Path Integral Methods
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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. -
Motion of the Reduced Density Operator
Motion of the Reduced Density Operator Nicholas Wheeler, Reed College Physics Department Spring 2009 Introduction. Quantum mechanical decoherence, dissipation and measurements all involve the interaction of the system of interest with an environmental system (reservoir, measurement device) that is typically assumed to possess a great many degrees of freedom (while the system of interest is typically assumed to possess relatively few degrees of freedom). The state of the composite system is described by a density operator ρ which in the absence of system-bath interaction we would denote ρs ρe, though in the cases of primary interest that notation becomes unavailable,⊗ since in those cases the states of the system and its environment are entangled. The observable properties of the system are latent then in the reduced density operator ρs = tre ρ (1) which is produced by “tracing out” the environmental component of ρ. Concerning the specific meaning of (1). Let n) be an orthonormal basis | in the state space H of the (open) system, and N) be an orthonormal basis s !| " in the state space H of the (also open) environment. Then n) N) comprise e ! " an orthonormal basis in the state space H = H H of the |(closed)⊗| composite s e ! " system. We are in position now to write ⊗ tr ρ I (N ρ I N) e ≡ s ⊗ | s ⊗ | # ! " ! " ↓ = ρ tr ρ in separable cases s · e The dynamics of the composite system is generated by Hamiltonian of the form H = H s + H e + H i 2 Motion of the reduced density operator where H = h I s s ⊗ e = m h n m N n N $ | s| % | % ⊗ | % · $ | ⊗ $ | m,n N # # $% & % &' H = I h e s ⊗ e = n M n N M h N | % ⊗ | % · $ | ⊗ $ | $ | e| % n M,N # # $% & % &' H = m M m M H n N n N i | % ⊗ | % $ | ⊗ $ | i | % ⊗ | % $ | ⊗ $ | m,n M,N # # % &$% & % &'% & —all components of which we will assume to be time-independent. -
Coherent State Path Integral for the Harmonic Oscillator 11
COHERENT STATE PATH INTEGRAL FOR THE HARMONIC OSCILLATOR AND A SPIN PARTICLE IN A CONSTANT MAGNETIC FLELD By MARIO BERGERON B. Sc. (Physique) Universite Laval, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1989 © MARIO BERGERON, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis . for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The definition and formulas for the harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism is also reviewed with its relation and the partition function of a sytem is also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level, and its relation with various regularizations is established. The use of harmonic oscillator coherent states and spin coherent states for the computation of the path integral for a particle of spin s put in a magnetic field is caried out in several ways, and a careful analysis of infinitesimal terms (in 1/N where TV is the number of time slices) is done explicitly. -
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. -
Coherent and Squeezed States on Physical Basis
PRAMANA © Printed in India Vol. 48, No. 3, __ journal of March 1997 physics pp. 787-797 Coherent and squeezed states on physical basis R R PURI Theoretical Physics Division, Central Complex, Bhabha Atomic Research Centre, Bombay 400 085, India MS received 6 April 1996; revised 24 December 1996 Abstract. A definition of coherent states is proposed as the minimum uncertainty states with equal variance in two hermitian non-commuting generators of the Lie algebra of the hamil- tonian. That approach classifies the coherent states into distinct classes. The coherent states of a harmonic oscillator, according to the proposed approach, are shown to fall in two classes. One is the familiar class of Glauber states whereas the other is a new class. The coherent states of spin constitute only one class. The squeezed states are similarly defined on the physical basis as the states that give better precision than the coherent states in a process of measurement of a force coupled to the given system. The condition of squeezing based on that criterion is derived for a system of spins. Keywords. Coherent state; squeezed state; Lie algebra; SU(m, n). PACS Nos 03.65; 02-90 1. Introduction The hamiltonian of a quantum system is generally a linear combination of a set of operators closed under the operation of commutation i.e. they constitute a Lie algebra. Since their introduction for a hamiltonian linear in harmonic oscillator (HO) oper- ators, the notion of coherent states [1] has been extended to other hamiltonians (see [2-9] and references therein). Physically, the HO coherent states are the minimum uncertainty states of the two quadratures of the bosonic operator with equal variance in the two. -
