DOCSLIB.ORG

Appendix a Time Reversal Invariance

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix a Time Reversal Invariance Appendix A Time Reversal Invariance Let us briefly discuss the time reversal property. In Chap. 4, we introduced the scattering and transfer matrices S and M. We have seen also that in a scattering process as the one shown in Fig. A.1, these matrices satisfy the following relations u ðx Þ u ðx Þ u ðx Þ u ðx Þ ol 1 ¼ S il 1 or 2 ¼ M il 1 ; ðA:1Þ uorðx2Þ uirðx2Þ uirðx2Þ uolðx1Þ the matrix S relates the incoming waves with the outgoing ones, while the matrix M connects the state vector at x1 with the state vector at x2. If we take the complex conjugate of the previous relations, we have à à à à uolðx1Þ Ã uilðx1Þ uorðx2Þ Ã uilðx1Þ Ã ¼ S à à ¼ M à : ðA:2Þ uorðx2Þ uirðx2Þ uirðx2Þ uolðx1Þ Let us now consider the time inversion operator T. The action of this operator changes the sign of time, thus the physical quantities depending linearly on time as the velocity and the wave number k ¼ p=h, change sign. Under the time inversion operation, particles that were moving to the right, move towards the left and vice versa. A wave function like eikx, under T, gets transformed into eÀikx. This is equivalent to complex conjugate. Therefore TuðxÞ¼uÃðxÞ: ðA:3Þ If the time inversion operator T acts on the system of Fig. A.1, we will have à à Tuilðx1Þ¼uilðx1Þ¼/olðx1Þ; Tuirðx2Þ¼uirðx2Þ¼/orðx2Þ; ðA:4Þ Ã Ã Tuolðx1Þ¼uolðx1Þ¼/ilðx1Þ; Tuorðx2Þ¼uorðx2Þ¼/irðx2Þ: ðA:5Þ As one can see in Fig. A.2, the wave functions that were incoming functions transform into outgoing functions and viceversa. Reversing time in the scattering system implies reversing time every where. This includes the interactions that define the scattering and transfer matrices. If the system remains invariant under time inversion, the matrices S and M should also remain invariant. Therefore, if a P. Pereyra, Fundamentals of Quantum Physics, Undergraduate Lecture Notes in Physics, 305 DOI: 10.1007/978-3-642-29378-8, Ó Springer-Verlag Berlin Heidelberg 2012 306 Appendix A: Time Reversal Invariance Fig. A.1 The incident wave V(x) ϕ ϕ or functions, and the reflected il and transmitted wave func- tions by a dispersion potential VðxÞ ϕ ϕ ol ir x1 x2 Fig. A.2 The incident wave functions, and the wave functions that are reflected and transmitted by the dis- persion potential VðxÞ system is time reversal invariant, we must have / ðx Þ / ðx Þ / ðx Þ / ðx Þ ol 1 ¼ S il 1 or 2 ¼ M il 1 : ðA:6Þ /orðx2Þ /irðx2Þ /irðx2Þ /olðx1Þ with S and M that we had in (A.1). These relations, written in terms of the wave functions u, take the form à à à à uilðx1Þ uolðx1Þ uirðx2Þ uolðx1Þ Ã ¼ S à à ¼ M à : ðA:7Þ uirðx2Þ uorðx2Þ uorðx2Þ uilðx1Þ If we multiply the first of these equations, from the left by SÀ1, and we observe that à à à à uirðx2Þ 01 uorðx2Þ uolðx1Þ 01 uilðx1Þ Ã ¼ à and à ¼ à ; uorðx2Þ 10 uirðx2Þ uilðx1Þ 10 uolðx1Þ ðA:8Þ it becomes clear, after comparing with (A.2), that 01 01 SÀ1 ¼ Sà and M ¼ MÃ: ðA:9Þ 10 10 For the transfer matrix, this means that dc aà bà ¼ : ðA:10Þ ba cà dà Therefore we conclude that, for systems that are time reversal invariant, the transfer matrices have the structure Appendix A: Time Reversal Invariance 