tional symmetry of space. This is easily demonstrated (10) and substituted into the Pauli equation (see above), to pro- for rotation around an axis (z) in the x,y plane, at a dis- duct a generalized HamiltonJacobi HJ equation tance r=(.2+y2)' from the origin. The angular momentum with respect to The spin vector is the origin is fi. M=m(~-yi) S = plnE sin yr, sin 0 cosyr, cos 8) and the time rate of change follows as The HJ equation differs from the classical equation for a spinning particle by two extra terms, the usual quantum potential, Q=- fi2v2~ Because mj; and miare components of force, 2mR and a spin-dependent quantum potential, fiZ Q. = -+(VO)~8m t sin2 8 (v~)') Both of these terms are inversely proportional to particle mass, which is thereby identified as the parameter that distinguishes between quantum and classical systems. Thus, In terms of the spin vector, 414 Journal of Chemical Education .
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.
    [Show full text]
  • Quantum Field Theory*
    Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
    [Show full text]
  • 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.
    [Show full text]
  • 5 the Dirac Equation and Spinors
    5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory.
    [Show full text]
  • 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 ?.
    [Show full text]
  • The Critical Casimir Effect in Model Physical Systems
    University of California Los Angeles The Critical Casimir Effect in Model Physical Systems A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Jonathan Ariel Bergknoff 2012 ⃝c Copyright by Jonathan Ariel Bergknoff 2012 Abstract of the Dissertation The Critical Casimir Effect in Model Physical Systems by Jonathan Ariel Bergknoff Doctor of Philosophy in Physics University of California, Los Angeles, 2012 Professor Joseph Rudnick, Chair The Casimir effect is an interaction between the boundaries of a finite system when fluctua- tions in that system correlate on length scales comparable to the system size. In particular, the critical Casimir effect is that which arises from the long-ranged thermal fluctuation of the order parameter in a system near criticality. Recent experiments on the Casimir force in binary liquids near critical points and 4He near the superfluid transition have redoubled theoretical interest in the topic. It is an unfortunate fact that exact models of the experi- mental systems are mathematically intractable in general. However, there is often insight to be gained by studying approximations and toy models, or doing numerical computations. In this work, we present a brief motivation and overview of the field, followed by explications of the O(2) model with twisted boundary conditions and the O(n ! 1) model with free boundary conditions. New results, both analytical and numerical, are presented. ii The dissertation of Jonathan Ariel Bergknoff is approved. Giovanni Zocchi Alex Levine Lincoln Chayes Joseph Rudnick, Committee Chair University of California, Los Angeles 2012 iii To my parents, Hugh and Esther Bergknoff iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 The Casimir Effect .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • A Peek Into Spin Physics
    A Peek into Spin Physics Dustin Keller University of Virginia Colloquium at Kent State Physics Outline ● What is Spin Physics ● How Do we Use It ● An Example Physics ● Instrumentation What is Spin Physics The Physics of exploiting spin - Spin in nuclear reactions - Nucleon helicity structure - 3D Structure of nucleons - Fundamental symmetries - Spin probes in beyond SM - Polarized Beams and Targets,... What is Spin Physics What is Spin Physics ● The Physics of exploiting spin : By using Polarized Observables Spin: The intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. The Spin quantum number is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. What is Spin Physics What Quantum Numbers? What is Spin Physics What Quantum Numbers? Internal or intrinsic quantum properties of