The Paradoxes of the Interaction-Free Measurements
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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. -
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. -
The Concept of Quantum State : New Views on Old Phenomena Michel Paty
The concept of quantum state : new views on old phenomena Michel Paty To cite this version: Michel Paty. The concept of quantum state : new views on old phenomena. Ashtekar, Abhay, Cohen, Robert S., Howard, Don, Renn, Jürgen, Sarkar, Sahotra & Shimony, Abner. Revisiting the Founda- tions of Relativistic Physics : Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht: Kluwer Academic Publishers, p. 451-478, 2003. halshs-00189410 HAL Id: halshs-00189410 https://halshs.archives-ouvertes.fr/halshs-00189410 Submitted on 20 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. « The concept of quantum state: new views on old phenomena », in Ashtekar, Abhay, Cohen, Robert S., Howard, Don, Renn, Jürgen, Sarkar, Sahotra & Shimony, Abner (eds.), Revisiting the Foundations of Relativistic Physics : Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht: Kluwer Academic Publishers, 451-478. , 2003 The concept of quantum state : new views on old phenomena par Michel PATY* ABSTRACT. Recent developments in the area of the knowledge of quantum systems have led to consider as physical facts statements that appeared formerly to be more related to interpretation, with free options. -
Lecture 9, P 1 Lecture 9: Introduction to QM: Review and Examples
Lecture 9, p 1 Lecture 9: Introduction to QM: Review and Examples S1 S2 Lecture 9, p 2 Photoelectric Effect Binding KE=⋅ eV = hf −Φ energy max stop Φ The work function: 3.5 Φ is the minimum energy needed to strip 3 an electron from the metal. 2.5 2 (v) Φ is defined as positive . 1.5 stop 1 Not all electrons will leave with the maximum V f0 kinetic energy (due to losses). 0.5 0 0 5 10 15 Conclusions: f (x10 14 Hz) • Light arrives in “packets” of energy (photons ). • Ephoton = hf • Increasing the intensity increases # photons, not the photon energy. Each photon ejects (at most) one electron from the metal. Recall: For EM waves, frequency and wavelength are related by f = c/ λ. Therefore: Ephoton = hc/ λ = 1240 eV-nm/ λ Lecture 9, p 3 Photoelectric Effect Example 1. When light of wavelength λ = 400 nm shines on lithium, the stopping voltage of the electrons is Vstop = 0.21 V . What is the work function of lithium? Lecture 9, p 4 Photoelectric Effect: Solution 1. When light of wavelength λ = 400 nm shines on lithium, the stopping voltage of the electrons is Vstop = 0.21 V . What is the work function of lithium? Φ = hf - eV stop Instead of hf, use hc/ λ: 1240/400 = 3.1 eV = 3.1eV - 0.21eV For Vstop = 0.21 V, eV stop = 0.21 eV = 2.89 eV Lecture 9, p 5 Act 1 3 1. If the workfunction of the material increased, (v) 2 how would the graph change? stop a. -
Introduction ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Li
This guide is designed to show someone with a basic undergraduate background in physics how to perform a presentation on the topic Introduction ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Page 1 of holography and how to demonstrate the making of a hologram. Although holography can be a very complex area of study, the material has been explained mostly qualitatively, so as to match the target audience’s background in mathematics and physics. The List of Equipment ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Page 2 target audience high school students in grades 9‐ 11 and so limited to no previous knowledge on optics, lasers or holograms is assumed. The presentation and demonstrations will teach the students about basic optics such as reflection, refraction, and interference. The Preparation for presentation ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Page 2‐3 students will learn the basic physics of lasers, with a focus on the apparatus set up of a laser and the necessity of a coherent light source for the creation of interference patterns. The last part of the presentation covers the two most popular types of holographic Presentation guideline ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Page 3‐7 production apparatuses which are reflection and transmission holograms respectively. Some historical background and fun facts are also included throughout the presentation. The module ends Demonstration guideline ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Page 7 with a live demonstration on how to make a hologram to give the students some exposure to the care that must be taken in experiments and to give them a taste of what higher level physics laboratories can include. The following flow chart summarizes the References ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Page 8 basics of how the presentation will flow. 1 First and foremost, you should find out the location you’ll Laptop be presenting at and when the presentation will take place. -
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. -
The Uncertainty Principle (Stanford Encyclopedia of Philosophy) Page 1 of 14
