12 Wavefunction Collapse
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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. -
Unit 1 Old Quantum Theory
UNIT 1 OLD QUANTUM THEORY Structure Introduction Objectives li;,:overy of Sub-atomic Particles Earlier Atom Models Light as clectromagnetic Wave Failures of Classical Physics Black Body Radiation '1 Heat Capacity Variation Photoelectric Effect Atomic Spectra Planck's Quantum Theory, Black Body ~diation. and Heat Capacity Variation Einstein's Theory of Photoelectric Effect Bohr Atom Model Calculation of Radius of Orbits Energy of an Electron in an Orbit Atomic Spectra and Bohr's Theory Critical Analysis of Bohr's Theory Refinements in the Atomic Spectra The61-y Summary Terminal Questions Answers 1.1 INTRODUCTION The ideas of classical mechanics developed by Galileo, Kepler and Newton, when applied to atomic and molecular systems were found to be inadequate. Need was felt for a theory to describe, correlate and predict the behaviour of the sub-atomic particles. The quantum theory, proposed by Max Planck and applied by Einstein and Bohr to explain different aspects of behaviour of matter, is an important milestone in the formulation of the modern concept of atom. In this unit, we will study how black body radiation, heat capacity variation, photoelectric effect and atomic spectra of hydrogen can be explained on the basis of theories proposed by Max Planck, Einstein and Bohr. They based their theories on the postulate that all interactions between matter and radiation occur in terms of definite packets of energy, known as quanta. Their ideas, when extended further, led to the evolution of wave mechanics, which shows the dual nature of matter -
Quantum Theory of the Hydrogen Atom
Quantum Theory of the Hydrogen Atom Chemistry 35 Fall 2000 Balmer and the Hydrogen Spectrum n 1885: Johann Balmer, a Swiss schoolteacher, empirically deduced a formula which predicted the wavelengths of emission for Hydrogen: l (in Å) = 3645.6 x n2 for n = 3, 4, 5, 6 n2 -4 •Predicts the wavelengths of the 4 visible emission lines from Hydrogen (which are called the Balmer Series) •Implies that there is some underlying order in the atom that results in this deceptively simple equation. 2 1 The Bohr Atom n 1913: Niels Bohr uses quantum theory to explain the origin of the line spectrum of hydrogen 1. The electron in a hydrogen atom can exist only in discrete orbits 2. The orbits are circular paths about the nucleus at varying radii 3. Each orbit corresponds to a particular energy 4. Orbit energies increase with increasing radii 5. The lowest energy orbit is called the ground state 6. After absorbing energy, the e- jumps to a higher energy orbit (an excited state) 7. When the e- drops down to a lower energy orbit, the energy lost can be given off as a quantum of light 8. The energy of the photon emitted is equal to the difference in energies of the two orbits involved 3 Mohr Bohr n Mathematically, Bohr equated the two forces acting on the orbiting electron: coulombic attraction = centrifugal accelleration 2 2 2 -(Z/4peo)(e /r ) = m(v /r) n Rearranging and making the wild assumption: mvr = n(h/2p) n e- angular momentum can only have certain quantified values in whole multiples of h/2p 4 2 Hydrogen Energy Levels n Based on this model, Bohr arrived at a simple equation to calculate the electron energy levels in hydrogen: 2 En = -RH(1/n ) for n = 1, 2, 3, 4, . -
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. -
Light and Matter Diffraction from the Unified Viewpoint of Feynman's
European J of Physics Education Volume 8 Issue 2 1309-7202 Arlego & Fanaro Light and Matter Diffraction from the Unified Viewpoint of Feynman’s Sum of All Paths Marcelo Arlego* Maria de los Angeles Fanaro** Universidad Nacional del Centro de la Provincia de Buenos Aires CONICET, Argentine *[email protected] **[email protected] (Received: 05.04.2018, Accepted: 22.05.2017) Abstract In this work, we present a pedagogical strategy to describe the diffraction phenomenon based on a didactic adaptation of the Feynman’s path integrals method, which uses only high school mathematics. The advantage of our approach is that it allows to describe the diffraction in a fully quantum context, where superposition and probabilistic aspects emerge naturally. Our method is based on a time-independent formulation, which allows modelling the phenomenon in geometric terms and trajectories in real space, which is an advantage from the didactic point of view. A distinctive aspect of our work is the description of the series of transformations and didactic transpositions of the fundamental equations that give rise to a common quantum framework for light and matter. This is something that is usually masked by the common use, and that to our knowledge has not been emphasized enough in a unified way. Finally, the role of the superposition of non-classical paths and their didactic potential are briefly mentioned. Keywords: quantum mechanics, light and matter diffraction, Feynman’s Sum of all Paths, high education INTRODUCTION This work promotes the teaching of quantum mechanics at the basic level of secondary school, where the students have not the necessary mathematics to deal with canonical models that uses Schrodinger equation. -