particles, which can be used to uniquely characterize What is Spin Physics What Quantum Numbers? Internal or intrinsic quantum properties of particles, which can be used to uniquely characterize These numbers describe values of conserved quantities in the dynamics of a quantum system What is Spin Physics But a particle is not a sphere and spin is solely a quantum-mechanical phenomena What is Spin Physics Stern-Gerlach: If spin had continuous values like the classical picture we would see it What is Spin Physics Stern-Gerlach: Instead we see spin has only two values in the field with opposite directions: or spin-up and spin-down What is Spin Physics W. Pauli (1925)
    [Show full text]
  • 1. Dirac Equation for Spin ½ Particles 2
    Advanced Particle Physics: III. QED III. QED for “pedestrians” 1. Dirac equation for spin ½ particles 2. Quantum-Electrodynamics and Feynman rules 3. Fermion-fermion scattering 4. Higher orders Literature: F. Halzen, A.D. Martin, “Quarks and Leptons” O. Nachtmann, “Elementarteilchenphysik” 1. Dirac Equation for spin ½ particles ∂ Idea: Linear ansatz to obtain E → i a relativistic wave equation w/ “ E = p + m ” ∂t r linear time derivatives (remove pr = −i ∇ negative energy solutions). Eψ = (αr ⋅ pr + β ⋅ m)ψ ∂ ⎛ ∂ ∂ ∂ ⎞ ⎜ ⎟ i ψ = −i⎜α1 ψ +α2 ψ +α3 ψ ⎟ + β mψ ∂t ⎝ ∂x1 ∂x2 ∂x3 ⎠ Solutions should also satisfy the relativistic energy momentum relation: E 2ψ = (pr 2 + m2 )ψ (Klein-Gordon Eq.) U. Uwer 1 Advanced Particle Physics: III. QED This is only the case if coefficients fulfill the relations: αiα j + α jαi = 2δij αi β + βα j = 0 β 2 = 1 Cannot be satisfied by scalar coefficients: Dirac proposed αi and β being 4×4 matrices working on 4 dim. vectors: ⎛ 0 σ ⎞ ⎛ 1 0 ⎞ σ are Pauli α = ⎜ i ⎟ and β = ⎜ ⎟ i 4×4 martices: i ⎜ ⎟ ⎜ ⎟ matrices ⎝σ i 0 ⎠ ⎝0 − 1⎠ ⎛ψ ⎞ ⎜ 1 ⎟ ⎜ψ ⎟ ψ = 2 ⎜ψ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ψ 4 ⎠ ⎛ ∂ r ⎞ i⎜ β ψ + βαr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ ⎛ 0 ∂ r ⎞ i⎜γ ψ + γr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ 0 i where γ = β and γ = βα i , i = 1,2,3 Dirac Equation: µ i γ ∂µψ − mψ = 0 ⎛ψ ⎞ ⎜ 1 ⎟ ⎜ψ 2 ⎟ Solutions ψ describe spin ½ (anti) particles: ψ = ⎜ψ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ψ 4 ⎠ Extremely 4 ⎛ µ ∂ ⎞ compressed j = 1...4 : ∑ ⎜∑ i ⋅ (γ ) jk µ − mδ jk ⎟ψ k description k =1⎝ µ ∂x ⎠ U.
    [Show full text]
  • Quantum Computing in the De Broglie-Bohm Pilot-Wave Picture
    Quantum Computing in the de Broglie-Bohm Pilot-Wave Picture Philipp Roser September 2010 Blackett Laboratory Imperial College London Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London Supervisor: Dr. Antony Valentini Internal Supervisor: Prof. Jonathan Halliwell Abstract Much attention has been drawn to quantum computing and the expo- nential speed-up in computation the technology would be able to provide. Various claims have been made about what aspect of quantum mechan- ics causes this speed-up. Formulations of quantum computing have tradi- tionally been made in orthodox (Copenhagen) and sometimes many-worlds quantum mechanics. We will aim to understand quantum computing in terms of de Broglie-Bohm Pilot-Wave Theory by considering different sim- ple systems that may function as a basic quantum computer. We will provide a careful discussion of Pilot-Wave Theory and evaluate criticisms of the theory. We will assess claims regarding what causes the exponential speed-up in the light of our analysis and the fact that Pilot-Wave The- ory is perfectly able to account for the phenomena involved in quantum computing. I, Philipp Roser, hereby confirm that this dissertation is entirely my own work. Where other sources have been used, these have been clearly referenced. 1 Contents 1 Introduction 3 2 De Broglie-Bohm Pilot-Wave Theory 8 2.1 Motivation . 8 2.2 Two theories of pilot-waves . 9 2.3 The ensemble distribution and probability . 18 2.4 Measurement . 22 2.5 Spin . 26 2.6 Objections and open questions . 33 2.7 Pilot-Wave Theory, Many-Worlds and Many-Worlds in denial .
    [Show full text]