DOCSLIB.ORG

Ground-State Energy of Two-Electron Atoms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ground-State Energy of Two-Electron Atoms THE JOURNAL OF CHEMICAL PHYSICS VOLUME 45, NUMBER 10 15 NOVEMBER 1966 Ground-State Energy of Two-Electron Atoms FRANK H. STILLINGER, JR. Bell Telephone Laboratories, Inc., Murray Hill, New Jersey (Received 29 June 1966) We examine the ground-state energy '(A) for two electrons bound to an infinite-mass point nucleus regarded as a function of the complex coupling constant A for the interelectron interaction. In the units used, the ground states of the homologous series H-, He, Li+, Be+ +, "', correspond, respectively, to A= 1, !, }, i, "', so the A power-series expansion of .(A) is equivalent to the familiar expansion in inverse nuclear charge Z-l. It is argued that the power series has a finite radius of convergence imposed by the existence of a branch point on the positive real axis at A=A*~1.1184 with exponent approximately 6/5. Furthermore, .(A*) apparently lies above the continuum limit, but still corresponds to a localized wavefunction. I. INTRODUCTION The final section V is devoted to a discussion of results and their implications for variational calculations INGLE-electron quantum-mechanical descriptions on two-electron systems, and to conjectures concerning of atoms and molecules have been developed and S many-electron systems and excited states. Also, an successfully exploited over the past several decades in Appendix is included giving details of two relevant elucidation of a wide variety of physical and chemical elementary variational calculations. phenomena. The primary remaining problem appears to be concerned with understanding the nature of electron II. ELEMENTARY CONSIDERATIONS correlation, and with practical techniques for taking electron correlation into account in large-molecule com­ The Schrodinger equation for the ground-state energy putations. E and wavefunction if;(rl, rz) for two electrons bound A necessary prerequisite for resolving the nature of to an infinite mass nucleus with atomic number Z is correlation in systems of large numbers of electrons obviously is the ability to understand the simpler cases. H(X')if;(rl, rz, X') = E(X')if; (rx, r2, X'), (1) 2 With this emphasis in mind, we have undertaken to H(X') = - (ft,2f2m.) (VI +W) elucidate certain aspects of the simplest nontrivial 2 l 1 2 (and physically important) case exhibiting electron -Ze (rl- +r2- ) + (X'e j r12), (2) correlation: the sequence of two-electron atoms, H-, where in anticipation of subsequent development we He, Li+, Be++, "', in their ground states. have formally included an electron-electron coupling In the next section we write the Hamiltonian for the constant X'. We note in passing that spin may be sequence in a common form in which only a variation suppressed since if; represents a singlet-state symmetric in coupling strength Xbetween the electrons is necessary with respect to interchange of electron positions rl and to pass from one member of the sequence to the next. rz. Introduce the following reductions: We then have the option of a global enquiry into the analytic nature of the ground-state energye(X) in the o=rjl, complex X plane, of which the perturbation series in X comprises only one aspect. In particular we may easily (3) investigate the process of allowing X-t- 00, and thereby e(X) =hZE(X')jm Z2e4, establish the asymptotics of e(X) in the neighborhood e of infinity (Sec. III). X=X'jZ, (4) The impressively accurate and extensive information currently available for the X perturbation series (more so that the eigenvalue equation (1) adopts the simpler commonly known as energy expansions in inverse nu­ form in terms of 01, (}2: clear charge for the homologous sequence) provides [i(V12+ V22) +PI-I+pz-L (VP12) +e(X) ] the basis for the analysis in Sec. IV. By means of the ratio test, the nearest singularity of e(X) to the origin X¢(Ol, (}2, X) =0. (5) is tentatively located on the positive real axis, and identified as a branch point. Subsequently, it is possible The interelectron repulsion may of course be removed to rewrite e(X) as a contribution from this nearest by setting X=O, and in this limit singularity, plus a manageable correction. Re-expansion e(O) = -1, then permits easy quantitative prediction of pertur­ bation coefficients of arbitrarily high order. ¢(Ol' (}2, X=O) = (1/11') exp( -Pl-pz). (6) 3623 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.112.66.66 On: Mon, 23 Dec 2013 03:26:24 3624 FRANK H. STILLINGER, JR. From the second Eq. (4) it is clear that the real systems is initially degree A. Since this second-order