# Path Integrals with Μ1(X) and Μ2(X) the ﬁrst and Second Moments of the Transition Probabilities

Total Page：16

File Type：pdf, Size：1020Kb

1 Introduction to path integrals with ¹1(x) and ¹2(x) the ¯rst and second moments of the transition probabilities. v. January 22, 2006 Another approach is to use path integrals. We will use Phys 719 - M. Hilke the example of a simple brownian motion (the random walk) to illustrate the concept of the path integral (or Wiener integral) in this context. CONTENT ² Classical stochastic dynamics DISCRETE RANDOM WALK ² Brownian motion (random walk) ² Quantum dynamics The discrete random walk describes a particle (or per- son) moving along ¯xed segments for ¯xed time intervals ² Free particle (of unit 1). The choice of direction for each new segment ² Particle in a potential is random. We will assume that only perpendicular di- rection are possible (4 possibilities in 2D and 6 in 3D). ² Driven harmonic oscillator The important quantity is the probability to reach x at 0 0 ² Semiclassical approximation time t, when the random walker started at x at time t : ² Statistical description (imaginary time) P (x; t; x0; t0): (3) ² Quantum dissipative systems For this probability the following normalization condi- tions apply: INTRODUCTION X 0 0 0 0 P (x; t; x ; t) = ±x;x0 and for (t > t ) P (x; t; x ; t ) = 1: Path integrals are used in a variety of ¯elds, including x stochastic dynamics, polymer physics, protein folding, (4) ¯eld theories, quantum mechanics, quantum ¯eld theo- This leads to ries, quantum gravity and string theory. The basic idea 1 X is to sum up all contributing paths. Here we will overview P (x; t + 1; x0; t0) = P (»; t; x0; t0); (5) the technique be starting on classical dynamics, in par- 2D h»i ticular, the random walk problem before we discuss the quantum case by looking at a particle in a potential, and where h»i represent the nearest neighbors of x. This is ¯nish with an example of a quantum dissipative system, solved by the Fourier transform: where this technique bares all its fruits. Z ¼ D 0 0 d k ikx ~ P (x; t; x ; t ) = D e P (k; t); (6) CLASSICAL STOCHASTIC DYNAMICS ¡¼ (2¼) The classical equation of motion for a particle in a which leads to ﬂuctuating environment is simply given by D 1 X 0 P~(k; t+1) = cos(k )P~(k; t) with P~(k; t0) = e¡ikx : dv D ¹ m = F (t) ¡ v=B (1) ¹=1 dt (7) where m is the mass, v the velocity and B describes the Hence, friction of the particle. This equation is the Langevin Ã !t¡t0 equation if F (t) is a ﬂuctuating force. These equa- Z ¼ D D d k 0 1 X tions are often di±cult to solve and one possibility is P (x; t; x0; t0) = eik(x¡x ) cos(k ) ; (2¼)D D ¹ to look at the probability density ½(x; t) instead. This ¡¼ ¹=1 approach leads to expressing ½ in terms of a master- (8) equation (Fokker-Planck) which takes on the following where we used 7 iteratively. Now, instead of using the form: probability we introduce the probability density p = P=aD, where a ia the step size, which will allow us to take the limit a ! 0, for the continuous case with the @½ @ @2 = ¡ (¹ ½) + (¹ ½) (2) variable change (k ! ak, x ! x=a). Hence, @t @x 1 @x2 2 2 Ã !(t¡t0)2D=a2 Z 1 D D d k 0 1 X p(x; t; x0; 0) = eik(x¡x ) cos(ak ) ; (9) (2¼)D D ¹ ¡1 ¹=1 | {z } 'e¡tk2 1 PD 2 2 where we used D ¹=1 cos(ak¹) ' (1 ¡ a k =2D + ¢ ¢ ¢ ). It is now possible to describe the evolution from an Hence, initial point (xi; 0) to a ¯nal point (xf ; t) by inserting an intermediate point (x1; t1) and integrating over it, i.e., Z 1 D 0 d k ik(x¡x0)¡tk2 p(x; t; x ; 0) = D e ¡1 (2¼) 0 2 1 ¡ (x¡x ) = e 4t ; (10) 4¼tD=2 which is the standard di®usion equation. The normaliza- Z D tion of the probability density requires p(xf ; t; xi; 0) = d x1p(xf ; t; x1; t1)p(x1; t1; xi; 0) (0 < t1 < tf ): Z (13) dDxp(x; t; x0; t0) = 1 (11) We can perform a similar decomposition for n interme- diate points, with xf = xn and x0 = xi. This then leads and to the summation over n paths, which is the main idea of the path integral. Thus, using (13) and (10) we lim p(x; t; x0; t0) = ±D(x ¡ x0) (12) obtain t!0 Z n¡1 2 Y D 1 Pn¡1 (xj+1¡xj ) d xj 1 ¡ p(x ; t; x ; 0) = e 4 j=0 tj+1¡tj : (14) f i 4¼(t ¡ t )D=2 D=2 j=1 j+1 j 4¼t1 Since Z nX¡1 (x ¡ x )2 t j+1 j = dsx_ 2; (15) t ¡ t j=0 j+1 j 0 we de¯ne the path integral as Z xf R ¡ 1 t dsx_ 2 p(xf ; t; xi; 0) = Dx(s)e 4 0 : (16) xi Here D denotes the multiple integrals de¯ned in (14) and represents the sum over all paths (Wiener integral). We will see in the next sections that a very similar expression