307 ab M ¼ : ðA:11Þ bà aà Notice that this holds when the wave number k on the right side is the same as on the left side of the scattering region. If that is not the case, these structures may change but (A.9) still holds. Appendix B Laguerre’s Polynomials With respect to the Laguerre polynomials, there is in the literature such a great diversity of definitions and notations that we feel justified to include this appendix, and deduce some basic relations and properties. The polynomials that Laguerre 0 studied are the polynomials LmðxÞ¼Lm ðxÞ defined by 1 dm L ðxÞ¼ ex xmeÀx; ðB:1Þ m m! dxm or as the expansion coefficients of the generating function no xt X1 G ðt; xÞ¼exp ¼ð1 À tÞ L ðxÞtm; ðB:2Þ L t 1 m À m¼0 and they are solutions of the differential equation d2y 1 dy m þ À 1 þ y ¼ 0: ðB:3Þ dx2 x dx x The Laguerre polynomials of lower orders are L0ðxÞ¼1; 1 À Á L ðxÞ¼ 1 À x ; 1 1! 1 À Á L ðxÞ¼ 1 À 4x þ x2 ; 2 2! 1 À Á L ðxÞ¼ 6 À 18x þ 9x2 À x3 ; 3 3! 1 À Á L ðxÞ¼ 24 À 96x þ 72x2 À 16x3 þ x4 : 4 4! Related to these polynomials we have the generalized (or associated) Laguerre l polynomials LmÀlðxÞ defined by P. Pereyra, Fundamentals of Quantum Physics, Undergraduate Lecture Notes in Physics, 309 DOI: 10.1007/978-3-642-29378-8, Ó Springer-Verlag Berlin Heidelberg 2012 310 Appendix B: Laguerre’s Polynomials 1 dk dmþk dk LkðxÞ¼ðÀ1Þk ex xmþkeÀx ¼ðÀ1Þk L ðxÞ: ðB:4Þ m ðm þ kÞ! dxk dxmþk dxk mþk These polynomials are also the coefficients of a generating function, and one can easily verify that a particular solution of the differential equation d2y dy k þ 1 x mðm À kÞ x þðk þ 1 À 2mÞ þ n þ À þ y ¼ 0 ðB:5Þ dx2 dx 2 4 x is Àx=2 m k yn ¼ e x Ln: ðB:6Þ In this way, if m ¼ k=2, the function Àx=2 k=2 k yn ¼ e x Ln; ðB:7Þ is a solution of k þ 1 x k2 xy00 þ y0 þ n þ À À y ¼ 0; ðB:8Þ n n 2 4 4x n that can be written as d d k þ 1 x k2 x y þ n þ À À y ¼ 0: ðB:9Þ dx dx n 2 4 4x n k A very useful representation of LmðxÞ is the Rodrigues formula Xm 1 dm ðm þ kÞ! xr LkðxÞ¼ exxÀk xmþkeÀx ¼ ð1Þr : ðB:10Þ m m! dxm m r !r! r k ! r¼0 ð À Þ ð þ Þ Some generalized Laguerre polynomials are: 0 L0ðxÞ¼1; 0 1 L1ðxÞ¼1 À x; L1ðxÞ¼2 À x; 1À Á 1À Á 1À Á L0ðxÞ¼ 2À4xþ x2 ; L1ðxÞ¼ 6À 6xþ x2 ; L2ðxÞ¼ 12À 8xþ x2 : 2 2 2 2 2 2 Like other orthogonal polynomials, the Laguerre polynomials satisfy a three- term recurrence relation. The relation that the Laguerre polynomials satisfy is k k k ðn þ 1ÞLnþ1ðxÞþ ðx À k À 2n À 1ÞLnðxÞþ ðn þ kÞLnÀ1ðxÞ¼ 0 n ¼ 1; 2; ... ðB:11Þ This recurrence relation can also be written as Appendix B: Laguerre’s Polynomials 311 k k k nLnðxÞþ ðx À k À 2n þ 1ÞLnÀ1ðxÞþ ðn þ k À 1ÞLnÀ2ðxÞ¼ 0 n ¼ 1; 2; ... ðB:12Þ With the help of these definitions and relations we will obtain two very useful results for the evaluation of normalization integrals and the calculation of expectation values for the Hydrogen atom. Orthogonality of Laguerre’s Polynomials If we multiply equation (B.9)byym, subtract the same equation (with the indices n and m interchanged), and integrate from 0 to 1, we have Z Z 1 1 dy 0 dy 0 ym xyn À yn xym dx þðn À mÞ ymyndx ¼ 0; 0 dx dx Z 0 1 ðB:13Þ 0 0 1 ðymxyn À ynxymÞ 0 þðn À mÞ ymyndx ¼ 0: 0 Àx=2 k=2 k k 1 x Àk dn nþk Àx As yn ¼ e x Ln, and as LnðxÞ¼n ! e x dxn x e is a polynomial of degree