Quantum wires, orthogonal polynomials and Diophantine approximation
|
|
- Rosamund Stafford
- 5 years ago
- Views:
Transcription
1 Quantum wires, orthogonal polynomials and Diophantine approximation
2 Introduction Quantum Mechanics (QM) is a linear theory Main idea behind Quantum Information (QI): use the superposition principle of QM to process faster Basic task in QI: transfer quantum states from one location to another communications between Alice and Bob algorithm (operations in a register) connection between quantum computer parts Quantum wires are needed to that end Goal of the talk: discuss one approach to the design of quantum wires Based on joint work with A. Zhedanov, papers can be found on arxiv
3 Constructing quantum wires is difficult. Need to connect microscopic objects (behaving as per QM) with macroscopic devices doing control Issues of control (micro-macro link) and of noise (decoherence) that control induces One strategy: use one-dimensional chains (more generally graphs) of coupled spin / particles Advantage: no need for control along the chain: the transfer exploits the dynamics of the chain (Noise reduction)
4 Another challenge: construct quantum wires with high fidelity i.e.: it is required that states are transferred between the two locations with probability equal or very close to When = : perfect state transfer (PST) When = : almost perfect state transfer (APST) Part I: PST and orthogonal polynomials Part II: APST and Diophantine approximation
5 spin chains = + (σ σ + + σ σ + )+ (σ + ) = = in (C ) + With insight, we are allowing the couplings and magnetic fields to depend on the site Pauli matrices σ = σ = σ =
6 Action on vectors σ = σ = σ = σ = σ = σ = σ σ = σ σ = σ σ = σ σ = (σ σ + σ σ ) = (σ σ + σ σ ) = Conservation of total -projection [ (σ + )] = = The eigenstates of split in subspaces labeled by the numbers of spins over the chain that are in state
7 State transfer in chain Most state transfer properties can be derived from the -excitation subspace Initially assume that the register is prepared in the state + = Introduce the unknown state ψ = α + β onto the first site ( = ). We want to recuperate ψ on last site ( = ) after some time Component α must go to final state α +. Automatic because + is an eigenstate of It suffices to consider Allows for restriction to -excitation states
8 -Excitation dynamics A natural basis for C + ( -excitation state space) =( ) = where the is in the th position Restriction of to the one-excitation subspace is given by the Jacobi matrix (Recall the action of Pauli matrices on,, etc.
9 Its action on the basis vectors Initial conditions = > = + = Note that the knowledge of prescribes as the couplings and the magnetic fields are then given
10 Condition for perfect state transfer (PST) and almost perfect state transfer (APST) = : ( spin up at =, all others down) QM time evolution: ( ) = Probability to find at site = the spin up that was at site = after some time is PST: ( )=, i.e. ( )= ( ) ( )= ( ) = = φ( )
11 APST: ( ) φ( ) < i.e. for APST we ask that ( ) be as close to as desired Problem: Find the matrix and hence the Hamiltonians such that the PST or the APST condition are satisfied Inverse spectral problems
12 Diagonalization and orthogonal polynomials Eigenbasis : = with = The are real and non-degenerate (i.e. = if = ) Expansion of basis over the other = = = Hence: = χ ( ) where χ ( ) are polynomials satisfying the recurrence relation and conditions + χ + ( )+ χ ( )+ χ ( )= χ ( ) χ = χ =
13 Bases and are orthonormal = δ = δ The matrix = χ ( ) is orthogonal: = δ and = δ = = Polynomials χ ( ) are orthonormal on the finite set of spectral points ω χ ( )χ ( )=δ = where the discrete weights ω = are normalized ω = = = =
14 Dual expansions: = = ω χ ( ) and = ω χ ( = ) Monic polynomials ( )= χ ( )= + ( ) Recurrence relation + ( )+ ( )+ ( )= ( ) where = > Orthogonality relations = ( ) ( )ω = δ =
