Degenerate Perturbation Theory
|
|
- Bernard Jayson Anthony
- 5 years ago
- Views:
Transcription
1 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 We begin with a Hamitonian H which can be decomposed into an operator H 0 with known eigenvectors and eigenvalues and a second perturbing piece V : H = H 0 + λv. (1) The eigenvectors and eigen values of H and H 0 are defined as follows: H 0 φ n = E n (0) φ n (2) H ψ n = E n ψ n. (3) For simplicity we will assume that all of the eigenvalues E n (0) are distinct with the exception of a group of N deg vectors which share the same eigenvalue E n (0) = E deg. (0) We will label these N deg degenerate eigenvectors as { φ ni } 1 i Ndeg and define to the set of N deg indices of these degenerate states as S deg = {n i } 1 i Ndeg. We should recognize that there may be many possible choices for the basis of eigenvectors which lie in this N deg subspace of degenerate states. However, this subspace itself is well-defined and is denoted V deg as is the projection operator P deg onto this subspace: P deg = N deg i=1 φ ni φ ni. (4) Our objective is to express the exact eigenvalues and eigenstates of H in terms of those of H 0 as a power series in the small parameter λ. In order to easily express such a relationship we need to coordinate the labeling of the perturbed ( ψ n ) and unperturbed ( φ n ) eigenstates. For n S deg this is 1
2 easy. We simply use the label n for the perturbed ψ n whose eigenvalue E n obeys: lim E(λ) n = E n (0). (5) λ 0 Since each E n (0) corresponds to a unique index n there is no ambiguity. However, for the N deg perturbed eigenstates whose energies approach deg as λ 0 there is no guarantee that their λ 0 limit will match up with a particular unperturbed eigenstate φ n. In general we expect that if we were to label the N deg states whose energies approach deg as { ψ ni } 1 i Ndeg,we would find: N deg lim ψ n i = C ji φ nj. (6) λ 0 j=1 Later in our procedure we will go back and redefine the unperturbed states φ ni so that the C matrix above is the unit matrix: C ji = δ ij which implies a smooth limit as λ 0 relating our ψ n and φ n, consisent with the perturbative expansion that we were able to derive in the case of no degeneracy. However, at present we do not have enough information to make this redefinition of the φ ni states and must take a less convenient approach in order to make an expansion in λ possible. We introduce a new basis for the N deg -dimensional subspace spanned by the perturbed eigenvectors ψ ni, denoted ψ ni which are no longer eigenvectors of H but instead obey the conditions: lim n λ 0 i = φ nj (7) ψ ni φ ni is real and positive (8) These conditions are analogues our standard condition that φ n ψ n is real and positive, needed to define the phase of the perturbed states ψ n. For simplicity of notation we will also define: ψ n = ψ n (9) when n S deg. We will learn more about the relationship between the two sets of states ψ ni and φ ni for 1 i N deg as we develop the perturbation series which expresses the former in terms of the latter. Our new states ψ ni are now easily related to our unperturbed states and almost diagonalize H: H ψ n = H n ψ, (10) 2
3 where { δn n or n H n = S deg (11) H n otherwise Now we have a problem that can be solved in perturbation theory and which has reduced H to a matrix which is entirely diagonal except for a finite N deg N deg block which must be diagonalized by hand. We will now show how the states ψ n and the nearly diagonal matrix H n can be worked out to first and second order in λ. The final diagonalization of the N deg N deg matrix H ni must be carred out by some method other than perturbation j theory. Just as in the non-degenerate case we start with the eigenvalue equation, Eq. (10), take its matrix element on the left with φ n and insert a complete sum over a complete set of unperturbed states between (H + λv ) and ψ n : φ ( H 0 + λv )( ) φ φ ψn = φ ψ H n. (12) Next we introduce expansions in λ for both H n and φ ψ n = S n: H,n = δ ne n (0) + λe (1) n + λ2 n (13) φ ψ n = δ n + λs (1) n + λ2 λs (2) n (14) and substitute them into Eq. (12). Note that the matrices E (1) n and n will be diagonal except for the same N deg N deg non-diagonal block which appears in H n. We obtain the following equation: ( (0) E n δ + λv )( δ n + λs (1) n + λ2 S (2) ) n (15) = ( δ + λs(1) n + λ2 S (2) )( δ ne n (0) + λe (1) n + λ2 n )). 2 First-order degenereate perturbation theory We can next determine the matrices S (1) n and collecting all terms that are first order in λ: and E(1) nn by starting with Eq. (15) S(1) n + V n = E (1) n + S(1) n E(0) n. (16) 3