n, the first term of the previous equation is zero at x ¼ 0ifk [ À 1. Thus Z 1 Àx k k k ðn À mÞ e x LnLmdx ¼ 0: ðB:14Þ 0 From here we conclude that Z 1 Àx k k k e x LnLmdx ¼ 0 if n 6¼ m: 0 Normalization of Laguerre’s Polynomials To obtain the normalization constant of the Laguerre polynomials we need the recurrence relations (B.11) and (B.12). If we consider the relation (B.11) k k multiplied by LnÀ1ðxÞ, subtract (B.12) multiplied by LnðxÞ, multiply the result by eÀxxk and integrate from 0 an 1 (using the orthogonality condition), we will obtain Z Z 1 Âà 1 Âà Àx k k 2 n þ k Àx k k 2 e x LnðxÞ dx ¼ e x LnÀ1ðxÞ dx: ðB:16Þ 0 n 0 One can repeat this relation n À 1 times to obtain 312 Appendix B: Laguerre’s Polynomials Z Z 1 Âà 1 Âà Àx k k 2 n þ k n À 1 þ k 2 þ k Àx k k 2 e x LnðxÞ dx ¼ ÁÁÁ e x L1ðxÞ dx: ðB:17Þ 0 n n À 1 2 0 k Since L1ðxÞ¼1 þ k À x it is possible to show that Z 1 Âà Àx k k 2 e x L1ðxÞ dx ¼ðk þ 1Þ! ðB:18Þ 0 Therefore Z 1 Âà Àx k k 2 ðn þ kÞ! e x LnðxÞ dx ¼ for n ¼ 2; 3; ... ðB:19Þ 0 n! It is possible to explicitly show that this formula is also valid for n ¼ 0 and n ¼ 1. Thus Z 1 Âà Àx k k 2 ðn þ kÞ! e x LnðxÞ dx ¼ for n ¼ 0; 1; 2; 3; ... ðB:20Þ 0 n! This integral and the orthogonality condition are summarized in the following equation Z 1 m! k Àx k k x e LmðxÞLm0 ðxÞdx ¼ dm;m0 : ðB:21Þ ðm þ kÞ! 0 Finally, if we consider the recurrence relation (B.11), we multiply it by k Àx k x e LnðxÞ and integrate from 0 to 1 we get: Z 1 kþ1 Àx k k ðn þ kÞ! x e LnðxÞLnðxÞdx ¼ðk þ 2n þ 1Þ : ðB:22Þ 0 n! References 1.
Load more

Recommended publications
  • LYCEN 7112 Février 1972 Notice 92Ft on the Generalized Exchange
    LYCEN 7112 Février 1972 notice 92ft On the Generalized Exchange Operators for SU(n) A. Partensky Institut de Physique Nucléaire, Université Claude Bernard de Lyon Institut National de Physique Nucléaire et de Physique des Particules 43, Bd du 11 novembre 1918, 69-Villeurbanne, France Abstract - By following the work of Biedenharn we have redefined the k-particles generalized exchange operators (g.e.o) and studied their properties. By a straightforward but cur-jersome calculation we have derived the expression, in terms of the SU(n) Casimir operators, of the ?., 3 or 4 particles g.e.o. acting on the A-particles states which span an irreducible representation of the grown SU(n). A striking and interesting result is that the eigenvalues of each of these g.e.o. do not depend on the n in SU(n) but only on the Young pattern associated with the irreducible representation considered. For a given g.e.o. , the eigenvalues corresponding to two conjugate Young patterns are the same except for the sign which depends on the parity of the g.e.o. considered. Three Appendices deal with some related problems and, more specifically, Appendix C 2 contains a method of obtaining the eigenvalues of Gel'fand invariants in a new and simple way. 1 . Introduction The interest attributed by physicists to the Lie groups has 3-4 been continuously growing since the pionner works of Weyl , Wigner and Racah • In particular, unitary groups have been used successfully in various domains of physics, although the physical reasons for their introduction have not always been completely explained.