exists in quantum mechanics. In classical systems this path integral is useful when there exist many ways to get form one point to the other (like in stochastic processes) and where it is then necessary to sum over all possible paths. Quantum Mechanics Feynman Path Integral ⇒ ⇒ with Written in eigenfunction basis: ⇒ Splitting the time evolution in 2: ⇒ Free particle: ⇒ using and ⇒ In terms of the classical action: (classical free path) ⇒ ⇒ Particle in a potential: { ⇒ using 0 and ⇒ c The Feynman path integral Driven harmonic oscillator: Classical path = Classical path+quantum fluctuations fluctuations ⇓ with ; Hence, Normalization: but and ⇒ Semiclassical approximation : 0 ⇒ ⇒ with Imaginary time (statistical properties): The equilibrium density operator: where is the quantum partition function But Shows that for hence with Applying this to the harmonic oscillator: Dissipative Systems: { Effective action: (the density matrix) (initially independent) (tracing out the environment) (tracing out the environment) ⇒ Path Integrals and Their Application to Dissipative Quantum Systems Gert-Ludwig Ingold Institut f¨ur Physik, Universit¨at Augsburg, D-86135 Augsburg to be published in “Coherent Evolution in Noisy Environments”, Lecture Notes in Physics, http://link.springer.de/series/lnpp/ c Springer Verlag, Berlin-Heidelberg-New York Path Integrals and Their Application to Dissipative Quantum Systems Gert-Ludwig Ingold Institut f¨ur Physik, Universit¨at Augsburg, D-86135 Augsburg, Germany 1 Introduction The coupling of a system to its environment is a recurrent subject in this collec- tion of lecture notes. The consequences of such a coupling are threefold. First of all, energy may irreversibly be transferred from the system to the environment thereby giving rise to the phenomenon of dissipation. In addition, the ﬂuctuating force exerted by the environment on the system causes ﬂuctuations of the system degree of freedom which manifest itself for example as Brownian motion. While these two eﬀects occur both for classical as well as quantum systems, there exists a third phenomenon which is speciﬁc to the quantum world. As a consequence of the entanglement between system and environmental degrees of freedom a coherent superposition of quantum states may be destroyed in a process referred to as decoherence. This eﬀect is of major concern if one wants to implement a quantum computer. Therefore, decoherence is discussed in detail in Chap. 5. Quantum computation, however, is by no means the only topic where the coupling to an environment is relevant. In fact, virtually no real system can be considered as completely isolated from its surroundings. Therefore, the phenom- ena listed in the previous paragraph play a role in many areas of physics and chemistry and a series of methods has been developed to address this situation. Some approaches like the master equations discussed in Chap. 2 are particularly well suited if the coupling to the environment is weak, a situation desired in quantum computing. On the other hand, in many solid state systems, the envi- ronmental coupling can be so strong that weak coupling theories are no longer valid. This is the regime where the path integral approach has proven to be very useful. It would be beyond the scope of this chapter even to attempt to give a com- plete overview of the use of path integrals in the description of dissipative quan- tum systems. In particular for a two-level system coupled to harmonic oscillator degrees of freedom, the so-called spin-boson model, quite a number of approx- imations have been developed which are useful in their respective parameter regimes. This chapter rather attempts to give an introduction to path integrals for readers unfamiliar with but interested in this method and its application to dissipative quantum systems. In this spirit, Sect. 2 gives an introduction to path integrals. Some aspects discussed in this section are not necessarily closely related to the problem of dissipative systems. They rather serve to illustrate the path integral approach and to convey to the reader the beauty and power of this approach. In Sect. 3 2 Gert-Ludwig Ingold we elaborate on the general idea of the coupling of a system to an environment. The path integral formalism is employed to eliminate the environmental degrees of freedom and thus to obtain an eﬀective description of the system degree of freedom. The results provide the basis for a discussion of the damped harmonic oscillator in Sect.