15 For = + : characteristic polynomial Important formula + ( )=( )( ) ( ) ω = ( ) + ( )) We assume < < Taking the number of exchanges and ordering into account + ( )=( )( ) ( )( + ) ( ) =( ) + + ( ) + ( )=( ) + ( ) ( (-1) factors because is smallest and for each step one less)
16 Necessary and sufficient conditions for PST The PST condition: = φ Expand and in terms of the eigenstates and equate coefficients. Using χ ( )= ω = φ = = ω χ ( ) Whence χ ( )= φ = Necessary and sufficient condition for PST
17 Another formulation Polynomial χ ( ) has real coefficients. Hence χ ( ) =± Interlacing property of zeros of orthogonal polynomials leads to an alternating rule: χ ( ) must have zeros between (the zeros of + ( )) PST condition χ ( )=( ) + = = φ π( + + ) Z = [ φ + π( + + )] Thus the PST condition amounts to the following relation between the neighboring eigenvalues + = π where are odd integers
18 This implies that the weights satisfy the equivalent condition ω = + ( ) > using ω = ( ) + ( ) + ( )=( ) + + ( ) ( ) χ ( )=( ) +
19 Summary Given a -excitation spectrum { } such that = + odd integers Knowing that + ( )=( )( ) ( ) The polynomials orthogonal for the weights ω = + ( ) define a Hamiltonian with. We shall provide an algorithm for constructing ( and thus H)
20 Mirror symmetry Reflected Jacobi matrix: = where is the reflection matrix =.. has the structure: with coefficients: =, = +
21 Corresponding monic orthogonal polynomials + ( )+ ( )+ ( )= ( ) The matrix has the same eigenvalues, = as Property of the weights ω ω = ( + ( )) for = A Jacobi matrix is mirror-symmetric if = = This is equivalent to ω = ω
22 This in in turn implies that ω = + ( ) > () Conversely, if ω =(), then ω = ω and = Conclusion: () or mirror symmetry = are equivalent necessary conditions for PST
23 Reconstruction algorithm for the matrix Given a spectrum { } satisfying the PST condition. We then know explicitly the polynomial + ( )=( )( ) ( ) The orthonormal polynomial χ ( ) is prescribed at + points χ ( )=( ) + = Hence χ ( ) can be explicitly found by Lagrange interpolation χ ( )= = ( ) + where are the Lagrange polynomials = =
24 Starting from monic + ( ) and ( ) it is possible to reconstruct ( ), = by the Euclidean algorithm Recall the -term recurrence relation: + ( )+ ( )+ ( )= ( ) 1 st step: divide the polynomial + ( ) by ( ): + ( )= ( )+ ( ) ( )= ( )= ( ) Next step: repeat this procedure with respect to polynomials ( ) and ( ) and find ( ), and Iteratively, we find all polynomials and all coefficients, = ( ) = This yields
25 New chains with PST from spectral surgery Can we obtain a spin chain with PST from a parent chain known to possess the property? What kind of spectral surgery is admissible? What are the resulting spin chains? Answer: Christoffel transform Remove : ( ) + ω = ( ) > = ( ) + ω = = ( ) =
26 This is equivalent to ω = const ( )ω > = which is a Christoffel transform Removing arbitrary ω = const ( )ω However, if = or, then ω is not positive for all How to circumvent: remove a pair of levels, + In general ω = const ( )( + )ω > ω = const ( )( ) ( )ω where must satisfy obvious conditions for positivity of ω
27 Under the Christoffel transform ω =( )ω, the transformation of the polynomials is: ( )= + ( ) ( ) = + ( ) ( ) Transformation of recurrence coefficients = = These formulas can be repeated step-by-step to obtain polynomials and recurrence coefficients, after Christoffel transforms To sum-up, CT s preserve the PST condition on weights and thus give PST chains directly. The inverse problem gets bypassed
28 Explicit examples: 1. Krawtchouk polynomials Main idea: start from a spectral grid obeying the PST condition and use the algorithm to determine uniquely Uniform grid: Krawtchouk polynomials = / = PST condition + = π is satisfied for = π =