1 Coherent States and Path Integral Quantiza- Tion. 1.1 Coherent States Let Q and P Be the Coordinate and Momentum Operators
1 Coherent states and path integral quantiza- tion. 1.1 Coherent States Let q and p be the coordinate and momentum operators. They satisfy the Heisenberg algebra, [q,p] = i~. Let us introduce the creation and annihilation operators a† and a, by their standard relations ~ q = a† + a (1) r2mω m~ω p = i a† a (2) r 2 − Let us consider a Hilbert space spanned by a complete set of harmonic os- cillator states n , with n =0,..., . Leta ˆ† anda ˆ be a pair of creation and annihilation operators{| i} acting on that∞ Hilbert space, and satisfying the commu- tation relations a,ˆ aˆ† =1 , aˆ†, aˆ† =0 , [ˆa, aˆ] = 0 (3) These operators generate the harmonic oscillators states n in the usual way, {| i} 1 n n = aˆ† 0 (4) | i √n! | i aˆ 0 = 0 (5) | i where 0 is the vacuum state of the oscillator. Let| usi denote by z the coherent state | i † z = ezaˆ 0 (6) | i | i z = 0 ez¯aˆ (7) h | h | where z is an arbitrary complex number andz ¯ is the complex conjugate. The coherent state z has the defining property of being a wave packet with opti- mal spread, i.e.,| ithe Heisenberg uncertainty inequality is an equality for these coherent states. How doesa ˆ act on the coherent state z ? | i ∞ n z n aˆ z = aˆ aˆ† 0 (8) | i n! | i n=0 X Since n n−1 a,ˆ aˆ† = n aˆ† (9) we get h i ∞ n z n n aˆ z = a,ˆ aˆ† + aˆ† aˆ 0 (10) | i n! | i n=0 X h i 1 Thus, we find ∞ n z n−1 aˆ z = n aˆ† 0 z z (11) | i n! | i≡ | i n=0 X Therefore z is a right eigenvector ofa ˆ and z is the (right) eigenvalue. -
Detailing Coherent, Minimum Uncertainty States of Gravitons, As
Journal of Modern Physics, 2011, 2, 730-751 doi:10.4236/jmp.2011.27086 Published Online July 2011 (http://www.SciRP.org/journal/jmp) Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits to Graviton Mass Andrew Beckwith1 Department of Physics, Chongqing University, Chongqing, China E-mail: [email protected] Received April 12, 2011; revised June 1, 2011; accepted June 13, 2011 Abstract We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW, and also gravitons. A side issue to consider is if gravitons can be configured as semi classical “particles”, which is akin to the Pilot model of Quantum Mechanics as embedded in a larger non linear “de- terministic” background. Keywords: Squeezed State, Graviton, GW, Pilot Model 1. Introduction condensed matter application. The string theory metho- dology is merely extending much the same thinking up to Gravitons may be de composed via an instanton-anti higher than four dimensional situations. instanton structure. i.e. that the structure of SO(4) gauge 1) Modeling of entropy, generally, as kink-anti-kinks theory is initially broken due to the introduction of pairs with N the number of the kink-anti-kink pairs. vacuum energy [1], so after a second-order phase tran- This number, N is, initially in tandem with entropy sition, the instanton-anti-instanton structure of relic production, as will be explained later, gravitons is reconstituted. This will be crucial to link 2) The tie in with entropy and gravitons is this: the two graviton production with entropy, provided we have structures are related to each other in terms of kinks and sufficiently HFGW at the origin of the big bang. -
Density Matrix Description of NMR
Density Matrix Description of NMR BCMB/CHEM 8190 Operators in Matrix Notation • It will be important, and convenient, to express the commonly used operators in matrix form • Consider the operator Iz and the single spin functions α and β - recall ˆ ˆ ˆ ˆ ˆ ˆ Ix α = 1 2 β Ix β = 1 2α Iy α = 1 2 iβ Iy β = −1 2 iα Iz α = +1 2α Iz β = −1 2 β α α = β β =1 α β = β α = 0 - recall the expectation value for an observable Q = ψ Qˆ ψ = ∫ ψ∗Qˆψ dτ Qˆ - some operator ψ - some wavefunction - the matrix representation is the possible expectation values for the basis functions α β α ⎡ α Iˆ α α Iˆ β ⎤ ⎢ z z ⎥ ⎢ ˆ ˆ ⎥ β ⎣ β Iz α β Iz β ⎦ ⎡ α Iˆ α α Iˆ β ⎤ ⎡1 2 α α −1 2 α β ⎤ ⎡1 2 0 ⎤ ⎡1 0 ⎤ ˆ ⎢ z z ⎥ 1 Iz = = ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ ⎢ ˆ ˆ ⎥ 1 2 β α −1 2 β β 0 −1 2 2 0 −1 ⎣ β Iz α β Iz β ⎦ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎣ ⎦ • This is convenient, as the operator is just expressed as a matrix of numbers – no need to derive it again, just store it in computer Operators in Matrix Notation • The matrices for Ix, Iy,and Iz are called the Pauli spin matrices ˆ ⎡ 0 1 2⎤ 1 ⎡0 1 ⎤ ˆ ⎡0 −1 2 i⎤ 1 ⎡0 −i ⎤ ˆ ⎡1 2 0 ⎤ 1 ⎡1 0 ⎤ Ix = ⎢ ⎥ = ⎢ ⎥ Iy = ⎢ ⎥ = ⎢ ⎥ Iz = ⎢ ⎥ = ⎢ ⎥ ⎣1 2 0 ⎦ 2 ⎣1 0⎦ ⎣1 2 i 0 ⎦ 2 ⎣i 0 ⎦ ⎣ 0 −1 2⎦ 2 ⎣0 −1 ⎦ • Express α , β , α and β as 1×2 column and 2×1 row vectors ⎡1⎤ ⎡0⎤ α = ⎢ ⎥ β = ⎢ ⎥ α = [1 0] β = [0 1] ⎣0⎦ ⎣1⎦ • Using matrices, the operations of Ix, Iy, and Iz on α and β , and the orthonormality relationships, are shown below ˆ 1 ⎡0 1⎤⎡1⎤ 1 ⎡0⎤ 1 ˆ 1 ⎡0 1⎤⎡0⎤ 1 ⎡1⎤ 1 Ix α = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = β Ix β = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = α 2⎣1 0⎦⎣0⎦ 2⎣1⎦ 2 2⎣1 0⎦⎣1⎦ 2⎣0⎦ 2 ⎡1⎤ ⎡0⎤ α α = [1 0]⎢ ⎥ -
Basics of Thermal Field Theory
September 2021 Basics of Thermal Field Theory A Tutorial on Perturbative Computations 1 Mikko Lainea and Aleksi Vuorinenb aAEC, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland bDepartment of Physics, University of Helsinki, P.O. Box 64, FI-00014 University of Helsinki, Finland Abstract These lecture notes, suitable for a two-semester introductory course or self-study, offer an elemen- tary and self-contained exposition of the basic tools and concepts that are encountered in practical computations in perturbative thermal field theory. Selected applications to heavy ion collision physics and cosmology are outlined in the last chapter. 1An earlier version of these notes is available as an ebook (Springer Lecture Notes in Physics 925) at dx.doi.org/10.1007/978-3-319-31933-9; a citable eprint can be found at arxiv.org/abs/1701.01554; the very latest version is kept up to date at www.laine.itp.unibe.ch/basics.pdf. Contents Foreword ........................................... i Notation............................................ ii Generaloutline....................................... iii 1 Quantummechanics ................................... 1 1.1 Path integral representation of the partition function . ....... 1 1.2 Evaluation of the path integral for the harmonic oscillator . ..... 6 2 Freescalarfields ..................................... 13 2.1 Path integral for the partition function . ... 13 2.2 Evaluation of thermal sums and their low-temperature limit . .... 16 2.3 High-temperatureexpansion. 23 3 Interactingscalarfields............................... ... 30 3.1 Principles of the weak-coupling expansion . 30 3.2 Problems of the naive weak-coupling expansion . .. 38 3.3 Proper free energy density to (λ): ultraviolet renormalization . 40 O 3 3.4 Proper free energy density to (λ 2 ): infraredresummation . -
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. -
ECE 604, Lecture 39
ECE 604, Lecture 39 Fri, April 26, 2019 Contents 1 Quantum Coherent State of Light 2 1.1 Quantum Harmonic Oscillator Revisited . 2 2 Some Words on Quantum Randomness and Quantum Observ- ables 4 3 Derivation of the Coherent States 5 3.1 Time Evolution of a Quantum State . 7 3.1.1 Time Evolution of the Coherent State . 7 4 More on the Creation and Annihilation Operator 8 4.1 Connecting Quantum Pendulum to Electromagnetic Oscillator . 10 Printed on April 26, 2019 at 23 : 35: W.C. Chew and D. Jiao. 1 ECE 604, Lecture 39 Fri, April 26, 2019 1 Quantum Coherent State of Light We have seen that a photon number state1 of a quantum pendulum do not have a classical correspondence as the average or expectation values of the position and momentum of the pendulum are always zero for all time for this state. Therefore, we have to seek a time-dependent quantum state that has the classical equivalence of a pendulum. This is the coherent state, which is the contribution of many researchers, most notably, George Sudarshan (1931{2018) and Roy Glauber (born 1925) in 1963. Glauber was awarded the Nobel prize in 2005. We like to emphasize again that the mode of an electromagnetic oscillation is homomorphic to the oscillation of classical pendulum. Hence, we first con- nect the oscillation of a quantum pendulum to a classical pendulum. Then we can connect the oscillation of a quantum electromagnetic mode to the classical electromagnetic mode and then to the quantum pendulum. 1.1 Quantum Harmonic Oscillator Revisited To this end, we revisit the quantum harmonic oscillator or the quantum pen- dulum with more mathematical depth. -
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.