Relativistic Quantum Mechanics
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A Study of Fractional Schrödinger Equation-Composed Via Jumarie Fractional Derivative
A Study of Fractional Schrödinger Equation-composed via Jumarie fractional derivative Joydip Banerjee1, Uttam Ghosh2a , Susmita Sarkar2b and Shantanu Das3 Uttar Buincha Kajal Hari Primary school, Fulia, Nadia, West Bengal, India email- [email protected] 2Department of Applied Mathematics, University of Calcutta, Kolkata, India; 2aemail : [email protected] 2b email : [email protected] 3 Reactor Control Division BARC Mumbai India email : [email protected] Abstract One of the motivations for using fractional calculus in physical systems is due to fact that many times, in the space and time variables we are dealing which exhibit coarse-grained phenomena, meaning that infinitesimal quantities cannot be placed arbitrarily to zero-rather they are non-zero with a minimum length. Especially when we are dealing in microscopic to mesoscopic level of systems. Meaning if we denote x the point in space andt as point in time; then the differentials dx (and dt ) cannot be taken to limit zero, rather it has spread. A way to take this into account is to use infinitesimal quantities as ()Δx α (and ()Δt α ) with 01<α <, which for very-very small Δx (and Δt ); that is trending towards zero, these ‘fractional’ differentials are greater that Δx (and Δt ). That is()Δx α >Δx . This way defining the differentials-or rather fractional differentials makes us to use fractional derivatives in the study of dynamic systems. In fractional calculus the fractional order trigonometric functions play important role. The Mittag-Leffler function which plays important role in the field of fractional calculus; and the fractional order trigonometric functions are defined using this Mittag-Leffler function. -
Introductory Lectures on Quantum Field Theory
Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties. -
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. -
WEYL SPINORS and DIRAC's ELECTRON EQUATION C William O
WEYL SPINORS AND DIRAC’SELECTRON EQUATION c William O. Straub, PhD Pasadena, California March 17, 2005 I’ve been planning for some time now to provide a simplified write-up of Weyl’s seminal 1929 paper on gauge invariance. I’m still planning to do it, but since Weyl’s paper covers so much ground I thought I would first address a discovery that he made kind of in passing that (as far as I know) has nothing to do with gauge-invariant gravitation. It involves the mathematical objects known as spinors. Although Weyl did not invent spinors, I believe he was the first to explore them in the context of Dirac’srelativistic electron equation. However, without a doubt Weyl was the first person to investigate the consequences of zero mass in the Dirac equation and the implications this has on parity conservation. In doing so, Weyl unwittingly anticipated the existence of a particle that does not respect the preservation of parity, an unheard-of idea back in 1929 when parity conservation was a sacred cow. Following that, I will use this opportunity to derive the Dirac equation itself and talk a little about its role in particle spin. Those of you who have studied Dirac’s relativistic electron equation may know that the 4-component Dirac spinor is actually composed of two 2-component spinors that Weyl introduced to physics back in 1929. The Weyl spinors have unusual parity properties, and because of this Pauli was initially very critical of Weyl’sanalysis because it postulated massless fermions (neutrinos) that violated the then-cherished notion of parity conservation. -
L the Probability Current Or Flux
Atomic and Molecular Quantum Theory Course Number: C561 L The Probability Current or flux 1. Consider the time-dependent Schr¨odinger Equation and its complex conjugate: 2 ∂ h¯ ∂2 ıh¯ ψ(x, t)= Hψ(x, t)= − 2 + V ψ(x, t) (L.26) ∂t " 2m ∂x # 2 2 ∂ ∗ ∗ h¯ ∂ ∗ −ıh¯ ψ (x, t)= Hψ (x, t)= − 2 + V ψ (x, t) (L.27) ∂t " 2m ∂x # 2. Multiply Eq. (L.26) by ψ∗(x, t), andEq. (L.27) by ψ(x, t): ∂ ψ∗(x, t)ıh¯ ψ(x, t) = ψ∗(x, t)Hψ(x, t) ∂t 2 2 ∗ h¯ ∂ = ψ (x, t) − 2 + V ψ(x, t) (L.28) " 2m ∂x # ∂ −ψ(x, t)ıh¯ ψ∗(x, t) = ψ(x, t)Hψ∗(x, t) ∂t 2 2 h¯ ∂ ∗ = ψ(x, t) − 2 + V ψ (x, t) (L.29) " 2m ∂x # 3. Subtract the two equations to obtain (see that the terms involving V cancels out) 2 2 ∂ ∗ h¯ ∗ ∂ ıh¯ [ψ(x, t)ψ (x, t)] = − ψ (x, t) 2 ψ(x, t) ∂t 2m " ∂x 2 ∂ ∗ − ψ(x, t) 2 ψ (x, t) (L.30) ∂x # or ∂ h¯ ∂ ∂ ∂ ρ(x, t)= − ψ∗(x, t) ψ(x, t) − ψ(x, t) ψ∗(x, t) (L.31) ∂t 2mı ∂x " ∂x ∂x # 4. If we make the variable substitution: h¯ ∂ ∂ J = ψ∗(x, t) ψ(x, t) − ψ(x, t) ψ∗(x, t) 2mı " ∂x ∂x # h¯ ∂ = I ψ∗(x, t) ψ(x, t) (L.32) m " ∂x # (where I [···] represents the imaginary part of the quantity in brackets) we obtain ∂ ∂ ρ(x, t)= − J (L.33) ∂t ∂x or in three-dimensions ∂ ρ(x, t)+ ∇·J =0 (L.34) ∂t Chemistry, Indiana University L-37 c 2003, Srinivasan S. -
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). -
Charge Conjugation Symmetry
Charge Conjugation Symmetry In the previous set of notes we followed Dirac's original construction of positrons as holes in the electron's Dirac sea. But the modern point of view is rather different: The Dirac sea is experimentally undetectable | it's simply one of the aspects of the physical ? vacuum state | and the electrons and the positrons are simply two related particle species. Moreover, the electrons and the positrons have exactly the same mass but opposite electric charges. Many other particle species exist in similar particle-antiparticle pairs. The particle and the corresponding antiparticle have exactly the same mass but opposite electric charges, as well as other conserved charges such as the lepton number or the baryon number. Moreover, the strong and the electromagnetic interactions | but not the weak interactions | respect the change conjugation symmetry which turns particles into antiparticles and vice verse, C^ jparticle(p; s)i = jantiparticle(p; s)i ; C^ jantiparticle(p; s)i = jparticle(p; s)i ; (1) − + + − for example C^ e (p; s) = e (p; s) and C^ e (p; s) = e (p; s) . In light of this sym- metry, deciding which particle species is particle and which is antiparticle is a matter of convention. For example, we know that the charged pions π+ and π− are each other's an- tiparticles, but it's up to our choice whether we call the π+ mesons particles and the π− mesons antiparticles or the other way around. In the Hilbert space of the quantum field theory, the charge conjugation operator C^ is a unitary operator which squares to 1, thus C^ 2 = 1 =) C^ y = C^ −1 = C^:; (2) ? In condensed matter | say, in a piece of semiconductor | we may detect the filled electron states by making them interact with the outside world. -
7 Quantized Free Dirac Fields
7 Quantized Free Dirac Fields 7.1 The Dirac Equation and Quantum Field Theory The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section that a consistent description of this theory cannot be done outside the framework of (local) relativistic Quantum Field Theory. The Dirac Equation (i∂/ m)ψ =0 ψ¯(i∂/ + m) = 0 (1) − can be regarded as the equations of motion of a complex field ψ. Much as in the case of the scalar field, and also in close analogy to the theory of non-relativistic many particle systems discussed in the last chapter, the Dirac field is an operator which acts on a Fock space. We have already discussed that the Dirac equation also follows from a least-action-principle. Indeed the Lagrangian i µ = [ψ¯∂ψ/ (∂µψ¯)γ ψ] mψψ¯ ψ¯(i∂/ m)ψ (2) L 2 − − ≡ − has the Dirac equation for its equation of motion. Also, the momentum Πα(x) canonically conjugate to ψα(x) is ψ δ † Πα(x)= L = iψα (3) δ∂0ψα(x) Thus, they obey the equal-time Poisson Brackets ψ 3 ψα(~x), Π (~y) P B = iδαβδ (~x ~y) (4) { β } − Thus † 3 ψα(~x), ψ (~y) P B = δαβδ (~x ~y) (5) { β } − † In other words the field ψα and its adjoint ψα are a canonical pair. This result follows from the fact that the Dirac Lagrangian is first order in time derivatives. Notice that the field theory of non-relativistic many-particle systems (for both fermions on bosons) also has a Lagrangian which is first order in time derivatives. -
Photon Wave Mechanics: a De Broglie - Bohm Approach
PHOTON WAVE MECHANICS: A DE BROGLIE - BOHM APPROACH S. ESPOSITO Dipartimento di Scienze Fisiche, Universit`adi Napoli “Federico II” and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli Mostra d’Oltremare Pad. 20, I-80125 Napoli Italy e-mail: [email protected] Abstract. We compare the de Broglie - Bohm theory for non-relativistic, scalar matter particles with the Majorana-R¨omer theory of electrodynamics, pointing out the impressive common pecu- liarities and the role of the spin in both theories. A novel insight into photon wave mechanics is envisaged. 1. Introduction Modern Quantum Mechanics was born with the observation of Heisenberg [1] that in atomic (and subatomic) systems there are directly observable quantities, such as emission frequencies, intensities and so on, as well as non directly observable quantities such as, for example, the position coordinates of an electron in an atom at a given time instant. The later fruitful developments of the quantum formalism was then devoted to connect observable quantities between them without the use of a model, differently to what happened in the framework of old quantum mechanics where specific geometrical and mechanical models were investigated to deduce the values of the observable quantities from a substantially non observable underlying structure. We now know that quantum phenomena are completely described by a complex- valued state function ψ satisfying the Schr¨odinger equation. The probabilistic in- terpretation of it was first suggested by Born [2] and, in the light of Heisenberg uncertainty principle, is a pillar of quantum mechanics itself. All the known experiments show that the probabilistic interpretation of the wave function is indeed the correct one (see any textbook on quantum mechanics, for 2 S. -