29 Recall that: ω = + ( ) = + ( ) = ( ) ( )( ) ( ) =!( )! This yields the (normalized) binomial distribution ω =! ( )!!
30 The corresponding OPs are the symmetric Krawtchouk polynomials with recurrence coefficients = = ( + ) First example found by Albanese et al. Krawtchouk polynomials are associated to Ehrenfest urn model Equivalence to random walk on hypercube
31 Explicit examples: 2. Para-Krawtchouk polynomials Bi-lattice: Para-Krawtchouk polynomials Take odd, consider for the -excitation spectrum the bi-lattice = + (γ )( ( ) ) = Bi-lattice because it is the union of two uniform grids separated by γ: = and + = + γ When γ =, uniform grid and Krawtchouk polynomials
32 We have + = γ + + = γetc. PST condition + = π odd integer is satisfied if γ = N odd and = π
33 The weights ( + γ/ ) ( ) ( γ/ ) ω = ( / )!( γ/ ) ( + γ/ ) ( ) (γ/ ) ω + = ( / )!( + γ/ ) with =( )/ and ( ) = ( + ) ( + ). One has ω = = = ω + = / Recurrence coefficients = + γ = ( + )(( + ) γ ) ( )( + )
34 A commercial break: the Bannai-Ito scheme It is nice when physically motivated problem lead to pieces of new mathematics Askey-Scheme of hypergeometric orthogonal polynomials
35
36
37 A commercial break: the Bannai-Ito scheme It is nice when physically motivated problem lead to pieces of new mathematics It is possible to systematically explore the orthogonality grids of the finite OPs to find new PST chains The OPs of the Askey scheme are bispectral. In addition to -term recurrence relation, they obey Differential Difference q-difference eigenvalue equation
38 Recently with A. Zhedanov, S. Tsujimoto and V.X. Genest, I explored the bispectral OPs that satisfy eigenvalue equations of Dunkl type Dunkl operators: differential (difference) operators + reflections e.g: µ = + µ ( ) ( )= ( ) At the top of this scheme: Bannai-Ito polynomials (BI) The BI polynomials have four parameters Introduced in 1984 in algebraic combinatorics We found them to be eigenfunctions of the most general operator of st order in Dunkl shifts (discrete)
39 The analog of the Askey scheme for Dunkl or is as follows; we call it the Bannai-Ito scheme polynomials
40
41 The analog of the Askey scheme for Dunkl or is as follows; we call it the Bannai-Ito scheme polynomials We found the para-krawtchouk polynomials in our search for PST chains We now understand better where they fit There are also PST chains associated with other Ops of the BI scheme
42 Part II: Almost perfect state transfer Question of practical importance: what is the robustness of the properties of theoretically designed chains with PST against errors? Many sources of error: imperfect input/output operations, fabrications defects, additional interactions, systematic biases, etc. Since measurements always have some imprecisions, it practically suffices to require that the transfer probability be as close to as we desire and hence to consider APST Question: Is APST more broadly realized in spin chains and stabler against perturbations?
43 APST condition: >, such that ( ) φ ( ) < Using the same occupation basis similarly found that the condition and eigenbasis, it is χ ( )=( ) + is also necessary for APST As we saw, this implies that = (mirror symmetry) ω = + ( ) As a result, (and ) is reconstructed from the spectrum with same algorithm as in PST case
44 Difference between PST and APST resides in the condition on the spectrum Recall: amplitude in : ( )= = ω ω χ ( )χ ( ) = ω χ ( ) = ω ( ) + ( ) i.e. ( ) is almost periodic function amounts to ( ) + φ ( ) since ω =
45 APST condition can be restated: δ >, φ and such that or π + φ < δ (mod π) δ < π + φ + π < δ integers that may depend on Condition on spectrum of for APST Question: what properties must the eigenvalues possess to ensure that and integers can be found so that this condition is verified?
46 In other words: given the set of real numbers = π φ, when is it possible to find values of for which is approximated with any accuracy in terms of integers by π? This is where Diophantine approximation comes in Definition A set α, =, of reals are linearly independent over the field of rationals if for any, the only rational values of such that are = = = α + α + + α =
47 The theorem of Kronecker is perfectly suited to answer our question which is: given δ and φ, when are there a and such that π + φ + π < δ Version 1: Assume that the reals, = are linearly independent over rational numbers. Let be fixed arbitrary reals. Then δ, and integers such that π < δ = Conclusion: provided that the are linearly independent, for every φ it is possible to find a time such that ( ) is as close to as we wish
48 In many cases, the relations are NOT linearly independent; i.e. exist ( ) + ( ) + + ( ) = = () where ( ) are integers ( integers equivalent to rationals) In such cases we can use the (generalized) Kronecker theorem Theorem Assume that the reals, = are all distinct and that there are non-trivial relations of type (), then π < δ holds δ if and only if the reals satisfy ( ) + ( ) + + ( ) (mod π)
49 Necessary and sufficient conditions for APST Let be the (assumed) distinct eigenvalues of -excitation restriction of If there are 6 relations ( ) + + ( ) = = 6 with ( ) = for some for every APST occurs if and only if 1 is mirror symmetric = 2 The linear relations = are compatible = = + ( ) (π φ) = (mod π) =
50 Spectral surgery and APST Almost obvious that APST survives spectral surgery Weights have same form as for PST Appropriate Christoffel transforms preserve their class and positivity What happens with the spectrum? If the were linearly independent over rationals, a reduced set will have the same property If there are linear relations, all that can happen under the removal of levels is that some of these relations will disappear In fact, spectral surgery could generate chains with APST from chains without the property