    [Show full text]
  • 5.1 Two-Particle Systems
    5.1 Two-Particle Systems We encountered a two-particle system in dealing with the addition of angular momentum. Let's treat such systems in a more formal way. The w.f. for a two-particle system must depend on the spatial coordinates of both particles as @Ψ well as t: Ψ(r1; r2; t), satisfying i~ @t = HΨ, ~2 2 ~2 2 where H = + V (r1; r2; t), −2m1r1 − 2m2r2 and d3r d3r Ψ(r ; r ; t) 2 = 1. 1 2 j 1 2 j R Iff V is independent of time, then we can separate the time and spatial variables, obtaining Ψ(r1; r2; t) = (r1; r2) exp( iEt=~), − where E is the total energy of the system. Let us now make a very fundamental assumption: that each particle occupies a one-particle e.s. [Note that this is often a poor approximation for the true many-body w.f.] The joint e.f. can then be written as the product of two one-particle e.f.'s: (r1; r2) = a(r1) b(r2). Suppose furthermore that the two particles are indistinguishable. Then, the above w.f. is not really adequate since you can't actually tell whether it's particle 1 in state a or particle 2. This indeterminacy is correctly reflected if we replace the above w.f. by (r ; r ) = a(r ) (r ) (r ) a(r ). 1 2 1 b 2 b 1 2 The `plus-or-minus' sign reflects that there are two distinct ways to accomplish this. Thus we are naturally led to consider two kinds of identical particles, which we have come to call `bosons' (+) and `fermions' ( ).
    [Show full text]
  • Chapter 4 MANY PARTICLE SYSTEMS
    Chapter 4 MANY PARTICLE SYSTEMS The postulates of quantum mechanics outlined in previous chapters include no restrictions as to the kind of systems to which they are intended to apply. Thus, although we have considered numerous examples drawn from the quantum mechanics of a single particle, the postulates themselves are intended to apply to all quantum systems, including those containing more than one and possibly very many particles. Thus, the only real obstacle to our immediate application of the postulates to a system of many (possibly interacting) particles is that we have till now avoided the question of what the linear vector space, the state vector, and the operators of a many-particle quantum mechanical system look like. The construction of such a space turns out to be fairly straightforward, but it involves the forming a certain kind of methematical product of di¤erent linear vector spaces, referred to as a direct or tensor product. Indeed, the basic principle underlying the construction of the state spaces of many-particle quantum mechanical systems can be succinctly stated as follows: The state vector à of a system of N particles is an element of the direct product space j i S(N) = S(1) S(2) S(N) ­ ­ ¢¢¢­ formed from the N single-particle spaces associated with each particle. To understand this principle we need to explore the structure of such direct prod- uct spaces. This exploration forms the focus of the next section, after which we will return to the subject of many particle quantum mechanical systems. 4.1 The Direct Product of Linear Vector Spaces Let S1 and S2 be two independent quantum mechanical state spaces, of dimension N1 and N2, respectively (either or both of which may be in…nite).
    [Show full text]
  • Phase Transitions and the Casimir Effect in Neutron Stars
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 12-2017 Phase Transitions and the Casimir Effect in Neutron Stars William Patrick Moffitt University of Tennessee, Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Other Physics Commons Recommended Citation Moffitt, William Patrick, "Phase Transitions and the Casimir Effect in Neutron Stars. " Master's Thesis, University of Tennessee, 2017. https://trace.tennessee.edu/utk_gradthes/4956 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by William Patrick Moffitt entitled "Phaser T ansitions and the Casimir Effect in Neutron Stars." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Physics. Andrew W. Steiner, Major Professor We have read this thesis and recommend its acceptance: Marianne Breinig, Steve Johnston Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Phase Transitions and the Casimir Effect in Neutron Stars A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville William Patrick Moffitt December 2017 Abstract What lies at the core of a neutron star is still a highly debated topic, with both the composition and the physical interactions in question.