Vibrational Quantum Number
Fundamentals in Biophotonics Quantum nature of atoms, molecules – matter Aleksandra Radenovic [email protected] EPFL – Ecole Polytechnique Federale de Lausanne Bioengineering Institute IBI 26. 03. 2018. Quantum numbers •The four quantum numbers-are discrete sets of integers or half- integers. –n: Principal quantum number-The first describes the electron shell, or energy level, of an atom –ℓ : Orbital angular momentum quantum number-as the angular quantum number or orbital quantum number) describes the subshell, and gives the magnitude of the orbital angular momentum through the relation Ll2 ( 1) –mℓ:Magnetic (azimuthal) quantum number (refers, to the direction of the angular momentum vector. The magnetic quantum number m does not affect the electron's energy, but it does affect the probability cloud)- magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field-Zeeman effect -s spin- intrinsic angular momentum Spin "up" and "down" allows two electrons for each set of spatial quantum numbers. The restrictions for the quantum numbers: – n = 1, 2, 3, 4, . – ℓ = 0, 1, 2, 3, . , n − 1 – mℓ = − ℓ, − ℓ + 1, . , 0, 1, . , ℓ − 1, ℓ – –Equivalently: n > 0 The energy levels are: ℓ < n |m | ≤ ℓ ℓ E E 0 n n2 Stern-Gerlach experiment If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by a different amount, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount. -
Origin of Probability in Quantum Mechanics and the Physical Interpretation of the Wave Function
Origin of Probability in Quantum Mechanics and the Physical Interpretation of the Wave Function Shuming Wen ( [email protected] ) Faculty of Land and Resources Engineering, Kunming University of Science and Technology. Research Article Keywords: probability origin, wave-function collapse, uncertainty principle, quantum tunnelling, double-slit and single-slit experiments Posted Date: November 16th, 2020 DOI: https://doi.org/10.21203/rs.3.rs-95171/v2 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Origin of Probability in Quantum Mechanics and the Physical Interpretation of the Wave Function Shuming Wen Faculty of Land and Resources Engineering, Kunming University of Science and Technology, Kunming 650093 Abstract The theoretical calculation of quantum mechanics has been accurately verified by experiments, but Copenhagen interpretation with probability is still controversial. To find the source of the probability, we revised the definition of the energy quantum and reconstructed the wave function of the physical particle. Here, we found that the energy quantum ê is 6.62606896 ×10-34J instead of hν as proposed by Planck. Additionally, the value of the quality quantum ô is 7.372496 × 10-51 kg. This discontinuity of energy leads to a periodic non-uniform spatial distribution of the particles that transmit energy. A quantum objective system (QOS) consists of many physical particles whose wave function is the superposition of the wave functions of all physical particles. The probability of quantum mechanics originates from the distribution rate of the particles in the QOS per unit volume at time t and near position r. Based on the revision of the energy quantum assumption and the origin of the probability, we proposed new certainty and uncertainty relationships, explained the physical mechanism of wave-function collapse and the quantum tunnelling effect, derived the quantum theoretical expression of double-slit and single-slit experiments. -
Testing the Limits of Quantum Mechanics the Physics Underlying