Recommended publications
• 1 Notation for States
The S-Matrix1 D. E. Soper2 University of Oregon Physics 634, Advanced Quantum Mechanics November 2000 1 Notation for states In these notes we discuss scattering nonrelativistic quantum mechanics. We will use states with the nonrelativistic normalization h~p |~ki = (2π)3δ(~p − ~k). (1) Recall that in a relativistic theory there is an extra factor of 2E on the right hand side of this relation, where E = [~k2 + m2]1/2. We will use states in the “Heisenberg picture,” in which states |ψ(t)i do not depend on time. Often in quantum mechanics one uses the Schr¨odinger picture, with time dependent states |ψ(t)iS. The relation between these is −iHt |ψ(t)iS = e |ψi. (2) Thus these are the same at time zero, and the Schrdinger states obey d i |ψ(t)i = H |ψ(t)i (3) dt S S In the Heisenberg picture, the states do not depend on time but the op- erators do depend on time. A Heisenberg operator O(t) is related to the corresponding Schr¨odinger operator OS by iHt −iHt O(t) = e OS e (4) Thus hψ|O(t)|ψi = Shψ(t)|OS|ψ(t)iS. (5) The Heisenberg picture is favored over the Schr¨odinger picture in the case of relativistic quantum mechanics: we don’t have to say which reference 1Copyright, 2000, D. E. Soper [email protected] 1 frame we use to deﬁne t in |ψ(t)iS. For operators, we can deal with local operators like, for instance, the electric ﬁeld F µν(~x, t).
[Show full text]
• 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]
• 22 Scattering in Supersymmetric Matter Chern-Simons Theories at Large N
2 2 scattering in supersymmetric matter ! Chern-Simons theories at large N Karthik Inbasekar 10th Asian Winter School on Strings, Particles and Cosmology 09 Jan 2016 Scattering in CS matter theories In QFT, Crossing symmetry: analytic continuation of amplitudes. Particle-antiparticle scattering: obtained from particle-particle scattering by analytic continuation. Naive crossing symmetry leads to non-unitary S matrices in U(N) Chern-Simons matter theories.[ Jain, Mandlik, Minwalla, Takimi, Wadia, Yokoyama] Consistency with unitarity required Delta function term at forward scattering. Modiﬁed crossing symmetry rules. Conjecture: Singlet channel S matrices have the form sin(πλ) = 8πpscos(πλ)δ(θ)+ i S;naive(s; θ) S πλ T S;naive: naive analytic continuation of particle-particle scattering. T Scattering in U(N) CS matter theories at large N Particle: fund rep of U(N), Antiparticle: antifund rep of U(N). Fundamental Fundamental Symm(Ud ) Asymm(Ue ) ⊗ ! ⊕ Fundamental Antifundamental Adjoint(T ) Singlet(S) ⊗ ! ⊕ C2(R1)+C2(R2)−C2(Rm) Eigenvalues of Anyonic phase operator νm = 2κ 1 νAsym νSym νAdj O ;ν Sing O(λ) ∼ ∼ ∼ N ∼ symm, asymm and adjoint channels- non anyonic at large N. Scattering in the singlet channel is eﬀectively anyonic at large N- naive crossing rules fail unitarity. Conjecture beyond large N: general form of 2 2 S matrices in any U(N) Chern-Simons matter theory ! sin(πνm) (s; θ) = 8πpscos(πνm)δ(θ)+ i (s; θ) S πνm T Universality and tests Delta function and modiﬁed crossing rules conjectured by Jain et al appear to be universal. Tests of the conjecture: Unitarity of the S matrix.