    [Show full text]
  • Time-Reversal Symmetry Breaking Abelian Chiral Spin Liquid in Mott Phases of Three-Component Fermions on the Triangular Lattice
    PHYSICAL REVIEW RESEARCH 2, 023098 (2020) Time-reversal symmetry breaking Abelian chiral spin liquid in Mott phases of three-component fermions on the triangular lattice C. Boos,1,2 C. J. Ganahl,3 M. Lajkó ,2 P. Nataf ,4 A. M. Läuchli ,3 K. Penc ,5 K. P. Schmidt,1 and F. Mila2 1Institute for Theoretical Physics, FAU Erlangen-Nürnberg, D-91058 Erlangen, Germany 2Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH 1015 Lausanne, Switzerland 3Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria 4Laboratoire de Physique et Modélisation des Milieux Condensés, Université Grenoble Alpes and CNRS, 25 avenue des Martyrs, 38042 Grenoble, France 5Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, H-1525 Budapest, P.O.B. 49, Hungary (Received 6 December 2019; revised manuscript received 20 March 2020; accepted 20 March 2020; published 29 April 2020) We provide numerical evidence in favor of spontaneous chiral symmetry breaking and the concomitant appearance of an Abelian chiral spin liquid for three-component fermions on the triangular lattice described by an SU(3) symmetric Hubbard model with hopping amplitude −t (t > 0) and on-site interaction U. This chiral phase is stabilized in the Mott phase with one particle per site in the presence of a uniform π flux per plaquette, and in the Mott phase with two particles per site without any flux. Our approach relies on effective spin models derived in the strong-coupling limit in powers of t/U for general SU(N ) and arbitrary uniform charge flux per plaquette, which are subsequently studied using exact diagonalizations and variational Monte Carlo simulations for N = 3, as well as exact diagonalizations of the SU(3) Hubbard model on small clusters.
    [Show full text]
  • 2 Quantum Theory of Spin Waves
    2 Quantum Theory of Spin Waves In Chapter 1, we discussed the angular momenta and magnetic moments of individual atoms and ions. When these atoms or ions are constituents of a solid, it is important to take into consideration the ways in which the angular momenta on different sites interact with one another. For simplicity, we will restrict our attention to the case when the angular momentum on each site is entirely due to spin. The elementary excitations of coupled spin systems in solids are called spin waves. In this chapter, we will introduce the quantum theory of these excita- tions at low temperatures. The two primary interaction mechanisms for spins are magnetic dipole–dipole coupling and a mechanism of quantum mechanical origin referred to as the exchange interaction. The dipolar interactions are of importance when the spin wavelength is very long compared to the spacing between spins, and the exchange interaction dominates when the spacing be- tween spins becomes significant on the scale of a wavelength. In this chapter, we focus on exchange-dominated spin waves, while dipolar spin waves are the primary topic of subsequent chapters. We begin this chapter with a quantum mechanical treatment of a sin- gle electron in a uniform field and follow it with the derivations of Zeeman energy and Larmor precession. We then consider one of the simplest exchange- coupled spin systems, molecular hydrogen. Exchange plays a crucial role in the existence of ordered spin systems. The ground state of H2 is a two-electron exchange-coupled system in an embryonic antiferromagnetic state.