51 Special case: the uniform spin chain = σ σ + + σ σ + = = and = and = This matrix is easily diagonalized (with Chebyshev polynomials) and the spectrum is ( + )π = + It is checked that PST occurs only for 6, i.e. for at most four spins. ( Hence the interest in the non-uniform chains)
52 APST, if it is to occur, will put constraints on + the number of spins which is the only parameter Godsil et al. have proved using the Kronecker theorem that APST occurs if and only if + = or + = where is a prime or if + = Those are the values of for which the are linearly independent over the rationals Occurrence of APST in testing However, waiting time chain can be related to primality grows with Puts in question the practicality of APST
53 Special case: the para-krawtchouk chain An example where state transfer can be achieved with arbitrary high fidelity in finite time = + = + γ = ( )/ PST for γ = /, odd Clearly we have the relations = + = + So, to have APST, the conditions = + = + must be satisfied for φ = (mod π) = π φ Compatible for
54 γ = ζ + ζ + ζ + If γ is irrational no further conditions on, and APST will occur Possible to estimate waiting time for given accuracy with Diophantine approximation methods Expand irrational γ in continued fractions with ζ integers The convergents / are rational numbers which provide approximations of γ We can choose an infinite set of convergents (intermediate if necessary) { / / } such that is odd
55 Let = γ It is known that < Choose = π Substitute in the amplitude ( ): ( )= ω ( ) + = and + = + γ, odd
56 We get ( )= ( )/ = ( )/ = ( )/ ω π ( ) + + = ( )/ ω + πγ = = ω + ω + ( +γ)π ( ) + + πγ = π( + / ) = π (since is odd) Thus with ( )/ = ω = ( )/ = ω + = / ( )= ( + π ) ( ) = (π / ) π
57 Consider γ = = = First appropriate convergents = with odd It is checked that the rd convergent / already give an accuracy of % for a waiting time = π
58 Conclusions Good idea of the design of quantum wires with Study the conditions for PST and APST. Formulation as inverse spectral problem Role of OP theory and Diophantine approximation Fertilization of OP theory Still many questions: other chains, higher dimensions (lattices), inverse spectral problem for block 3-diagonal matrices, stability of APST under perturbations, etc. spin chains
PADÉ INTERPOLATION TABLE AND BIORTHOGONAL RATIONAL FUNCTIONS
ELLIPTIC INTEGRABLE SYSTEMS PADÉ INTERPOLATION TABLE AND BIORTHOGONAL RATIONAL FUNCTIONS A.S. ZHEDANOV Abstract. We study recurrence relations and biorthogonality properties for polynomials and rational
More informationIncompatibility Paradoxes
Chapter 22 Incompatibility Paradoxes 22.1 Simultaneous Values There is never any difficulty in supposing that a classical mechanical system possesses, at a particular instant of time, precise values of
More informationarxiv: v1 [math-ph] 26 May 2017
arxiv:1705.09737v1 [math-ph] 26 May 2017 An embedding of the Bannai Ito algebra in U (osp(1, 2)) and 1 polynomials Pascal Baseilhac Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350, Fédération
More informationThe Quantum Heisenberg Ferromagnet
The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,
More informationPage 404. Lecture 22: Simple Harmonic Oscillator: Energy Basis Date Given: 2008/11/19 Date Revised: 2008/11/19
Page 404 Lecture : Simple Harmonic Oscillator: Energy Basis Date Given: 008/11/19 Date Revised: 008/11/19 Coordinate Basis Section 6. The One-Dimensional Simple Harmonic Oscillator: Coordinate Basis Page
More information3 Symmetry Protected Topological Phase
Physics 3b Lecture 16 Caltech, 05/30/18 3 Symmetry Protected Topological Phase 3.1 Breakdown of noninteracting SPT phases with interaction Building on our previous discussion of the Majorana chain and
More informationThe Postulates of Quantum Mechanics Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case: 3-D case
The Postulates of Quantum Mechanics Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about the
More informationHaydock s recursive solution of self-adjoint problems. Discrete spectrum
Haydock s recursive solution of self-adjoint problems. Discrete spectrum Alexander Moroz Wave-scattering.com wavescattering@yahoo.com January 3, 2015 Alexander Moroz (WS) Recursive solution January 3,
More informationConsistent Histories. Chapter Chain Operators and Weights
Chapter 10 Consistent Histories 10.1 Chain Operators and Weights The previous chapter showed how the Born rule can be used to assign probabilities to a sample space of histories based upon an initial state
More informationAddition of Angular Momenta
Addition of Angular Momenta What we have so far considered to be an exact solution for the many electron problem, should really be called exact non-relativistic solution. A relativistic treatment is needed
More informationA Field Extension as a Vector Space
Chapter 8 A Field Extension as a Vector Space In this chapter, we take a closer look at a finite extension from the point of view that is a vector space over. It is clear, for instance, that any is a linear
More informationBrief review of Quantum Mechanics (QM)
Brief review of Quantum Mechanics (QM) Note: This is a collection of several formulae and facts that we will use throughout the course. It is by no means a complete discussion of QM, nor will I attempt
More informationQuantum Mechanics Solutions. λ i λ j v j v j v i v i.
Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =
More informationTime Independent Perturbation Theory Contd.