Spin-1/2 Fermions in Quantum Field Theory
Spin-1/2 fermions in quantum field theory µ First, recall that 4-vectors transform under Lorentz transformations, Λ ν, as p′ µ = Λµ pν, where Λ SO(3,1) satisfies Λµ g Λρ = g .∗ A Lorentz ν ∈ ν µρ λ νλ transformation corresponds to a rotation by θ about an axis nˆ [θ~ θnˆ] and ≡ a boost, ζ~ = vˆ tanh−1 ~v , where ~v is the corresponding velocity. Under the | | same Lorentz transformation, a generic field transforms as: ′ ′ Φ (x )= MR(Λ)Φ(x) , where M exp iθ~·J~ iζ~·K~ are N N representation matrices of R ≡ − − × 1 1 the Lorentz group. Defining J~+ (J~ + iK~ ) and J~− (J~ iK~ ), ≡ 2 ≡ 2 − i j ijk k i j [J± , J±]= iǫ J± , [J± , J∓] = 0 . Thus, the representations of the Lorentz algebra are characterized by (j1,j2), 1 1 where the ji are half-integers. (0, 0) is a scalar and (2, 2) is a four-vector. Of 1 1 interest to us here are the spinor representations (2, 0) and (0, 2). ∗ In our conventions, gµν = diag(1 , −1 , −1 , −1). (1, 0): M = exp i θ~·~σ 1ζ~·~σ , butalso (M −1)T = iσ2M(iσ2)−1 2 −2 − 2 (0, 1): [M −1]† = exp i θ~·~σ + 1ζ~·~σ , butalso M ∗ = iσ2[M −1]†(iσ2)−1 2 −2 2 since (iσ2)~σ(iσ2)−1 = ~σ∗ = ~σT − − Transformation laws of 2-component fields ′ β ξα = Mα ξβ , ′ α −1 T α β ξ = [(M ) ] β ξ , ′† α˙ −1 † α˙ † β˙ ξ = [(M ) ] β˙ ξ , ˙ ξ′† =[M ∗] βξ† . -
Time Reversal Symmetry
Contents Contents i List of Figures ii List of Tables iii 7 Time Reversal 1 7.1 The Poincar´eGroup ........................................ 1 7.1.1 Space inversion and time-reversal ............................ 1 7.1.2 Representations of the Poincar´eLie algebra ...................... 2 7.1.3 Whither time-reversal? .................................. 3 7.2 Antilinearity : The Solution to All Our Problems ........................ 4 7.2.1 Properties of antilinear operators ............................ 4 7.2.2 Position and momentum eigenstates .......................... 6 7.2.3 Change of basis for time-reversal ............................ 7 7.2.4 Time reversal with spin .................................. 8 7.2.5 Kramers degeneracy ................................... 9 7.2.6 Externalfields ....................................... 9 7.3 Time Reversal and Point Group Symmetries .......................... 10 7.3.1 Complex conjugate representations ........................... 11 7.3.2 Generalized time-reversal ................................ 12 7.4 Consequences of Time-Reversal ................................. 13 i ii CONTENTS 7.4.1 Selection rules and time-reversal ............................ 13 7.4.2 Onsager reciprocity .................................... 14 7.5 Colorgroups ............................................ 16 7.5.1 Magnetic Bravais lattices and magnetic space groups ................. 19 7.5.2 Corepresentations of color groups ............................ 23 7.6 Appendix : The Foldy-Wouthuysen Transformation ..................... -
Electron Spin and Probability Current Density in Quantum Mechanics W
Electron spin and probability current density in quantum mechanics W. B. Hodgea) and S. V. Migirditch Department of Physics, Davidson College, Davidson, North Carolina 28035 W. C. Kerrb) Olin Physical Laboratory, Wake Forest University, Winston-Salem, North Carolina 27109 (Received 13 August 2013; accepted 26 February 2014) This paper analyzes how the existence of electron spin changes the equation for the probability current density in the quantum-mechanical continuity equation. A spinful electron moving in a potential energy field experiences the spin-orbit interaction, and that additional term in the time- dependent Schrodinger€ equation places an additional spin-dependent term in the probability current density. Further, making an analogy with classical magnetostatics hints that there may be an additional magnetization current contribution. This contribution seems not to be derivable from a non-relativistic time-dependent Schrodinger€ equation, but there is a procedure described in the quantum mechanics textbook by Landau and Lifschitz to obtain it. We utilize and extend their procedure to obtain this magnetization term, which also gives a second derivation of the spin-orbit term. We conclude with an evaluation of these terms for the ground state of the hydrogen atom with spin-orbit interaction. The magnetization contribution is generally the larger one, except very near an atomic nucleus. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4868094] I. INTRODUCTION occurs early in the course (and the textbook) when only the simplest form of the time-dependent Schrodinger€ equation has The probability interpretation of the wave function w of a been introduced.