DOCSLIB.ORG

Path Integral Quantization in Quantum Mechanics and in Quantum Field

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Path Integral Quantization in Quantum Mechanics and in Quantum Field 5 Path Integrals in Quantum Mechanics and Quantum Field Theory In chapter 4 we discussed the Hilbert space picture of QuantumMechanics and Quantum Field Theory for the case of a free relativistic scalar fields. Here we will present the Path Integral picture of Quantum Mechanics and of relativistic scalar field theories. The Path Integral picture is important for two reasons. First, it offers an alternative, complementary, picture of Quantum Mechanics in which the role of the classical limit is apparent. Secondly, it gives a direct route to the study regimes where perturbation theory is either inadequate or fails com- pletely. In Quantum mechanics a standard approach to such problems is the WKB approximation, of Wentzel, Kramers and Brillouin. However, as it happens, it is extremely difficult (if not impossible) the generalize the WKB approximation to a Quantum Field Theory. Instead, the non-perturbative treatment of the Feynman path integral, which in Quantum Mechanics is equivalent to WKB, is generalizable to non-perturbative problems in Quan- tum Field Theory. In this chapter we will use path integrals only for bosonic systems, such as scalar fields. In subsequent chapters we willalsogivea full treatment of the path integral, including its applications to fermionic fields, abelian and non-abelian gauge fields, classical statistical mechanics, and non-relativistic many body systems. There is a huge literature on path integrals, going back to theoriginal papers by Dirac (Dirac, 1933), and particularly Feynman’s 1942 PhD Thesis (Feynman, 2005), and his review paper (Feynman, 1948). Popular textbooks on path integrals include the classic by Feynman and Hibbs (Feynman and Hibbs, 1965), and Schulman’s book (Schulman, 1981), among many others. 5.1 Path Integrals and Quantum Mechanics 115 5.1 Path Integrals and Quantum Mechanics Consider a simple quantum mechanical system whose dynamics can be de- scribed by a generalized coordinate operatorq ˆ.Wewanttocomputethe amplitude F qf ,tf qi,ti = qf ,tf qi,ti (5.1) known as the Wightman function. This function represents theamplitude ( ∣ ) ⟨ ∣ ⟩ to find the system at coordinate qf at the final time tf knowing that it was at coordinate qi at the initial time ti.TheamplitudeF qf ,tf qi,ti is just amatrixelementoftheevolutionoperator ˆ − ( ∣ ) iH ti tf h̵ F qf ,tf qi,ti = qf e qi (5.2) ( )/ q ,t = , q ,t = q,t Let us set, for simplicity,( i∣ i ) 0 ⟨0 and∣ f f ∣ ⟩ .Then,fromthe definition of this matrix element, we find out that it obeys ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ lim F q,t 0, 0 = q 0 = δ q (5.3) t→0 Furthermore, after some algebra( ∣ we) also⟨ find∣ ⟩ that( ) ∂F ∂ ∂ −iHtˆ h ih =ih q,t 0, 0 = ih q e ̵ 0 ̵ ∂t ̵ ∂t ̵ ∂t − ˆ / = ˆ iHt h̵ q He⟨ ∣ ⟩0 ⟨ ∣ ∣ ⟩ ′ / ′ ′ −iHtˆ h = dq q Hˆ q q e ̵ 0 (5.4) ⟨"∣ ∣ ⟩ / where we have used that, since⟨ ∣q ∣ is⟩⟨ a complete∣ ∣ set⟩ of states, the identity operator I has the expansion, called the resolution of the identity, {∣ ⟩} ′ ′ ′ I = dq q q (5.5) " Here we have assumed that the states are∣ ⟩⟨ orthonormal,∣ ′ ′ q q = δ q − q (5.6) Hence, ⟨ ∣ ⟩ ( ) ∂ ′ ′ ′ ih F q,t 0, 0 = dq q Hˆ q F q ,t 0, 0 ≡ Hˆ F q,t 0, 0 (5.7) ̵ ∂t " q In other words,( ∣F q,t) 0, 0 is⟨ the∣ solution∣ ⟩ ( of∣ the) Schr¨odinger( ∣ Equation) that satisfies the initial condition of Eq. (5.3). For this reason,theamplitude F q,t 0, 0 is called( the∣ Schr¨odinger) Propagator. ( ∣ ) 116 Path Integrals in Quantum Mechanics and Quantum Field Theory q f qf ,tf ( ) q′ q′,t′ q ( ) i qi,ti ′ t t ti( ) t f Figure 5.1 The amplitude to go from qi,ti to qf ,tf is a sum of products ′ ′ of amplitudes through the intermediate states q ,t . ∣ ⟩ ∣ ⟩ ∣ ⟩ The superposition principle tells us that the amplitude to find the system in the final state at the final time is the sum of amplitudes of theform ′ ′ ′ ′ ′ F q ,t q ,t = dq q ,t q ,t q ,t q ,t (5.8) f f i i " f f i i ′ where the system( is in∣ an arbitrary) set⟨ of states∣ ⟩⟨ at an∣ interme⟩ diate time t . Here we represented this situation by inserting the identityoperatorI at ′ the intermediate time t in the form of the resolution of the identity of Eq. (5.8). Let us next define a partition of the time interval ti,tf into N sub- intervals each of length ∆t, [ ] tf − ti = N∆t (5.9) Let tj ,withj = 0,...,N+ 1, denote a set of points in the interval ti,tf , such that { } [ ] ti = t0 ≤ t1 ≤ ...≤ tN ≤ tN+1 = tf (5.10) Clearly, tk = t0 + k∆t,fork = 1,...,N+ 1. By repeating the procedure used in Eq.(5.8) of inserting the resolution of the identity at theintermediate times tk ,wefind F q ,t q ,t = dq ...dq q ,t q ,t q ,t q ,t × ... {f }f i i " 1 N f f N N N N N−1 N−1 × ... q ,t q ,t ... q ,t q ,t ( ∣ ) ⟨ ∣ j j ⟩⟨j−1 j−1 ∣ 1 1 ⟩i i (5.11) ⟨ ∣ ⟩ ⟨ ∣ ⟩ 5.1 Path Integrals and Quantum Mechanics 117 Each factor qj,tj qj−1,tj−1 in Eq.(5.11) has the form − ˆ − − ˆ = iH tj tj−1 h̵ ≡ iH∆t h̵ qj,tj qj−⟨1,tj−1∣ qj e ⟩ qj−1 qj e qj−1 (5.12) ( )/ / In the limit N → ∞,with tf − ti fixed and finite, the interval ∆t becomes ⟨ ∣ ⟩ ⟨ ∣ ∣ ⟩ ⟨ ∣ ∣ ⟩ infinitesimally small and ∆t → 0. Hence, as N → ∞ we can approximate the expression for qj,tj qj∣−1,tj−1∣ in Eq.(5.12) as follows − q ,t q ,t = q e iH∆t h̵ q j j j−1 ⟨j−1 ∣ j ⟩ j−1 ∆/ t ˆ 2 = qj I − i H + O ∆t qj−1 ⟨ ∣ ⟩ ⟨ ∣ h̵ ∣ ⟩ t = ∆ ˆ 2 ⟨δ q∣$j − qj−1 − i ((qj H)q)j−%∣1 + O⟩ ∆t h̵ (5.13) ( ) ⟨ ∣ ∣ ⟩ (( ) ) which becomes asymptotically exact as N → ∞. q(t) qf qi t t ti ∆t f Figure 5.2 A history q t of the system. We can also introduce at each intermediate( ) time tj acompletesetof momentum eigenstates p using their resolution of the identity ∞ = {∣ ⟩}I dp p p (5.14) "−∞ Recall that the overlap between the states∣ ⟩⟨q ∣and p is 1 = ipq h̵ q p e ∣ ⟩ ∣ ⟩ (5.15) 2πh̵ / For a typical Hamiltonian of⟨ ∣ the⟩ form& pˆ2 Hˆ = + V qˆ (5.16) 2m ( ) 118 Path Integrals in Quantum Mechanics and Quantum Field Theory its matrix elements are ∞ 2 dpj − pj ˆ ipj qj qj−1 h̵ qj H qj−1 = e + V qj (5.17) "−∞ 2πh 2m ̵ ( )/ Within the⟨ ∣ same∣ level⟩ of approximation we can' also write( )( dpj pj qj + qj−1 qj,tj qj−1,tj−1 ≈ exp i qj − qj−1 − ∆tH pj, " 2πh̵ h̵ 2 (5.18) where⟨ we∣ have introduce⟩ the “mid-point) * ( rule” which) amounts* tothereplace-++, q → 1 q + q H p, q ment j 2 j j−1 inside the Hamiltonian .Puttingeverything together we find that the matrix element qf ,tf qi,ti becomes - . ( ) N ∞ N+1 ⟨ dpj ∣ ⟩ qf ,tf qi,ti = lim dqj N→∞ " # " # 2πh j=1 −∞ j=1 ̵ ⟨ ∣ ⟩ N+1 i qj + qj−1 exp p q − q − ∆tH p , h $ j j j−1 j 2 ⎧ ̵ j=1 ⎫ ⎪ ⎪ ⎨⎪ ) ( ) * +,⎬⎪(5.19) ⎪ ⎪ ⎩⎪ ⎭⎪ Therefore, in the limit N → ∞,holding ti − tf fixed, the amplitude qf ,tf qi,ti is given by the (formal) expression ∣ ∣ i tf ⟨ ∣ ⟩ dt pq˙ − H p, q h " q ,t q ,t = DpDqe̵ ti (5.20) f f i i " [ ( )] where we have⟨ used∣ the⟩ notation N dpjdqj DpDq ≡ lim (5.21) N→∞ # 2πh j=1 ̵ which defines the integration measure. The functions, or configurations, q t ,p t must satisfy the initial and final conditions = = ( ( ) ( )) q ti qi,qtf qf (5.22) Thus the matrix element (qf),tf qi,ti can( be) expressed as a sum over his- tories in phase space.Theweightofeachhistoryistheexponentialfactor of Eq. (5.20). Notice that⟨ the quantity∣ ⟩ in brackets it is just the Lagrangian L = pq˙ − H p, q (5.23) ( ) 5.1 Path Integrals and Quantum Mechanics 119 Thus the matrix element is just i S q,p q ,t q,t = DpDqeh̵ (5.24) f f " ( ) where S q,p is the action⟨ ∣ of each⟩ history q t ,p t .Alsonoticethat the sum (or integral) runs over independent functions q t and p t which are not required( ) to satisfy any constraint (apart( ( ) from( )) the initial and final conditions) and, in particular they are not the solution( ) of the equations( ) of motion. Expressions of these type are known as path-integrals.Theyare also called functional integrals, since the integration measure is a sum over a space of functions,insteadofafieldofnumbersasinaconventionalintegral. Using a Gaussian integral of the form (which involves an analytic contin- uation) 2 p ∆t ∆t 2 ∞ dp i pq˙ − m i q˙ e 2m h = e 2h (5.25) " 2πh ̵ 2πih∆t ̵ −∞ ̵ 6 ̵ ( ) we can integrate out explicitly the momenta in the path-integral and find a formula that involves only the histories of the coordinate alone. Notice that there are no initial and final conditions on the momenta since the initial and final states have well defined positions. The result is i tf dt L q,q˙ h " q ,t q ,t = Dqe̵ ti (5.26) f f i i " ( ) which is known as⟨ the Feynman∣ ⟩ Path Integral (Feynman, 2005, 1948). Here L q,q˙ is the Lagrangian, 1 L q,q˙ = mq˙2 − V q (5.27) ( ) 2 and the sum over histories(q t ) is restricted( by) the boundary conditions q ti = qi and q tf = qf .
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]
  • 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]
  • Arxiv:Cond-Mat/0203258V1 [Cond-Mat.Str-El] 12 Mar 2002 AS 71.10.-W,71.27.+A PACS: Pnpolmi Oi Tt Hsc.Iscoeconnection Close Its High- of Physics
    Large-N expansion based on the Hubbard-operator path integral representation and its application to the t J model − Adriana Foussats and Andr´es Greco Facultad de Ciencias Exactas Ingenier´ıa y Agrimensura and Instituto de F´ısica Rosario (UNR-CONICET). Av.Pellegrini 250-2000 Rosario-Argentina. (October 29, 2018) In the present work we have developed a large-N expansion for the t − J model based on the path integral formulation for Hubbard-operators. Our large-N expansion formulation contains diagram- matic rules, in which the propagators and vertex are written in term of Hubbard operators. Using our large-N formulation we have calculated, for J = 0, the renormalized O(1/N) boson propagator. We also have calculated the spin-spin and charge-charge correlation functions to leading order 1/N. We have compared our diagram technique and results with the existing ones in the literature. PACS: 71.10.-w,71.27.+a I. INTRODUCTION this constrained theory leads to the commutation rules of the Hubbard-operators. Next, by using path-integral The role of electronic correlations is an important and techniques, the correlation functional and effective La- open problem in solid state physics. Its close connection grangian were constructed. 1 with the phenomena of high-Tc superconductivity makes In Ref.[ 11], we found a particular family of constrained this problem relevant in present days. Lagrangians and showed that the corresponding path- One of the most popular models in the context of high- integral can be mapped to that of the slave-boson rep- 13,5 Tc superconductivity is the t J model.
    [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]
  • On the Emergence of Time in Quantum Gravity
    Imperial/TP/98–99/23 On the Emergence of Time in Quantum Gravity1 J. Butterfield2 All Souls College Oxford OX1 4AL and C.J. Isham3 The Blackett Laboratory Imperial College of Science, Technology & Medicine South Kensington London SW7 2BZ 13 December 1998 Abstract We discuss from a philosophical perspective the way in which the normal concept of time might be said to ‘emerge’ in a quantum the- ory of gravity. After an introduction, we briefly discuss the notion of arXiv:gr-qc/9901024v1 8 Jan 1999 emergence, without regard to time (Section 2). We then introduce the search for a quantum theory of gravity (Section 3); and review some general interpretative issues about space, time and matter (Section 4). We then discuss the emergence of time in simple quantum ge- ometrodynamics, and in the Euclidean approach (Section 5). Section 6 concludes. 1To appear in The Arguments of Time, ed. J. Butterfield, Oxford University Press, 1999. 2email: [email protected]; jeremy.butterfi[email protected] 3email: [email protected] 1 Introduction The discovery of a satisfactory quantum theory of gravity has been widely regarded as the Holy Grail of theoretical physics for some forty years. In this essay, we will discuss a philosophical aspect of the search for such a theory that bears on our understanding of time: namely, the senses in which our standard ideas of time, and more generally spacetime, might be not fundamental to reality, but instead ‘emergent’ as an approximately valid concept on sufficiently large scales. In taking up this topic, our aim in part is to advertise to philosophers of time an unexplored area that promises to be fruitful.
    [Show full text]
  • Quantum Vacuum Energy Density and Unifying Perspectives Between Gravity and Quantum Behaviour of Matter
    Annales de la Fondation Louis de Broglie, Volume 42, numéro 2, 2017 251 Quantum vacuum energy density and unifying perspectives between gravity and quantum behaviour of matter Davide Fiscalettia, Amrit Sorlib aSpaceLife Institute, S. Lorenzo in Campo (PU), Italy corresponding author, email: [email protected] bSpaceLife Institute, S. Lorenzo in Campo (PU), Italy Foundations of Physics Institute, Idrija, Slovenia email: [email protected] ABSTRACT. A model of a three-dimensional quantum vacuum based on Planck energy density as a universal property of a granular space is suggested. This model introduces the possibility to interpret gravity and the quantum behaviour of matter as two different aspects of the same origin. The change of the quantum vacuum energy density can be considered as the fundamental medium which determines a bridge between gravity and the quantum behaviour, leading to new interest- ing perspectives about the problem of unifying gravity with quantum theory. PACS numbers: 04. ; 04.20-q ; 04.50.Kd ; 04.60.-m. Key words: general relativity, three-dimensional space, quantum vac- uum energy density, quantum mechanics, generalized Klein-Gordon equation for the quantum vacuum energy density, generalized Dirac equation for the quantum vacuum energy density. 1 Introduction The standard interpretation of phenomena in gravitational fields is in terms of a fundamentally curved space-time. However, this approach leads to well known problems if one aims to find a unifying picture which takes into account some basic aspects of the quantum theory. For this reason, several authors advocated different ways in order to treat gravitational interaction, in which the space-time manifold can be considered as an emergence of the deepest processes situated at the fundamental level of quantum gravity.
    [Show full text]
  • A More Direct Representation for Complex Relativity
    A more direct representation for complex relativity D.H. Delphenich ∗∗∗ Physics Department, Bethany College, Lindsborg, KS 67456, USA Received 22 Nov 2006, revised 5 July 2007, accepted 5 July 2007 by F. W. Hehl Key words Complex relativity, pre-metric electromagnetism, almost-complex structures, 3- spinors. PACS 02.10.Xm, 02.40.Dr, 03.50.De, 04.50.+h An alternative to the representation of complex relativity by self-dual complex 2-forms on the spacetime manifold is presented by assuming that the bundle of real 2-forms is given an almost- complex structure. From this, one can define a complex orthogonal structure on the bundle of 2- forms, which results in a more direct representation of the complex orthogonal group in three complex dimensions. The geometrical foundations of general relativity are then presented in terms of the bundle of oriented complex orthogonal 3-frames on the bundle of 2-forms in a manner that essentially parallels their construction in terms of self-dual complex 2-forms. It is shown that one can still discuss the Debever-Penrose classification of the Riemannian curvature tensor in terms of the representation presented here. Contents 1 Introduction …………………………………………………………………………. 1 2 The complex orthogonal group …………………………………………………….. 3 3 The geometry of bivectors ………………………………………………………….. 4 3.1 Real vector space structure on A 2……………………………………………….. 5 3.2 Complex vector space structure on A 2…………………………………………... 8 3.3 Relationship to the formalism of self-dual bivectors……………………………. 12 4 Associated principal bundles for the vector bundle of 2-forms ………………….. 16 5 Levi-Civita connection on the bundle SO (3, 1)( M)………………………………… 19 6 Levi-Civita connection on the bundle SO (3; C)( M)……………………………….