Time Independent Perturbation Theory Contd. A summary of the machinery for the Perturbation theory: H = H o + H p ; H 0 n >= E n n >; H Ψ n >= E n Ψ n > E n = E n + E n ; E n = < n H p n > + < m H p n
More informationThe 1+1-dimensional Ising model
Chapter 4 The 1+1-dimensional Ising model The 1+1-dimensional Ising model is one of the most important models in statistical mechanics. It is an interacting system, and behaves accordingly. Yet for a variety
More informationRepresentation theory and quantum mechanics tutorial Spin and the hydrogen atom
Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition
More information5.5. Representations. Phys520.nb Definition N is called the dimensions of the representations The trivial presentation
Phys50.nb 37 The rhombohedral and hexagonal lattice systems are not fully compatible with point group symmetries. Knowing the point group doesn t uniquely determine the lattice systems. Sometimes we can
More informationStatistical Interpretation
Physics 342 Lecture 15 Statistical Interpretation Lecture 15 Physics 342 Quantum Mechanics I Friday, February 29th, 2008 Quantum mechanics is a theory of probability densities given that we now have an
More informationSpin Chains for Perfect State Transfer and Quantum Computing. January 17th 2013 Martin Bruderer
Spin Chains for Perfect State Transfer and Quantum Computing January 17th 2013 Martin Bruderer Overview Basics of Spin Chains Engineering Spin Chains for Qubit Transfer Inverse Eigenvalue Problem spinguin
More information-state problems and an application to the free particle
-state problems and an application to the free particle Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 2013 3 September, 2013 Outline 1 Outline 2 The Hilbert space 3 A free particle 4 Keywords
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationThe Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case:
The Schrodinger Equation and Postulates Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about
More informationPHY305: Notes on Entanglement and the Density Matrix
PHY305: Notes on Entanglement and the Density Matrix Here follows a short summary of the definitions of qubits, EPR states, entanglement, the density matrix, pure states, mixed states, measurement, and
More informationDegenerate Perturbation Theory. 1 General framework and strategy
Physics G6037 Professor Christ 12/22/2015 Degenerate Perturbation Theory The treatment of degenerate perturbation theory presented in class is written out here in detail. The appendix presents the underlying
More informationTridiagonal pairs in algebraic graph theory
University of Wisconsin-Madison Contents Part I: The subconstituent algebra of a graph The adjacency algebra The dual adjacency algebra The subconstituent algebra The notion of a dual adjacency matrix
More information(1.1) In particular, ψ( q 1, m 1 ; ; q N, m N ) 2 is the probability to find the first particle
Chapter 1 Identical particles 1.1 Distinguishable particles The Hilbert space of N has to be a subspace H = N n=1h n. Observables Ân of the n-th particle are self-adjoint operators of the form 1 1 1 1
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationChapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found.
Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. In applying quantum mechanics to 'real' chemical problems, one is usually faced with a Schrödinger
More informationLS coupling. 2 2 n + H s o + H h f + H B. (1) 2m
LS coupling 1 The big picture We start from the Hamiltonian of an atomic system: H = [ ] 2 2 n Ze2 1 + 1 e 2 1 + H s o + H h f + H B. (1) 2m n e 4πɛ 0 r n 2 4πɛ 0 r nm n,m Here n runs pver the electrons,
More information1 Algebra of State Vectors
J. Rothberg October 6, Introduction to Quantum Mechanics: Part Algebra of State Vectors What does the State Vector mean? A state vector is not a property of a physical system, but rather represents an
More information0 belonging to the unperturbed Hamiltonian H 0 are known
Time Independent Perturbation Theory D Perturbation theory is used in two qualitatively different contexts in quantum chemistry. It allows one to estimate (because perturbation theory is usually employed
More informationQM and Angular Momentum
Chapter 5 QM and Angular Momentum 5. Angular Momentum Operators In your Introductory Quantum Mechanics (QM) course you learned about the basic properties of low spin systems. Here we want to review that
More information+E v(t) H(t) = v(t) E where v(t) is real and where v 0 for t ±.