perturbation H-, He, Li+, Be+ +, ... , in their ground states corre- to the nondegenerate ground state is necessarily nega­ spond to A=l,!, -1, -1, ... tive, E" (A) is negative, so we conclude the second If E(A) is regarded as a power-series expansion in A, property of E(A): it has downward curvature. There­ fore, the positive quantity E' (A) decreases with A, and E(A) = !: EnAn. (7) we have then systematically verified the intuitively n=O clear notion that the ground-state wavefunction should The coefficient En is then equivalent to the nth-order increase in spatial extension as A increases. perturbation energy to the state described by Eq. (6), where the interelectron repulsion is treated as the III. BEHAVIOR AS 'A~ - 00 perturbation. It is known that Expansion (7) has a One of the advantages of using the interelectronic finite radius of convergence,l,2 with the immediate con­ coupling A as variable in investigating the ground-state sequence that the En cannot increase in magnitude with energy, as compared with inverse nuclear charge Z-I, n any faster than exponentially. Besides the value is that negative real values of the parameter have a EO= -1, it is easy to carry out the first-order pertur­ clear meaning. As A passes through zero from positive 3 bation calculation with the result El =i. Although the to negative values, the electron-electron repulsion con­ coefficient E2 is necessarily negative (as any second­ verts to an attraction. Although CP«()I, ()2, A) exhibits the order energy to a nondegenerate ground state must be), tendency for electrons to avoid one another when A>O, its precise value is unknown, and it has not even been it must likewise reflect their tendency to stay close established whether E2 is a rational number.4 However, together for A<O, which is another manifestation of in spite of the lack of rigorously exact information electron correlation. L 6 concerning the En, the Hylleraas Scherr-Knight vari­ The situation is especially simple in the limit as ational perturbation approach to the two-electron prob­ A~- 00. The very strong attraction between the elec­ lem has succeeded in providing undoubtedly very trons in this asymptotic regime will bind these two accurate numerical estimates for the first few coeffi­ particles together to form a "di-electron," i.e., sub­ cients, whose values provide the basis for the analysis stantially a point particle with twice the mass and in Sec. IV. charge of a single electron, and then this di-electron Two final elementary properties of E(A) may next will be bound to the positively charged nucleus in a be noted. First, if A is varied by a sufficiently small suitably scaled hydrogenic (ls) orbital. amount oA such that the linear estimate It is easy to arrive at the behavior of E(A) as A~- 00, (8) an extreme limit of electron correlation. The Hamil­ tonian in Eq. (2) may be rewritten as follows: suffices, then formal treatment of the extra Coulomb H(A) =H(O) (A) +H(l), repulsion between the electrons, (oA) I P12, as a pertur­ bation obviously identifies E' (A) in the form H(O)(A) = -!(V?+V22) + (AlpI2) - (2Ip) , E' (A) = (A 11/ P121 A), (9) H(I) = (2Ip) - (llpI) - (1/P2) , 1 a matrix element of P12- in the ground state for degree ()=!«()I+()2) . (to) of coupling A. Now, Expression (9) is clearly a positive number, so we trivially establish that E(A) is a mono­ Retention of just H(O) (A) is equivalent to treatment of tonically increasing function of coupling strength A. the di-electron strictly as a point particle insofar as its But additionally the fact that slope E' (A) is the expec­ binding to the nucleus is concerned, and the corre­ tation value of P12-1 allows identification of [E' (A) J-l sponding eigenvalue problem, as a measure of spatial extension of the ground-state wavefunction cp«()l, ()2, A). In similar fashion E"(A) H(O)(A)cp(O)«()I, ()2, A) =E(O) (A)cp(O) (VI, ()2, A), (11) may be regarded as essentially the second-order pert~r­ can be readily solved since transformation to di-electron bation energy for perturbation oAf P12 when the couplIng center of mass «() and relative «()12) coordinates effects 1 T. Kato, J. Fac. Sci. Univ. Tokyo Sec. I 6, 145 (1951). separation of variables. One finds (A < 0) • A. Froman and G. G. Hall, J. Mol. Spectry. 7, 410 (1961). 3 L. Pauling and E. B. Wilson, Introduction to Quantum Mechan- cp(O) «()I, ()2, 'A) = (8/11.1) exp( -4p) [( -A) Ij2 (211' )IJ ics (McGraw-Hill Book Co., ~nc., New York, 1935), pp. 446;-447. 4 Since closed-form expressIOns have recently become available X exp(!ApI2); (12) for single-particle Green's functions in Coul0I?b fields [e.g., J. Schwinger, J. Math.