The Uncertainty Principle (Stanford Encyclopedia of Philosophy) Page 1 of 14 Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia Free The Uncertainty Principle First published Mon Oct 8, 2001; substantive revision Mon Jul 3, 2006 Quantum mechanics is generally regarded as the physical theory that is our best candidate for a fundamental and universal description of the physical world. The conceptual framework employed by this theory differs drastically from that of classical physics. Indeed, the transition from classical to quantum physics marks a genuine revolution in our understanding of the physical world. One striking aspect of the difference between classical and quantum physics is that whereas classical mechanics presupposes that exact simultaneous values can be assigned to all physical quantities, quantum mechanics denies this possibility, the prime example being the position and momentum of a particle. According to quantum mechanics, the more precisely the position (momentum) of a particle is given, the less precisely can one say what its momentum (position) is. This is (a simplistic and preliminary formulation of) the quantum mechanical uncertainty principle for position and momentum. The uncertainty principle played an important role in many discussions on the philosophical implications of quantum mechanics, in particular in discussions on the consistency of the so-called Copenhagen interpretation, the interpretation endorsed by the founding fathers Heisenberg and Bohr. This should not suggest that the uncertainty principle is the only aspect of the conceptual difference between classical and quantum physics: the implications of quantum mechanics for notions as (non)-locality, entanglement and identity play no less havoc with classical intuitions. -
Entanglement of the Uncertainty Principle
Open Journal of Mathematics and Physics | Volume 2, Article 127, 2020 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/9jhwx | published: 26 Jul 2020 | https://ojmp.org EX [original insight] Diamond Open Access Entanglement of the Uncertainty Principle Open Quantum Collaboration∗† August 15, 2020 Abstract We propose that position and momentum in the uncertainty principle are quantum entangled states. keywords: entanglement, uncertainty principle, quantum gravity The most updated version of this paper is available at https://osf.io/9jhwx/download Introduction 1. [1] 2. Logic leads us to conclude that quantum gravity–the theory of quan- tum spacetime–is the description of entanglement of the quantum states of spacetime and of the extra dimensions (if they really ex- ist) [2–7]. ∗All authors with their affiliations appear at the end of this paper. †Corresponding author: [email protected] | Join the Open Quantum Collaboration 1 Quantum Gravity 3. quantum gravity quantum spacetime entanglement of spacetime 4. Quantum spaceti=me might be in a sup=erposition of the extra dimen- sions [8]. Notation 5. UP Uncertainty Principle 6. UP= the quantum state of the uncertainty principle ∣ ⟩ = Discussion on the notation 7. The universe is mathematical. 8. The uncertainty principle is part of our universe, thus, it is described by mathematics. 9. A physical theory must itself be described within its own notation. 10. Thereupon, we introduce UP . ∣ ⟩ Entanglement 11. UP α1 x1p1 α2 x2p2 α3 x3p3 ... 12. ∣xi ⟩u=ncer∣tainty⟩ +in po∣ sitio⟩n+ ∣ ⟩ + 13. pi = uncertainty in momentum 14. xi=pi xi pi xi pi ∣ ⟩ = ∣ ⟩ ∣ ⟩ = ∣ ⟩ ⊗ ∣ ⟩ 2 Final Remarks 15. -
Path Integral for the Hydrogen Atom
Path Integral for the Hydrogen Atom Solutions in two and three dimensions Vägintegral för Väteatomen Lösningar i två och tre dimensioner Anders Svensson Faculty of Health, Science and Technology Physics, Bachelor Degree Project 15 ECTS Credits Supervisor: Jürgen Fuchs Examiner: Marcus Berg June 2016 Abstract The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude. Sammanfattning V¨agintegral-formuleringen av kvantmekanik generaliserar minsta-verkanprincipen fr˚anklassisk meka- nik. Feynmans v¨agintegral kan ses som en summa ¨over alla m¨ojligav¨agaren partikel kan ta mellan tv˚a givna ¨andpunkterA och B, d¨arvarje v¨agbidrar till summan med en fasfaktor inneh˚allandeden klas- siska verkan f¨orv¨agen.Den resulterande summan ger propagatorn, sannolikhetsamplituden att partikeln g˚arfr˚anA till B. -
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. -
Two-State Systems
1 TWO-STATE SYSTEMS Introduction. Relative to some/any discretely indexed orthonormal basis |n) | ∂ | the abstract Schr¨odinger equation H ψ)=i ∂t ψ) can be represented | | | ∂ | (m H n)(n ψ)=i ∂t(m ψ) n ∂ which can be notated Hmnψn = i ∂tψm n H | ∂ | or again ψ = i ∂t ψ We found it to be the fundamental commutation relation [x, p]=i I which forced the matrices/vectors thus encountered to be ∞-dimensional. If we are willing • to live without continuous spectra (therefore without x) • to live without analogs/implications of the fundamental commutator then it becomes possible to contemplate “toy quantum theories” in which all matrices/vectors are finite-dimensional. One loses some physics, it need hardly be said, but surprisingly much of genuine physical interest does survive. And one gains the advantage of sharpened analytical power: “finite-dimensional quantum mechanics” provides a methodological laboratory in which, not infrequently, the essentials of complicated computational procedures can be exposed with closed-form transparency. Finally, the toy theory serves to identify some unanticipated formal links—permitting ideas to flow back and forth— between quantum mechanics and other branches of physics. Here we will carry the technique to the limit: we will look to “2-dimensional quantum mechanics.” The theory preserves the linearity that dominates the full-blown theory, and is of the least-possible size in which it is possible for the effects of non-commutivity to become manifest. 2 Quantum theory of 2-state systems We have seen that quantum mechanics can be portrayed as a theory in which • states are represented by self-adjoint linear operators ρ ; • motion is generated by self-adjoint linear operators H; • measurement devices are represented by self-adjoint linear operators A. -
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.