. Brick in a Square Well REMEMBER: THIS PROBLEM AND THOSE BELOW SHOULD NOT BE HANDED IN. THEY WILL NOT BE GRADED. THEY ARE INTENDED AS A STUDY GUIDE TO HELP YOU UNDERSTAND TIME DEPENDENT PERTURBATION THEORY
More informationSSH Model. Alessandro David. November 3, 2016
SSH Model Alessandro David November 3, 2016 Adapted from Lecture Notes at: https://arxiv.org/abs/1509.02295 and from article: Nature Physics 9, 795 (2013) Motivations SSH = Su-Schrieffer-Heeger Polyacetylene
More informationSymmetries, Groups, and Conservation Laws
Chapter Symmetries, Groups, and Conservation Laws The dynamical properties and interactions of a system of particles and fields are derived from the principle of least action, where the action is a 4-dimensional
More informationThe quantum state as a vector
The quantum state as a vector February 6, 27 Wave mechanics In our review of the development of wave mechanics, we have established several basic properties of the quantum description of nature:. A particle
More informationNewton s Method and Localization
Newton s Method and Localization Workshop on Analytical Aspects of Mathematical Physics John Imbrie May 30, 2013 Overview Diagonalizing the Hamiltonian is a goal in quantum theory. I would like to discuss
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
More informationKitaev honeycomb lattice model: from A to B and beyond
Kitaev honeycomb lattice model: from A to B and beyond Jiri Vala Department of Mathematical Physics National University of Ireland at Maynooth Postdoc: PhD students: Collaborators: Graham Kells Ahmet Bolukbasi
More informationOrthogonality. 6.1 Orthogonal Vectors and Subspaces. Chapter 6
Chapter 6 Orthogonality 6.1 Orthogonal Vectors and Subspaces Recall that if nonzero vectors x, y R n are linearly independent then the subspace of all vectors αx + βy, α, β R (the space spanned by x and
More informationA classification of sharp tridiagonal pairs. Tatsuro Ito, Kazumasa Nomura, Paul Terwilliger
Tatsuro Ito Kazumasa Nomura Paul Terwilliger Overview This talk concerns a linear algebraic object called a tridiagonal pair. We will describe its features such as the eigenvalues, dual eigenvalues, shape,
More informationTopological order from quantum loops and nets
Topological order from quantum loops and nets Paul Fendley It has proved to be quite tricky to T -invariant spin models whose quasiparticles are non-abelian anyons. 1 Here I ll describe the simplest (so
More informationLie Theory in Particle Physics
Lie Theory in Particle Physics Tim Roethlisberger May 5, 8 Abstract In this report we look at the representation theory of the Lie algebra of SU(). We construct the general finite dimensional irreducible
More informationLucas Polynomials and Power Sums
Lucas Polynomials and Power Sums Ulrich Tamm Abstract The three term recurrence x n + y n = (x + y (x n + y n xy (x n + y n allows to express x n + y n as a polynomial in the two variables x + y and xy.
More information1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationRecursion Systems and Recursion Operators for the Soliton Equations Related to Rational Linear Problem with Reductions
GMV The s Systems and for the Soliton Equations Related to Rational Linear Problem with Reductions Department of Mathematics & Applied Mathematics University of Cape Town XIV th International Conference
More informationReal-Space Renormalization Group (RSRG) Approach to Quantum Spin Lattice Systems
WDS'11 Proceedings of Contributed Papers, Part III, 49 54, 011. ISBN 978-80-7378-186-6 MATFYZPRESS Real-Space Renormalization Group (RSRG) Approach to Quantum Spin Lattice Systems A. S. Serov and G. V.
More informationNANOSCALE SCIENCE & TECHNOLOGY
. NANOSCALE SCIENCE & TECHNOLOGY V Two-Level Quantum Systems (Qubits) Lecture notes 5 5. Qubit description Quantum bit (qubit) is an elementary unit of a quantum computer. Similar to classical computers,
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 7, February 1, 2006
Chem 350/450 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 006 Christopher J. Cramer ecture 7, February 1, 006 Solved Homework We are given that A is a Hermitian operator such that
More informationQuantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 4 Postulates of Quantum Mechanics I In today s lecture I will essentially be talking
More informationThe Bannai-Ito algebra and some applications
Journal of Physics: Conference Series PAPER OPEN ACCESS The Bannai-Ito algebra and some applications To cite this article: Hendrik De Bie et al 015 J. Phys.: Conf. Ser. 597 01001 View the article online
More informationSpectra of Semidirect Products of Cyclic Groups
Spectra of Semidirect Products of Cyclic Groups Nathan Fox 1 University of Minnesota-Twin Cities Abstract The spectrum of a graph is the set of eigenvalues of its adjacency matrix A group, together with
More informationLecture 6: Quantum error correction and quantum capacity
Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde The quantum capacity theorem is one of the most important theorems in quantum hannon theory. It is a fundamentally quantum theorem
More informationPhysics 239/139 Spring 2018 Assignment 2 Solutions
University of California at San Diego Department of Physics Prof. John McGreevy Physics 39/139 Spring 018 Assignment Solutions Due 1:30pm Monday, April 16, 018 1. Classical circuits brain-warmer. (a) Show
More informationI. Perturbation Theory and the Problem of Degeneracy[?,?,?]