Testing the limits of quantum mechanics The physics underlying non-relativistic quantum mechanics can be summed up in two postulates. Postulate 1 is very precise, and says that the wave function of a quantum system evolves according to the Schrodinger equation, which is a linear and deterministic equation. Postulate 2 has an entirely different flavor, and can be roughly stated as follows: when the quantum system interacts with a classical measuring apparatus, its wave function collapses, from being in a superposition of the eigenstates of the measured observable, to being in just one of the eigenstates. The outcome of the measurement is random and cannot be predicted; the quantum system collapses to one or the other eigenstates, with a probability that is proportional to the squared modulus of the wave function for that eigenstate. This is the Born probability rule. Since quantum theory is extremely successful, and not contradicted by any experiment to date, one can simply accept the 2nd postulate as such, and let things be. On the other hand, ever since the birth of quantum theory, some physicists have been bothered by this postulate. The following troubling questions arise. How exactly is a classical measuring apparatus defined? How large must a quantum system be, before it can be called classical? The Schrodinger equation, which in principle is supposed to apply to all physical systems, whether large or small, does not answer this question. In particular, the equation does not explain why the measuring apparatus, say a pointer, is never seen in a quantum superposition of the two states `pointer to the left’ and `pointer to the right’? And if the equation does apply to the (quantum system + apparatus) as a whole, why are the outcomes random? Why does collapse, which apparently violates linear superposition, take place? Where have the probabilities come from, in a deterministic equation (with precise initial conditions), and why do they obey the Born rule? This set of questions generally goes under the name `the quantum measurement problem’. -
The Quantum Mechanical Model of the Atom
The Quantum Mechanical Model of the Atom Quantum Numbers In order to describe the probable location of electrons, they are assigned four numbers called quantum numbers. The quantum numbers of an electron are kind of like the electron’s “address”. No two electrons can be described by the exact same four quantum numbers. This is called The Pauli Exclusion Principle. • Principle quantum number: The principle quantum number describes which orbit the electron is in and therefore how much energy the electron has. - it is symbolized by the letter n. - positive whole numbers are assigned (not including 0): n=1, n=2, n=3 , etc - the higher the number, the further the orbit from the nucleus - the higher the number, the more energy the electron has (this is sort of like Bohr’s energy levels) - the orbits (energy levels) are also called shells • Angular momentum (azimuthal) quantum number: The azimuthal quantum number describes the sublevels (subshells) that occur in each of the levels (shells) described above. - it is symbolized by the letter l - positive whole number values including 0 are assigned: l = 0, l = 1, l = 2, etc. - each number represents the shape of a subshell: l = 0, represents an s subshell l = 1, represents a p subshell l = 2, represents a d subshell l = 3, represents an f subshell - the higher the number, the more complex the shape of the subshell. The picture below shows the shape of the s and p subshells: (notice the electron clouds) • Magnetic quantum number: All of the subshells described above (except s) have more than one orientation. -
4 Nuclear Magnetic Resonance
Chapter 4, page 1 4 Nuclear Magnetic Resonance Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet. Since then, the splitting of energy levels proportional to an external magnetic field has been called the "Zeeman effect". The "Zeeman resonance effect" causes magnetic resonances which are classified under radio frequency spectroscopy (rf spectroscopy). In these resonances, the transitions between two branches of a single energy level split in an external magnetic field are measured in the megahertz and gigahertz range. In 1944, Jevgeni Konstantinovitch Savoiski discovered electron paramagnetic resonance. Shortly thereafter in 1945, nuclear magnetic resonance was demonstrated almost simultaneously in Boston by Edward Mills Purcell and in Stanford by Felix Bloch. Nuclear magnetic resonance was sometimes called nuclear induction or paramagnetic nuclear resonance. It is generally abbreviated to NMR. So as not to scare prospective patients in medicine, reference to the "nuclear" character of NMR is dropped and the magnetic resonance based imaging systems (scanner) found in hospitals are simply referred to as "magnetic resonance imaging" (MRI). 4.1 The Nuclear Resonance Effect Many atomic nuclei have spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is L =+h II(1). (4.01) The component in the direction of an applied field is Lz = Iz h ≡ m h. (4.02) The external field is usually defined along the z-direction. The magnetic quantum number is symbolized by Iz or m and can have 2I +1 values: Iz ≡ m = −I, −I+1, ..., I−1, I. -
5 the Dirac Equation and Spinors
5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory.