Lecture 5: Operators and the Schrödinger Equation
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
1 Notation for States
The S-Matrix1 D. E. Soper2 University of Oregon Physics 634, Advanced Quantum Mechanics November 2000 1 Notation for states In these notes we discuss scattering nonrelativistic quantum mechanics. We will use states with the nonrelativistic normalization h~p |~ki = (2π)3δ(~p − ~k). (1) Recall that in a relativistic theory there is an extra factor of 2E on the right hand side of this relation, where E = [~k2 + m2]1/2. We will use states in the “Heisenberg picture,” in which states |ψ(t)i do not depend on time. Often in quantum mechanics one uses the Schr¨odinger picture, with time dependent states |ψ(t)iS. The relation between these is −iHt |ψ(t)iS = e |ψi. (2) Thus these are the same at time zero, and the Schrdinger states obey d i |ψ(t)i = H |ψ(t)i (3) dt S S In the Heisenberg picture, the states do not depend on time but the op- erators do depend on time. A Heisenberg operator O(t) is related to the corresponding Schr¨odinger operator OS by iHt −iHt O(t) = e OS e (4) Thus hψ|O(t)|ψi = Shψ(t)|OS|ψ(t)iS. (5) The Heisenberg picture is favored over the Schr¨odinger picture in the case of relativistic quantum mechanics: we don’t have to say which reference 1Copyright, 2000, D. E. Soper [email protected] 1 frame we use to define t in |ψ(t)iS. For operators, we can deal with local operators like, for instance, the electric field F µν(~x, t). -
Symmetry Conditions on Dirac Observables
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 2, 671–676 Symmetry Conditions on Dirac Observables Heinz Otto CORDES Department of Mathematics, University of California, Berkeley, CA 94720 USA E-mail: [email protected] Using our Dirac invariant algebra we attempt a mathematically and philosophically clean theory of Dirac observables, within the environment of the 4 books [10,9,11,4]. All classical objections to one-particle Dirac theory seem removed, while also some principal objections to von Neumann’s observable theory may be cured. Heisenberg’s uncertainty principle appears improved. There is a clean and mathematically precise pseudodifferential Foldy–Wouthuysen transform, not only for the supersymmeytric case, but also for general (C∞-) potentials. 1 Introduction The free Dirac Hamiltonian H0 = αD+β is a self-adjoint square root of 1−∆ , with the Laplace HS − 1 operator ∆. The free Schr¨odinger√ Hamiltonian 0 =1 2 ∆ (where “1” is the “rest energy” 2 1 mc ) seems related to H0 like 1+x its approximation 1+ 2 x – second partial sum of its Taylor expansion. From that aspect the early discoverers of Quantum Mechanics were lucky that the energies of the bound states of hydrogen (a few eV) are small, compared to the mass energy of an electron (approx. 500 000 eV), because this should make “Schr¨odinger” a good approximation of “Dirac”. But what about the continuous spectrum of both operators – governing scattering theory. Note that today big machines scatter with energies of about 10,000 – compared to the above 1 = mc2. So, from that aspect the precise scattering theory of both Hamiltonians (with potentials added) should give completely different results, and one may have to put ones trust into one of them, because at most one of them can be applicable. -
Math 353 Lecture Notes Intro to Pdes Eigenfunction Expansions for Ibvps
Math 353 Lecture Notes Intro to PDEs Eigenfunction expansions for IBVPs J. Wong (Fall 2020) Topics covered • Eigenfunction expansions for PDEs ◦ The procedure for time-dependent problems ◦ Projection, independent evolution of modes 1 The eigenfunction method to solve PDEs We are now ready to demonstrate how to use the components derived thus far to solve the heat equation. First, two examples to illustrate the proces... 1.1 Example 1: no source; Dirichlet BCs The simplest case. We solve ut =uxx; x 2 (0; 1); t > 0 (1.1a) u(0; t) = 0; u(1; t) = 0; (1.1b) u(x; 0) = f(x): (1.1c) The eigenvalues/eigenfunctions are (as calculated in previous sections) 2 2 λn = n π ; φn = sin nπx; n ≥ 1: (1.2) Assuming the solution exists, it can be written in the eigenfunction basis as 1 X u(x; t) = cn(t)φn(x): n=0 Definition (modes) The n-th term of this series is sometimes called the n-th mode or Fourier mode. I'll use the word frequently to describe it (rather than, say, `basis function'). 1 00 Substitute into the PDE (1.1a) and use the fact that −φn = λnφ to obtain 1 X 0 (cn(t) + λncn(t))φn(x) = 0: n=1 By the fact the fφng is a basis, it follows that the coefficient for each mode satisfies the ODE 0 cn(t) + λncn(t) = 0: Solving the ODE gives us a `general' solution to the PDE with its BCs, 1 X −λnt u(x; t) = ane φn(x): n=1 The remaining coefficients are determined by the IC, u(x; 0) = f(x): To match to the solution, we need to also write f(x) in the basis: 1 Z 1 X hf; φni f(x) = f φ (x); f = = 2 f(x) sin nπx dx: (1.3) n n n hφ ; φ i n=1 n n 0 Then from the initial condition, we get u(x; 0) = f(x) 1 1 X X =) cn(0)φn(x) = fnφn(x) n=1 n=1 =) cn(0) = fn for all n ≥ 1: Now everything has been solved - we are done! The solution to the IBVP (1.1) is 1 X −n2π2t u(x; t) = ane sin nπx with an given by (1.5): (1.4) n=1 Alternatively, we could state the solution as follows: The solution is 1 X −λnt u(x; t) = fne φn(x) n=1 with eigenfunctions/values φn; λn given by (1.2) and fn by (1.3). -