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • 8 the Variational Principle
    8 The Variational Principle 8.1 Approximate solution of the Schroedinger equation If we can’t find an analytic solution to the Schroedinger equation, a trick known as the varia- tional principle allows us to estimate the energy of the ground state of a system. We choose an unnormalized trial function Φ(an) which depends on some variational parameters, an and minimise hΦ|Hˆ |Φi E[a ] = n hΦ|Φi with respect to those parameters. This gives an approximation to the wavefunction whose accuracy depends on the number of parameters and the clever choice of Φ(an). For more rigorous treatments, a set of basis functions with expansion coefficients an may be used. The proof is as follows, if we expand the normalised wavefunction 1/2 |φ(an)i = Φ(an)/hΦ(an)|Φ(an)i in terms of the true (unknown) eigenbasis |ii of the Hamiltonian, then its energy is X X X ˆ 2 2 E[an] = hφ|iihi|H|jihj|φi = |hφ|ii| Ei = E0 + |hφ|ii| (Ei − E0) ≥ E0 ij i i ˆ where the true (unknown) ground state of the system is defined by H|i0i = E0|i0i. The inequality 2 arises because both |hφ|ii| and (Ei − E0) must be positive. Thus the lower we can make the energy E[ai], the closer it will be to the actual ground state energy, and the closer |φi will be to |i0i. If the trial wavefunction consists of a complete basis set of orthonormal functions |χ i, each P i multiplied by ai: |φi = i ai|χii then the solution is exact and we just have the usual trick of expanding a wavefunction in a basis set.
    [Show full text]
  • Recent Experimental Progress of Fractional Quantum Hall Effect: 5/2 Filling State and Graphene
    Recent Experimental Progress of Fractional Quantum Hall Effect: 5/2 Filling State and Graphene X. Lin, R. R. Du and X. C. Xie International Center for Quantum Materials, Peking University, Beijing, People’s Republic of China 100871 ABSTRACT The phenomenon of fractional quantum Hall effect (FQHE) was first experimentally observed 33 years ago. FQHE involves strong Coulomb interactions and correlations among the electrons, which leads to quasiparticles with fractional elementary charge. Three decades later, the field of FQHE is still active with new discoveries and new technical developments. A significant portion of attention in FQHE has been dedicated to filling factor 5/2 state, for its unusual even denominator and possible application in topological quantum computation. Traditionally FQHE has been observed in high mobility GaAs heterostructure, but new materials such as graphene also open up a new area for FQHE. This review focuses on recent progress of FQHE at 5/2 state and FQHE in graphene. Keywords: Fractional Quantum Hall Effect, Experimental Progress, 5/2 State, Graphene measured through the temperature dependence of I. INTRODUCTION longitudinal resistance. Due to the confinement potential of a realistic 2DEG sample, the gapped QHE A. Quantum Hall Effect (QHE) state has chiral edge current at boundaries. Hall effect was discovered in 1879, in which a Hall voltage perpendicular to the current is produced across a conductor under a magnetic field. Although Hall effect was discovered in a sheet of gold leaf by Edwin Hall, Hall effect does not require two-dimensional condition. In 1980, quantum Hall effect was observed in two-dimensional electron gas (2DEG) system [1,2].
    [Show full text]
  • Quantum Mechanics by Numerical Simulation of Path Integral
    Quantum Mechanics by Numerical Simulation of Path Integral Author: Bruno Gim´enezUmbert Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Abstract: The Quantum Mechanics formulation of Feynman is based on the concept of path integrals, allowing to express the quantum transition between two space-time points without using the bra and ket formalism in the Hilbert space. A particular advantage of this approach is the ability to provide an intuitive representation of the classical limit of Quantum Mechanics. The practical importance of path integral formalism is being a powerful tool to solve quantum problems where the analytic solution of the Schr¨odingerequation is unknown. For this last type of physical systems, the path integrals can be calculated with the help of numerical integration methods suitable for implementation on a computer. Thus, they provide the development of arbitrarily accurate solutions. This is particularly important for the numerical simulation of strong interactions (QCD) which cannot be solved by a perturbative treatment. This thesis will focus on numerical techniques to calculate path integral on some physical systems of interest. I. INTRODUCTION [1]. In the first section of our writeup, we introduce the basic concepts of path integral and numerical simulation. Feynman's space-time approach based on path inte- Next, we discuss some specific examples such as the har- grals is not too convenient for attacking practical prob- monic oscillator. lems in non-relativistic Quantum Mechanics. Even for the simple harmonic oscillator it is rather cumbersome to evaluate explicitly the relevant path integral. However, II. QUANTUM MECHANICS BY PATH INTEGRAL his approach is extremely gratifying from a conceptual point of view.