[Show full text]
• An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
[Show full text]
• Old Supersymmetry As New Mathematics
old supersymmetry as new mathematics PILJIN YI Korea Institute for Advanced Study with help from Sungjay Lee Atiyah-Singer Index Theorem ~ 1963 1975 ~ Bogomolnyi-Prasad-Sommerfeld (BPS) Calabi-Yau ~ 1978 1977 ~ Supersymmetry Calibrated Geometry ~ 1982 1982 ~ Index Thm by Path Integral (Alvarez-Gaume) (Harvey & Lawson) 1985 ~ Calabi-Yau Compactification 1988 ~ Mirror Symmetry 1992~ 2d Wall-Crossing / tt* (Cecotti & Vafa etc) Homological Mirror Symmetry ~ 1994 1994 ~ 4d Wall-Crossing (Seiberg & Witten) (Kontsevich) 1998 ~ Wall-Crossing is Bound State Dissociation (Lee & P.Y.) Stability & Derived Category ~ 2000 2000 ~ Path Integral Proof of Mirror Symmetry (Hori & Vafa) Wall-Crossing Conjecture ~ 2008 2008 ~ Konstevich-Soibelman Explained (conjecture by Kontsevich & Soibelman) (Gaiotto & Moore & Neitzke) 2011 ~ KS Wall-Crossing proved via Quatum Mechanics (Manschot , Pioline & Sen / Kim , Park, Wang & P.Y. / Sen) 2012 ~ S2 Partition Function as tt* (Jocker, Kumar, Lapan, Morrison & Romo /Gomis & Lee) quantum and geometry glued by superstring theory when can we perform path integrals exactly ? counting geometry with supersymmetric path integrals quantum and geometry glued by superstring theory Einstein this theory famously resisted quantization, however on the other hand, five superstring theories, with a consistent quantum gravity inside, live in 10 dimensional spacetime these superstring theories say, spacetime is composed of 4+6 dimensions with very small & tightly-curved (say, Calabi-Yau) 6D manifold sitting at each and every point of
[Show full text]
• Motion of the Reduced Density Operator
Motion of the Reduced Density Operator Nicholas Wheeler, Reed College Physics Department Spring 2009 Introduction. Quantum mechanical decoherence, dissipation and measurements all involve the interaction of the system of interest with an environmental system (reservoir, measurement device) that is typically assumed to possess a great many degrees of freedom (while the system of interest is typically assumed to possess relatively few degrees of freedom). The state of the composite system is described by a density operator ρ which in the absence of system-bath interaction we would denote ρs ρe, though in the cases of primary interest that notation becomes unavailable,⊗ since in those cases the states of the system and its environment are entangled. The observable properties of the system are latent then in the reduced density operator ρs = tre ρ (1) which is produced by “tracing out” the environmental component of ρ. Concerning the speciﬁc meaning of (1). Let n) be an orthonormal basis | in the state space H of the (open) system, and N) be an orthonormal basis s !| " in the state space H of the (also open) environment. Then n) N) comprise e ! " an orthonormal basis in the state space H = H H of the |(closed)⊗| composite s e ! " system. We are in position now to write ⊗ tr ρ I (N ρ I N) e ≡ s ⊗ | s ⊗ | # ! " ! " ↓ = ρ tr ρ in separable cases s · e The dynamics of the composite system is generated by Hamiltonian of the form H = H s + H e + H i 2 Motion of the reduced density operator where H = h I s s ⊗ e = m h n m N n N \$ | s| % | % ⊗ | % · \$ | ⊗ \$ | m,n N # # \$% & % &' H = I h e s ⊗ e = n M n N M h N | % ⊗ | % · \$ | ⊗ \$ | \$ | e| % n M,N # # \$% & % &' H = m M m M H n N n N i | % ⊗ | % \$ | ⊗ \$ | i | % ⊗ | % \$ | ⊗ \$ | m,n M,N # # % &\$% & % &'% & —all components of which we will assume to be time-independent.