Math 356 Lecture Notes Intro to Pdes Eigenfunction Expansions for Ibvps
MATH 356 LECTURE NOTES INTRO TO PDES EIGENFUNCTION EXPANSIONS FOR IBVPS J. WONG (FALL 2019) Topics covered • Eigenfunction expansions for PDEs ◦ The procedure for time-dependent problems ◦ Projection, independent evolution of modes ◦ Separation of variables • The wave equation ◦ Standard solution, standing waves ◦ Example: tuning the vibrating string (resonance) • Interpreting solutions (what does the series mean?) ◦ Relevance of Fourier convergence theorem ◦ Smoothing for the heat equation ◦ Non-smoothing for the wave equation 3 1. The eigenfunction method to solve PDEs We are now ready to demonstrate how to use the components derived thus far to solve the heat equation. The procedure here for the heat equation will extend nicely to a variety of other problems. For now, consider an initial boundary value problem of the form ut = −Lu + h(x; t); x 2 (a; b); t > 0 hom. BCs at a and b (1.1) u(x; 0) = f(x) We seek a solution in terms of the eigenfunction basis X u(x; t) = cn(t)φn(x) n by finding simple ODEs to solve for the coefficients cn(t): This form of the solution is called an eigenfunction expansion for u (or `eigenfunction series') and each term cnφn(x) is a mode (or `Fourier mode' or `eigenmode'). Part 1: find the eigenfunction basis. The first step is to compute the basis. The eigenfunctions we need are the solutions to the eigenvalue problem Lφ = λφ, φ(x) satisfies the BCs for u: (1.2) By the theorem in ??, there is a sequence of eigenfunctions fφng with eigenvalues fλng that form an orthogonal basis for L2[a; b] (i.e. -
Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 December 2019 doi:10.20944/preprints201912.0044.v1 Peer-reviewed version available at Universe 2020, 6, 24; doi:10.3390/universe6020024 1 Review 2 Rotation and Spin and Position Operators in 3 Relativistic Gravity and Quantum Electrodynamics 4 R.F. O’Connell 5 Department of Physics and Astronomy, Louisiana State University; Baton Rouge, LA 70803-4001, USA 6 Correspondence: [email protected]; 225-578-6848 7 Abstract: First, we examine how spin is treated in special relativity and the necessity of introducing 8 spin supplementary conditions (SSC) and how they are related to the choice of a center-of-mass of a 9 spinning particle. Next, we discuss quantum electrodynamics and the Foldy-Wouthuysen 10 transformation which we note is a position operator identical to the Pryce-Newton-Wigner position 11 operator. The classical version of the operators are shown to be essential for the treatment of 12 classical relativistic particles in general relativity, of special interest being the case of binary 13 systems (black holes/neutron stars) which emit gravitational radiation. 14 Keywords: rotation; spin; position operators 15 16 I.Introduction 17 Rotation effects in relativistic systems involve many new concepts not needed in 18 non-relativistic classical physics. Some of these are quantum mechanical (where the emphasis is on 19 “spin”). Thus, our emphasis will be on quantum electrodynamics (QED) and both special and 20 general relativity. Also, as in [1] , we often use “spin” in the generic sense of meaning “internal” spin 21 in the case of an elementary particle and “rotation” in the case of an elementary particle. -
Relational Quantum Mechanics and Probability M
Relational Quantum Mechanics and Probability M. Trassinelli To cite this version: M. Trassinelli. Relational Quantum Mechanics and Probability. Foundations of Physics, Springer Verlag, 2018, 10.1007/s10701-018-0207-7. hal-01723999v3 HAL Id: hal-01723999 https://hal.archives-ouvertes.fr/hal-01723999v3 Submitted on 29 Aug 2018 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. Noname manuscript No. (will be inserted by the editor) Relational Quantum Mechanics and Probability M. Trassinelli the date of receipt and acceptance should be inserted later Abstract We present a derivation of the third postulate of Relational Quan- tum Mechanics (RQM) from the properties of conditional probabilities. The first two RQM postulates are based on the information that can be extracted from interaction of different systems, and the third postulate defines the prop- erties of the probability function. Here we demonstrate that from a rigorous definition of the conditional probability for the possible outcomes of different measurements, the third postulate is unnecessary and the Born's rule naturally emerges from the first two postulates by applying the Gleason's theorem. We demonstrate in addition that the probability function is uniquely defined for classical and quantum phenomena. -