    [Show full text]
  • Observables in Inhomogeneous Ground States at Large Global Charge
    IPMU18-0069 Observables in Inhomogeneous Ground States at Large Global Charge Simeon Hellerman1, Nozomu Kobayashi1,2, Shunsuke Maeda1,2, and Masataka Watanabe1,2 1Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 2Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 133-0022, Japan Abstract As a sequel to previous work, we extend the study of the ground state configu- ration of the D = 3, Wilson-Fisher conformal O(4) model. In this work, we prove that for generic ratios of two charge densities, ρ1=ρ2, the ground-state configuration is inhomogeneous and that the inhomogeneity expresses itself towards longer spatial periods. This is the direct extension of the similar statements we previously made for ρ =ρ 1. We also compute, at fixed set of charges, ρ ; ρ , the ground state en- 1 2 1 2 ergy and the two-point function(s) associated with this inhomogeneous configuration 3=2 on the torus. The ground state energy was found to scale (ρ1 + ρ2) , as dictated by dimensional analysis and similarly to the case of the O(2) model. Unlike the case of the O(2) model, the ground also strongly violates cluster decomposition in the large-volume, fixed-density limit, with a two-point function that is negative definite at antipodal points of the torus at leading order at large charge. arXiv:1804.06495v1 [hep-th] 17 Apr 2018 Contents 1 Introduction3 2 Goldstone counting and the (in)homogeneity of large-charge ground states5 1 A comment on helical symmetries and chemical potentials .
    [Show full text]
  • Quantum Mechanics in One Dimension
    Solved Problems on Quantum Mechanics in One Dimension Charles Asman, Adam Monahan and Malcolm McMillan Department of Physics and Astronomy University of British Columbia, Vancouver, British Columbia, Canada Fall 1999; revised 2011 by Malcolm McMillan Given here are solutions to 15 problems on Quantum Mechanics in one dimension. The solutions were used as a learning-tool for students in the introductory undergraduate course Physics 200 Relativity and Quanta given by Malcolm McMillan at UBC during the 1998 and 1999 Winter Sessions. The solutions were prepared in collaboration with Charles Asman and Adam Monaham who were graduate students in the Department of Physics at the time. The problems are from Chapter 5 Quantum Mechanics in One Dimension of the course text Modern Physics by Raymond A. Serway, Clement J. Moses and Curt A. Moyer, Saunders College Publishing, 2nd ed., (1997). Planck's Constant and the Speed of Light. When solving numerical problems in Quantum Mechanics it is useful to note that the product of Planck's constant h = 6:6261 × 10−34 J s (1) and the speed of light c = 2:9979 × 108 m s−1 (2) is hc = 1239:8 eV nm = 1239:8 keV pm = 1239:8 MeV fm (3) where eV = 1:6022 × 10−19 J (4) Also, ~c = 197:32 eV nm = 197:32 keV pm = 197:32 MeV fm (5) where ~ = h=2π. Wave Function for a Free Particle Problem 5.3, page 224 A free electron has wave function Ψ(x; t) = sin(kx − !t) (6) •Determine the electron's de Broglie wavelength, momentum, kinetic energy and speed when k = 50 nm−1.