[Show full text]
• Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany
Generating Functionals Functional Integration Renormalization Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany 2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective ﬁeld theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 1: Basics of Quantum Field Theory Thomas Mannel Theoretische Physik I, Universität Siegen f q f et Yerevan, October 2014 T. Mannel, Siegen University Effective Quantum
[Show full text]
• Configuration Interaction Study of the Ground State of the Carbon Atom
Configuration Interaction Study of the Ground State of the Carbon Atom María Belén Ruiz* and Robert Tröger Department of Theoretical Chemistry Friedrich-Alexander-University Erlangen-Nürnberg Egerlandstraße 3, 91054 Erlangen, Germany In print in Advances Quantum Chemistry Vol. 76: Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems 30th July 2017 Abstract Configuration Interaction (CI) calculations on the ground state of the C atom are carried out using a small basis set of Slater orbitals [7s6p5d4f3g]. The configurations are selected according to their contribution to the total energy. One set of exponents is optimized for the whole expansion. Using some computational techniques to increase efficiency, our computer program is able to perform partially-parallelized runs of 1000 configuration term functions within a few minutes. With the optimized computer programme we were able to test a large number of configuration types and chose the most important ones. The energy of the 3P ground state of carbon atom with a wave function of angular momentum L=1 and ML=0 and spin eigenfunction with S=1 and MS=0 leads to -37.83526523 h, which is millihartree accurate. We discuss the state of the art in the determination of the ground state of the carbon atom and give an outlook about the complex spectra of this atom and its low-lying states. Keywords: Carbon atom; Configuration Interaction; Slater orbitals; Ground state *Corresponding author: e-mail address: [email protected] 1 1. Introduction The spectrum of the isolated carbon atom is the most complex one among the light atoms. The ground state of carbon atom is a triplet 3P state and its low-lying excited states are singlet 1D, 1S and 1P states, more stable than the corresponding triplet excited ones 3D and 3S, against the Hund’s rule of maximal multiplicity.
[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]
• Analogies in Polymer Physics
Crimson Publishers Opinion Wings to the Research Analogies in Polymer Physics Sergey Panyukov* P N Lebedev Physics Institute, Russian Academy of Sciences, Russia Opinion Traditional methods for describing long polymer chains are so different from classical solid-state physics methods that it is difficult for scientists from different fields to find a seminar, Academician Ginzburg (not yet a Nobel laureate) got up and left the room: “This is common language. When I reported on my first work on polymer physics at a general physical not physics”. “Phew, what a nasty thing!”-this was a short commentary on another polymer work of the well-known theoretical physicist in the solid-state field. Of course, practical needs and the very time have somewhat changed the attitude towards the physics of polymers. But and slang of other areas of physics. even now, the best way to bury your work in polymer physics is to directly use the methods *Corresponding author: Sergey in elegant (or not so) analytical formulas, are replaced by the charts generated by computer Over the past fifty years, the methods of physics have changed a lot. New ideas, packaged simulations for carefully selected system parameters. People have learned to simulate RussiaPanyukov, P N Lebedev Physics Institute, Russian Academy of Sciences, Moscow, anything if they can allocate enough money for new computer clusters. But unfortunately, this Submission: does not always give a deeper understanding of physics, for which it is traditionally sufficient Published: July 06, 2020 to formulate a simplified model of a “spherical horse in a vacuum”. Complicate is simple, July 15, 2020 simplify is difficult, as Meyer’s law states.
[Show full text]
• FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk deﬁnes a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we deﬁne kxk = phx, xi . Lemma 1.1 (Schwarz inequality).
[Show full text]
• The Emergence of a Classical World in Bohmian Mechanics
The Emergence of a Classical World in Bohmian Mechanics Davide Romano Lausanne, 02 May 2014 Structure of QM • Physical state of a quantum system: - (normalized) vector in Hilbert space: state-vector - Wavefunction: the state-vector expressed in the coordinate representation Structure of QM • Physical state of a quantum system: - (normalized) vector in Hilbert space: state-vector; - Wavefunction: the state-vector expressed in the coordinate representation; • Physical properties of the system (Observables): - Observable ↔ hermitian operator acting on the state-vector or the wavefunction; - Given a certain operator A, the quantum system has a definite physical property respect to A iff it is an eigenstate of A; - The only possible measurement’s results of the physical property corresponding to the operator A are the eigenvalues of A; - Why hermitian operators? They have a real spectrum of eigenvalues → the result of a measurement can be interpreted as a physical value. Structure of QM • In the general case, the state-vector of a system is expressed as a linear combination of the eigenstates of a generic operator that acts upon it: 퐴 Ψ = 푎 Ψ1 + 푏|Ψ2 • When we perform a measure of A on the system |Ψ , the latter randomly collapses or 2 2 in the state |Ψ1 with probability |푎| or in the state |Ψ2 with probability |푏| . Structure of QM • In the general case, the state-vector of a system is expressed as a linear combination of the eigenstates of a generic operator that acts upon it: 퐴 Ψ = 푎 Ψ1 + 푏|Ψ2 • When we perform a measure of A on the system |Ψ , the latter randomly collapses or 2 2 in the state |Ψ1 with probability |푎| or in the state |Ψ2 with probability |푏| .
[Show full text]