    [Show full text]
  • Physics 461 / Quantum Mechanics I
    Physics 461 / Quantum Mechanics I P.E. Parris Department of Physics University of Missouri-Rolla Rolla, Missouri 65409 January 10, 2005 CONTENTS 1 Introduction 7 1.1WhatisQuantumMechanics?......................... 7 1.1.1 WhatisMechanics?.......................... 7 1.1.2 PostulatesofClassicalMechanics:.................. 8 1.2TheDevelopmentofWaveMechanics..................... 10 1.3TheWaveMechanicsofSchrödinger..................... 13 1.3.1 Postulates of Wave Mechanics for a Single Spinless Particle . 13 1.3.2 Schrödinger’sMechanicsforConservativeSystems......... 16 1.3.3 The Principle of Superposition and Spectral Decomposition . 17 1.3.4 TheFreeParticle............................ 19 1.3.5 Superpositions of Plane Waves and the Fourier Transform . 23 1.4Appendix:TheDeltaFunction........................ 26 2 The Formalism of Quantum Mechanics 31 2.1 Postulate I: SpecificationoftheDynamicalState.............. 31 2.1.1 PropertiesofLinearVectorSpaces.................. 31 2.1.2 Additional Definitions......................... 33 2.1.3 ContinuousBasesandContinuousSets................ 34 2.1.4 InnerProducts............................. 35 2.1.5 ExpansionofaVectoronanOrthonormalBasis.......... 38 2.1.6 Calculation of Inner Products Using an Orthonormal Basis . 39 2.1.7 ThePositionRepresentation..................... 40 2.1.8 TheWavevectorRepresentation.................... 41 2.2PostulateII:ObservablesofQuantumMechanicalSystems......... 43 2.2.1 OperatorsandTheirProperties.................... 43 2.2.2 MultiplicativeOperators.......................
    [Show full text]
  • The Cellular Automaton Interpretation of Quantum Mechanics Arxiv:1405.1548V3 [Quant-Ph] 21 Dec 2015
    The Cellular Automaton Interpretation of Quantum Mechanics Gerard 't Hooft Institute for Theoretical Physics Utrecht University Postbox 80.195 3508 TD Utrecht, the Netherlands e-mail: [email protected] internet: http://www.staff.science.uu.nl/˜hooft101/ When investigating theories at the tiniest conceivable scales in nature, almost all researchers today revert to the quantum language, accepting the verdict from the Copenhagen doctrine that the only way to describe what is going on will always involve states in Hilbert space, controlled by operator equations. Returning to classical, that is, non quantum mechanical, descriptions will be forever impossible, unless one accepts some extremely contrived theoretical constructions that may or may not reproduce the quantum mechanical phenomena observed in experiments. Dissatisfied, this author investigated how one can look at things differently. This book is an overview of older material, but also contains many new observations and calculations. Quantum mechanics is looked upon as a tool, not as a theory. Examples are displayed of models that are classical in essence, but can be analysed by the use of quantum techniques, and we argue that even the Standard Model, together with gravitational interactions, might be viewed as a quantum mechanical approach to analyse a system that could be classical at its core. We explain how such thoughts can conceivably be reconciled with Bell's theorem, and how the usual ob- jections voiced against the notion of `superdeterminism' can be overcome, at least in principle. Our proposal would eradicate the collapse problem and the measurement problem. Even the arXiv:1405.1548v3 [quant-ph] 21 Dec 2015 existence of an \arrow of time" can perhaps be explained in a more elegant way than usual.
    [Show full text]
  • Lecture Notes
    M.Phys Option in Theoretical Physics: C6 John Chalker and Andre Lukas Physics Department, Oxford University 2010-2011 2 Chapter 1 Path Integrals in Quantum Mechanics 1.1 Mathematical Background In this first part of the chapter, we will introduce functionals which are one of the main tools in modern theoretical physics and explain how to differentiate and integrate them. In the second part, these techniques will be applied to re-formulate quantum mechanics in terms of functional integrals (path integrals). 1.1.1 Functionals What is a functional? Let us first consider a real function f :[a,b] R, mapping all elements x [a,b] in the domain [a,b] to real numbers f(x). A functional is similar to a function→ in that it maps all elements∈ in a certain domain to real numbers, however, the nature of its domain is very different. Instead of acting on all points of an interval or some other subset of the real numbers, the domain of functionals consists of (suitably chosen) classes of functions itself. In other words, given some class f of functions, a functional F : f R assigns to each such f a real number F [f]. { } { } → Differentiation and integration are the two main operationson which the calculus of (real or complex) functions is based. We would like to understand how to perform analogous operations on functionals, that is, we would like to make sense of the expressions δF [f] , f F [f] (1.1) δf D Z for functional differentiation and integration. Here, we denote a functional derivative by δ/δf, replacing the ordinary d/dx or ∂/∂x for functions, and the measure of a functional integral by f, replacing dx in the case of integrals over functions.
    [Show full text]
  • Multi-Particle States 31.1 Multi-Particle Wave Functions
    Physics 342 Lecture 31 Multi-Particle States Lecture 31 Physics 342 Quantum Mechanics I Friday, April 16th, 2010 Just as quantum mechanics in one dimensions is meant to motivate and inform (and, in some specific cases, to model), quantum mechanics applied to a single particle in an external field is a fundamentally incomplete, albeit useful, picture. After all, an electron is not alone in the universe with a Coulomb field. Something is generating the Coulomb field, and we are honor bound to treat the entirety of the system, not just the electron in isolation. Indeed, we can establish the relative correctness of an electron in isolation only by considering its relation to other electrons and particles. This is the task we now undertake: A description of wavefunctions of sys- tems. Particles in isolation is itself a classical concept { fundamentally, quantum particles and more generally, quantum fields are all part of the \master wave-function of the universe". Think of the program of multi- particle classical mechanics: We have a position vector r(t) 2 IR3N for N particles, and the goal is to find the motion of each particle from Newton's second law (or some Euler-Lagrange equivalent). In the quantum mechanical analogue of this problem, the wave function is still the goal, but we don't get a vector 2 IR3N wave function, instead we want a one-dimensional wave function that depends on 3N coordinates (plus time, of course). 31.1 Multi-Particle Wave Functions It is easy to think about what we mean by multiple particles interacting in quantum mechanics.
    [Show full text]
  • Symmetries in Quantum Mechanics
    Symmetries in Quantum Mechanics Summer School on Quantum Mechanics (July 29 - August 2, 2019) Islamic University of Science and Technology, Awantipora, Pulwama, Jammu & Kashmir Govind S. Krishnaswami, Chennai Mathematical Institute www.cmi.ac.in/∼govind/teaching/awantipora-qm-sch19 [email protected], August 11, 2019 Contents 1 References 1 2 Generators of transformations, symmetries and conservation laws 2 2.1 Generators of space translations...................................................2 2.2 Generator of time translation and time evolution operator.....................................2 2.3 Angular momentum as the generator of rotations..........................................3 2.4 Example of rotations generated by orbital and spin angular momentum..............................3 2.5 Symmetries and conservation laws..................................................5 3 Parity: A discrete symmetry 6 4 Symmetry among identical particles: bosons & fermions 7 4.1 Identical and indistinguishable particles...............................................7 4.2 Two spin half electrons in Helium.................................................. 11 4.3 Coulomb and exchange interactions.................................................. 12 4.4 Electronic configuration of light elements and periodic table.................................... 13 5 Time reversal 14 5.1 Time reversal in classical mechanics................................................. 14 5.2 Time reversal in quantum mechanics................................................. 15 1 References 1. E Wichmann, Quantum Physics, Berkeley Physics Course, Vol 4. 2. R P Feynman, R B Leighton and M Sands, Feynman Lectures on Physics, Vol 3. 3. H S Mani and G K Mehta, Introduction to Modern Physics. 4. B Dutta-Roy, Elements of Quantum Mechanics. 5. J J Sakurai and J J Napolitano, Modern Quantum Mechanics, 2nd Ed. 6. L I Schiff, Quantum Mechanics, 3rd Ed. McGraw-Hill (1968). 7. P J E Peebles, Quantum Mechanics, Princeton Univ Press (1992). 8. B H Bransden and C J Joachain, Quantum Mechanics, 2nd Ed., Pearson (2000).
    [Show full text]
  • Microscopic Theory of Nuclear Fission: a Review
    Reports on Progress in Physics REVIEW Microscopic theory of nuclear fission: a review To cite this article: N Schunck and L M Robledo 2016 Rep. Prog. Phys. 79 116301 Manuscript version: Accepted Manuscript Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors” This Accepted Manuscript is © © 2016 IOP Publishing Ltd. During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after the 12 month embargo period. After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by-nc-nd/3.0 Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version.
    [Show full text]