    [Show full text]
  • 22.51 Course Notes, Chapter 9: Harmonic Oscillator
    9. Harmonic Oscillator 9.1 Harmonic Oscillator 9.1.1 Classical harmonic oscillator and h.o. model 9.1.2 Oscillator Hamiltonian: Position and momentum operators 9.1.3 Position representation 9.1.4 Heisenberg picture 9.1.5 Schr¨odinger picture 9.2 Uncertainty relationships 9.3 Coherent States 9.3.1 Expansion in terms of number states 9.3.2 Non-Orthogonality 9.3.3 Uncertainty relationships 9.3.4 X-representation 9.4 Phonons 9.4.1 Harmonic oscillator model for a crystal 9.4.2 Phonons as normal modes of the lattice vibration 9.4.3 Thermal energy density and Specific Heat 9.1 Harmonic Oscillator We have considered up to this moment only systems with a finite number of energy levels; we are now going to consider a system with an infinite number of energy levels: the quantum harmonic oscillator (h.o.). The quantum h.o. is a model that describes systems with a characteristic energy spectrum, given by a ladder of evenly spaced energy levels. The energy difference between two consecutive levels is ∆E. The number of levels is infinite, but there must exist a minimum energy, since the energy must always be positive. Given this spectrum, we expect the Hamiltonian will have the form 1 n = n + ~ω n , H | i 2 | i where each level in the ladder is identified by a number n. The name of the model is due to the analogy with characteristics of classical h.o., which we will review first. 9.1.1 Classical harmonic oscillator and h.o.
    [Show full text]
  • The Paradoxes of the Interaction-Free Measurements
    The Paradoxes of the Interaction-free Measurements L. Vaidman Centre for Quantum Computation, Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford 0X1 3PU, England; School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel Reprint requests to Prof. L. V.; E-mail: [email protected] Z. Naturforsch. 56 a, 100-107 (2001); received January 12, 2001 Presented at the 3rd Workshop on Mysteries, Puzzles and Paradoxes in Quantum Mechanics, Gargnano, Italy, September 17-23, 2000. Interaction-free measurements introduced by Elitzur and Vaidman [Found. Phys. 23,987 (1993)] allow finding infinitely fragile objects without destroying them. Paradoxical features of these and related measurements are discussed. The resolution of the paradoxes in the framework of the Many-Worlds Interpretation is proposed. Key words: Interaction-free Measurements; Quantum Paradoxes. I. Introduction experiment” proposed by Wheeler [22] which helps to define the context in which the above claims, that The interaction-free measurements proposed by the measurements are interaction-free, are legitimate. Elitzur and Vaidman [1,2] (EVIFM) led to numerousSection V is devoted to the variation of the EV IFM investigations and several experiments have been per­ proposed by Penrose [23] which, instead of testing for formed [3- 17]. Interaction-free measurements are the presence of an object in a particular place, tests very paradoxical. Usually it is claimed that quantum a certain property of the object in an interaction-free measurements, in contrast to classical measurements, way. Section VI introduces the EV IFM procedure for invariably cause a disturbance of the system.
    [Show full text]
  • Wave Mechanics (PDF)
    WAVE MECHANICS B. Zwiebach September 13, 2013 Contents 1 The Schr¨odinger equation 1 2 Stationary Solutions 4 3 Properties of energy eigenstates in one dimension 10 4 The nature of the spectrum 12 5 Variational Principle 18 6 Position and momentum 22 1 The Schr¨odinger equation In classical mechanics the motion of a particle is usually described using the time-dependent position ix(t) as the dynamical variable. In wave mechanics the dynamical variable is a wave- function. This wavefunction depends on position and on time and it is a complex number – it belongs to the complex numbers C (we denote the real numbers by R). When all three dimensions of space are relevant we write the wavefunction as Ψ(ix, t) C . (1.1) ∈ When only one spatial dimension is relevant we write it as Ψ(x, t) C. The wavefunction ∈ satisfies the Schr¨odinger equation. For one-dimensional space we write ∂Ψ 2 ∂2 i (x, t) = + V (x, t) Ψ(x, t) . (1.2) ∂t −2m ∂x2 This is the equation for a (non-relativistic) particle of mass m moving along the x axis while acted by the potential V (x, t) R. It is clear from this equation that the wavefunction must ∈ be complex: if it were real, the right-hand side of (1.2) would be real while the left-hand side would be imaginary, due to the explicit factor of i. Let us make two important remarks: 1 1. The Schr¨odinger equation is a first order differential equation in time. This means that if we prescribe the wavefunction Ψ(x, t0) for all of space at an arbitrary initial time t0, the wavefunction is determined for all times.
    [Show full text]