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 19 THE VAN VLECK TRANSFORMATION IN PERTURBATION THEORY 1 Although frequently it is desirable to carry a perturbation treatment to second or third
More informationEfficient time evolution of one-dimensional quantum systems
Efficient time evolution of one-dimensional quantum systems Frank Pollmann Max-Planck-Institut für komplexer Systeme, Dresden, Germany Sep. 5, 2012 Hsinchu Problems we will address... Finding ground states
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationDensity Matrices. Chapter Introduction
Chapter 15 Density Matrices 15.1 Introduction Density matrices are employed in quantum mechanics to give a partial description of a quantum system, one from which certain details have been omitted. For
More informationQuantum Mechanics II: Examples
Quantum Mechanics II: Examples Michael A. Nielsen University of Queensland Goals: 1. To apply the principles introduced in the last lecture to some illustrative examples: superdense coding, and quantum
More informationPerformance Evaluation of Generalized Polynomial Chaos
Performance Evaluation of Generalized Polynomial Chaos Dongbin Xiu, Didier Lucor, C.-H. Su, and George Em Karniadakis 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA, gk@dam.brown.edu
More informationPhysics 215 Quantum Mechanics 1 Assignment 1
Physics 5 Quantum Mechanics Assignment Logan A. Morrison January 9, 06 Problem Prove via the dual correspondence definition that the hermitian conjugate of α β is β α. By definition, the hermitian conjugate
More informationarxiv: v1 [cond-mat.stat-mech] 5 Oct 2015
The Coulomb potential V (r) = /r and other radial problems on the Bethe lattice Olga Petrova and Roderich Moessner Max Planck Institute for the Physics of Complex Systems, 087 Dresden, Germany We study
More informationProblem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1
Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate
More informationGenerators for Continuous Coordinate Transformations
Page 636 Lecture 37: Coordinate Transformations: Continuous Passive Coordinate Transformations Active Coordinate Transformations Date Revised: 2009/01/28 Date Given: 2009/01/26 Generators for Continuous
More information10 Time-Independent Perturbation Theory
S.K. Saiin Oct. 6, 009 Lecture 0 0 Time-Independent Perturbation Theory Content: Non-degenerate case. Degenerate case. Only a few quantum mechanical problems can be solved exactly. However, if the system
More information1 Quantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More information9. Integral Ring Extensions
80 Andreas Gathmann 9. Integral ing Extensions In this chapter we want to discuss a concept in commutative algebra that has its original motivation in algebra, but turns out to have surprisingly many applications
More informationAdiabatic quantum computation a tutorial for computer scientists
Adiabatic quantum computation a tutorial for computer scientists Itay Hen Dept. of Physics, UCSC Advanced Machine Learning class UCSC June 6 th 2012 Outline introduction I: what is a quantum computer?
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationQuantum Mechanics Solutions
Quantum Mechanics Solutions (a (i f A and B are Hermitian, since (AB = B A = BA, operator AB is Hermitian if and only if A and B commute So, we know that [A,B] = 0, which means that the Hilbert space H
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationThe Postulates of Quantum Mechanics
p. 1/23 The Postulates of Quantum Mechanics We have reviewed the mathematics (complex linear algebra) necessary to understand quantum mechanics. We will now see how the physics of quantum mechanics fits
More informationChiral Haldane-SPT phases of SU(N) quantum spin chains in the adjoint representation
Chiral Haldane-SPT phases of SU(N) quantum spin chains in the adjoint representation Thomas Quella University of Cologne Presentation given on 18 Feb 2016 at the Benasque Workshop Entanglement in Strongly
More informationLecture 6 Quantum Mechanical Systems and Measurements
Lecture 6 Quantum Mechanical Systems and Measurements Today s Program: 1. Simple Harmonic Oscillator (SHO). Principle of spectral decomposition. 3. Predicting the results of measurements, fourth postulate
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More informationLeonard triples of q-racah type
University of Wisconsin-Madison Overview This talk concerns a Leonard triple A, B, C of q-racah type. We will describe this triple, using three invertible linear maps called W, W, W. As we will see, A
More informationarxiv:hep-th/ Jan 95
Exact Solution of Long-Range Interacting Spin Chains with Boundaries SPhT/95/003 arxiv:hep-th/9501044 1 Jan 95 D. Bernard*, V. Pasquier and D. Serban Service de Physique Théorique y, CE Saclay, 91191 Gif-sur-Yvette,
More informationEigenvectors and Hermitian Operators
7 71 Eigenvalues and Eigenvectors Basic Definitions Let L be a linear operator on some given vector space V A scalar λ and a nonzero vector v are referred to, respectively, as an eigenvalue and corresponding