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. -
An S-Matrix for Massless Particles Arxiv:1911.06821V2 [Hep-Th]
An S-Matrix for Massless Particles Holmfridur Hannesdottir and Matthew D. Schwartz Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract The traditional S-matrix does not exist for theories with massless particles, such as quantum electrodynamics. The difficulty in isolating asymptotic states manifests itself as infrared divergences at each order in perturbation theory. Building on insights from the literature on coherent states and factorization, we construct an S-matrix that is free of singularities order-by-order in perturbation theory. Factorization guarantees that the asymptotic evolution in gauge theories is universal, i.e. independent of the hard process. Although the hard S-matrix element is computed between well-defined few particle Fock states, dressed/coherent states can be seen to form as intermediate states in the calculation of hard S-matrix elements. We present a framework for the perturbative calculation of hard S-matrix elements combining Lorentz-covariant Feyn- man rules for the dressed-state scattering with time-ordered perturbation theory for the asymptotic evolution. With hard cutoffs on the asymptotic Hamiltonian, the cancella- tion of divergences can be seen explicitly. In dimensional regularization, where the hard cutoffs are replaced by a renormalization scale, the contribution from the asymptotic evolution produces scaleless integrals that vanish. A number of illustrative examples are given in QED, QCD, and N = 4 super Yang-Mills theory. arXiv:1911.06821v2 [hep-th] 24 Aug 2020 Contents 1 Introduction1 2 The hard S-matrix6 2.1 SH and dressed states . .9 2.2 Computing observables using SH ........................... 11 2.3 Soft Wilson lines . 14 3 Computing the hard S-matrix 17 3.1 Asymptotic region Feynman rules . -
Identical Particles
8.06 Spring 2016 Lecture Notes 4. Identical particles Aram Harrow Last updated: May 19, 2016 Contents 1 Fermions and Bosons 1 1.1 Introduction and two-particle systems .......................... 1 1.2 N particles ......................................... 3 1.3 Non-interacting particles .................................. 5 1.4 Non-zero temperature ................................... 7 1.5 Composite particles .................................... 7 1.6 Emergence of distinguishability .............................. 9 2 Degenerate Fermi gas 10 2.1 Electrons in a box ..................................... 10 2.2 White dwarves ....................................... 12 2.3 Electrons in a periodic potential ............................. 16 3 Charged particles in a magnetic field 21 3.1 The Pauli Hamiltonian ................................... 21 3.2 Landau levels ........................................ 23 3.3 The de Haas-van Alphen effect .............................. 24 3.4 Integer Quantum Hall Effect ............................... 27 3.5 Aharonov-Bohm Effect ................................... 33 1 Fermions and Bosons 1.1 Introduction and two-particle systems Previously we have discussed multiple-particle systems using the tensor-product formalism (cf. Section 1.2 of Chapter 3 of these notes). But this applies only to distinguishable particles. In reality, all known particles are indistinguishable. In the coming lectures, we will explore the mathematical and physical consequences of this. First, consider classical many-particle systems. If a single particle has state described by position and momentum (~r; p~), then the state of N distinguishable particles can be written as (~r1; p~1; ~r2; p~2;:::; ~rN ; p~N ). The notation (·; ·;:::; ·) denotes an ordered list, in which different posi tions have different meanings; e.g. in general (~r1; p~1; ~r2; p~2)6 = (~r2; p~2; ~r1; p~1). 1 To describe indistinguishable particles, we can use set notation. -
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. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
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.