    [Show full text]
  • 1 the LOCALIZED QUANTUM VACUUM FIELD D. Dragoman
    1 THE LOCALIZED QUANTUM VACUUM FIELD D. Dragoman – Univ. Bucharest, Physics Dept., P.O. Box MG-11, 077125 Bucharest, Romania, e-mail: [email protected] ABSTRACT A model for the localized quantum vacuum is proposed in which the zero-point energy of the quantum electromagnetic field originates in energy- and momentum-conserving transitions of material systems from their ground state to an unstable state with negative energy. These transitions are accompanied by emissions and re-absorptions of real photons, which generate a localized quantum vacuum in the neighborhood of material systems. The model could help resolve the cosmological paradox associated to the zero-point energy of electromagnetic fields, while reclaiming quantum effects associated with quantum vacuum such as the Casimir effect and the Lamb shift; it also offers a new insight into the Zitterbewegung of material particles. 2 INTRODUCTION The zero-point energy (ZPE) of the quantum electromagnetic field is at the same time an indispensable concept of quantum field theory and a controversial issue (see [1] for an excellent review of the subject). The need of the ZPE has been recognized from the beginning of quantum theory of radiation, since only the inclusion of this term assures no first-order temperature-independent correction to the average energy of an oscillator in thermal equilibrium with blackbody radiation in the classical limit of high temperatures. A more rigorous introduction of the ZPE stems from the treatment of the electromagnetic radiation as an ensemble of harmonic quantum oscillators. Then, the total energy of the quantum electromagnetic field is given by E = åk,s hwk (nks +1/ 2) , where nks is the number of quantum oscillators (photons) in the (k,s) mode that propagate with wavevector k and frequency wk =| k | c = kc , and are characterized by the polarization index s.
    [Show full text]
  • On the Rational Relationship Between Heisenberg Principle and General
    CORE Metadata, citation and similar papers at core.ac.uk Provided by CERN Document Server On the Rational Relationship between Heisenberg Principle and General Relativity Xiao Jianhua Henan Polytechnic University, Jiaozuo City, Henan, P.R.C. 454000 Abstract: The research shows that the Heisenberg principle is the logic results of general relativity principle. If inertia coordinator system is used, the general relativity will logically be derived from the Heisenberg principle. The intrinsic relation between the quantum mechanics and general relativity is broken by introducing pure-imaginary time to explain the Lorentz transformation. Therefore, this research shows a way to establish an unified field theory of physics. PACS: 01.55.+b, 03.65.-w, 04.20.-q, 04.20.Ky Keywords: Heisenberg principle, general relativity, commoving coordinator, Riemannian geometry, gauge field 1. Introduction There are many documents to treat the relationship between the Newton mechanics and general relativity. There are also many documents to treat the relationship between the quantum mechanics and general relativity. For some researchers, the related research shows that an unified field theory be possible. For others, the actual efforts to establish an unified field theory for physics have shown that such an unified field theory is totally non-possible. The physics world is divided into three kingdoms. Hence, the title of this paper may make the reader think: See! Another nut! However, the unification of matter motion is the basic relieves for science researchers. The real problem is to find the way. This is the topic of this paper. The paper will firstly derive the Heisenberg principle from general relativity principle.
    [Show full text]
  • The Heisenberg Uncertainty Principle*
    OpenStax-CNX module: m58578 1 The Heisenberg Uncertainty Principle* OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract By the end of this section, you will be able to: • Describe the physical meaning of the position-momentum uncertainty relation • Explain the origins of the uncertainty principle in quantum theory • Describe the physical meaning of the energy-time uncertainty relation Heisenberg's uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately. 1 Momentum and Position To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x- direction. The particle moves with a constant velocity u and momentum p = mu. According to de Broglie's relations, p = }k and E = }!. As discussed in the previous section, the wave function for this particle is given by −i(! t−k x) −i ! t i k x k (x; t) = A [cos (! t − k x) − i sin (! t − k x)] = Ae = Ae e (1) 2 2 and the probability density j k (x; t) j = A is uniform and independent of time. The particle is equally likely to be found anywhere along the x-axis but has denite values of wavelength and wave number, and therefore momentum.
    [Show full text]
  • (Aka Second Quantization) 1 Quantum Field Theory
    221B Lecture Notes Quantum Field Theory (a.k.a. Second Quantization) 1 Quantum Field Theory Why quantum field theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum field theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content. 1.1 Limitations of Multi-body Schr¨odinger Wave Func- tion We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) difference. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The final state contains yet another particle, photon in this case. But the Schr¨odinger equation is a differential equation acting on the arguments of the Schr¨odingerwave function, and can never change the number of arguments.
    [Show full text]