More informationConstruction of spinors in various dimensions
Construction of spinors in various dimensions Rhys Davies November 23 2011 These notes grew out of a desire to have a nice Majorana representation of the gamma matrices in eight Euclidean dimensions I
More informationLecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators
Lecture Spin, orbital, and total angular momentum 70.00 Mechanics Very brief background MATH-GA In 9, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform
More informationarxiv:quant-ph/ v2 24 Dec 2003
Quantum Entanglement in Heisenberg Antiferromagnets V. Subrahmanyam Department of Physics, Indian Institute of Technology, Kanpur, India. arxiv:quant-ph/0309004 v2 24 Dec 2003 Entanglement sharing among
More informationCONSTRAINED PERCOLATION ON Z 2
CONSTRAINED PERCOLATION ON Z 2 ZHONGYANG LI Abstract. We study a constrained percolation process on Z 2, and prove the almost sure nonexistence of infinite clusters and contours for a large class of probability
More informationMatching Polynomials of Graphs
Spectral Graph Theory Lecture 25 Matching Polynomials of Graphs Daniel A Spielman December 7, 2015 Disclaimer These notes are not necessarily an accurate representation of what happened in class The notes
More informationv(r i r j ) = h(r i )+ 1 N
Chapter 1 Hartree-Fock Theory 1.1 Formalism For N electrons in an external potential V ext (r), the many-electron Hamiltonian can be written as follows: N H = [ p i i=1 m +V ext(r i )]+ 1 N N v(r i r j
More informationSymmetries for fun and profit
Symmetries for fun and profit Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 August 28, 2008 Sourendu Gupta (TIFR Graduate School) Symmetries for fun and profit QM I 1 / 20 Outline 1 The isotropic
More informationarxiv:quant-ph/ v1 21 Nov 2003
Analytic solutions for quantum logic gates and modeling pulse errors in a quantum computer with a Heisenberg interaction G.P. Berman 1, D.I. Kamenev 1, and V.I. Tsifrinovich 2 1 Theoretical Division and
More informationLecture 45: The Eigenvalue Problem of L z and L 2 in Three Dimensions, ct d: Operator Method Date Revised: 2009/02/17 Date Given: 2009/02/11
Page 757 Lecture 45: The Eigenvalue Problem of L z and L 2 in Three Dimensions, ct d: Operator Method Date Revised: 2009/02/17 Date Given: 2009/02/11 The Eigenvector-Eigenvalue Problem of L z and L 2 Section
More informationIntroduction to the Mathematics of the XY -Spin Chain
Introduction to the Mathematics of the XY -Spin Chain Günter Stolz June 9, 2014 Abstract In the following we present an introduction to the mathematical theory of the XY spin chain. The importance of this
More informationPhysics 221A Fall 2018 Notes 22 Bound-State Perturbation Theory
Copyright c 2018 by Robert G. Littlejohn Physics 221A Fall 2018 Notes 22 Bound-State Perturbation Theory 1. Introduction Bound state perturbation theory applies to the bound states of perturbed systems,
More informationFermat s Last Theorem for Regular Primes
Fermat s Last Theorem for Regular Primes S. M.-C. 22 September 2015 Abstract Fermat famously claimed in the margin of a book that a certain family of Diophantine equations have no solutions in integers.
More informationQuantum Information Types
qitd181 Quantum Information Types Robert B. Griffiths Version of 6 February 2012 References: R. B. Griffiths, Types of Quantum Information, Phys. Rev. A 76 (2007) 062320; arxiv:0707.3752 Contents 1 Introduction
More informationQuantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 6 Postulates of Quantum Mechanics II (Refer Slide Time: 00:07) In my last lecture,
More informationSpanning and Independence Properties of Finite Frames
Chapter 1 Spanning and Independence Properties of Finite Frames Peter G. Casazza and Darrin Speegle Abstract The fundamental notion of frame theory is redundancy. It is this property which makes frames
More informationExact diagonalization methods
Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Exact diagonalization methods Anders W. Sandvik, Boston University Representation of states in the computer bit
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationPhysics 221A Fall 2017 Notes 27 The Variational Method
Copyright c 2018 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 27 The Variational Method 1. Introduction Very few realistic problems in quantum mechanics are exactly solvable, so approximation methods
More informationDegenerate Perturbation Theory
Physics G6037 Professor Christ 12/05/2014 Degenerate Perturbation Theory The treatment of degenerate perturbation theory presented in class is written out here in detail. 1 General framework and strategy
More informationIntermission: Let s review the essentials of the Helium Atom
PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the
More informationPHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep
Problem set 1 Due Sep 15 2005 1. Let V be the set of all complex valued functions of a real variable θ, that are periodic with period 2π. That is u(θ + 2π) = u(θ), for all u V. (1) (i) Show that this V
More information