4 If either n or are not in the set S deg of degenerate states, then the consequeces are exactly as we found in non-degenerate perturbation theory. For n this equation can be solved for S (1) n without any need for a nonzero off-diagonal element E (1) n n. When n = E (1) nn is determined and S nn (1) is left unconstrained. However, as in the non-degenerate case our condition im(s nn (1) ) and the unit norm of the state ψ n combine to determine S nn (1) =0. Thus, to first order in λ and for either n S deg or / S deg we have: E (1) n = δ nv nn (17) S (1) V n n = n. (18) E n (0) For both n = n i and = n j in S deg, then S n (1) cancels between the left andn right sides of Eq. (16) and we are left with a formula for which there is a non-diagonal matrix E n (1) connecting degenerate states: E (1) n = V n (19) and through first order the matrix H n = ψ H ψ n is given by: H n = δ n n + λe (1) n, (20) where E (1) n is a diagonal matrix with diagonal element V nn when either n or is lies outside of S deg while if n, S deg then E (1) n is the N deg N deg matrix V n. The last step is to perform a unitary transformation on the N deg vectors ψ ni 1 i Ndeg which span this N deg -dimensional space to diagonalize this N deg N deg matrix. In the discussion of second order degenerate perturbation theory below we will assume that this diagonalization has been performed so that in our transformed basis: E n (1) = V n = V ni n i δ n. (21) for 1 i, j N deg. Because the matrix S n (1) drops out of Eq. (16), it has not been determined. For the first-order, non-degenerate case only S nn (1) was left undetermined at this step. However, our phase conventions for the perturbed states ψ n and the orthogonality of those states required S nn (1) = 0. The analogous choice is available here. As in the non-degenerate case we must ensure that 4
5 our new states ψ n are orthonormal which will be achieved if the transformation matrix S is unitary: S (S ) n = δ n. (22) To first order this equation becomes a condition on S (1) n: S (1) +(S ) n =0. (23) One can see that this equation is automatically obeyed if n or are not elements of S deg but must be imposed if both n and belong to S deg : S (1) n +(S ) n =0. (24) Thus, we are permited to use an arbitrary anti-hermitian matirx S (1) n,if we wish. This amounts to permitting a first-order unitary rotation among our N deg degnerate states φ ni. Since such a first order shift would serve no purpose, we adopt this simplest convention S (1) n =0 1 i, j N deg. (25) 3 Second-order degenereate perturbation theory Extending this proceedure to second order is a simple repetition of the steps just taken. However, the off-diagonal elements of H n which appear will now be second order terms term which must be computed and are less obvious than the simple off-diagonal elements of V n. Just as before we start with our general equation, Eq (15) and pick out the terms which are of order λ 2 : S(2) n + V S(1) n n = n + S (1) E(1) n n + S (2) n ne n (0). (26) Just as in the first-order case, this is easily solved for S (2) n and the diagonal, second order energy matrix n determined if either n or n is not an element of S deg : n = δ n V n S (1) n = δ n 5 n V n V n n (27)
6 S (2) n = = 1 n 1 n ( V S(1) n n ) S(1) E(1) n n n V n V n n V n E(0) n V nn. (28) For the case when both = n i and n = n j lie in the degenerate set S deg, the second order matrices S n (2) in Eq. (26) will cancel between the left- and right-hand sides and cannot be used to remove the off-diagonal terms. In this case Eq. (26) can be written as an equation for the N deg N deg, second-order matrix E n (2) : n = V ni S(1) n j S (1) n i E(1) n j (29) = V ni S(1) n j S (1) n V nj n j (30) using = n i and n = n j with 1 i, j N deg and assuming that the first order matrix E (1) n j has already been diagonalized as part of the first-order solution. We should consider three possibilities: 1. There are no second-order terms connecting the degenerate states. In this case the full matrix n will have only diagonal terms and we have succeeded in diagonalizing H through second order in λ. 2. The degeneracy has been lifted at first order. In this case all of the elements of the now diagonal matrix V n are distinct so V ni n i = V nj n j implies that i = j. In this case the fundamental problem posed by degenerate perturbation theory, how to determine which combination of the unperturbated states φ ni correspond to the perturbed states has been resolved and we should expect no further difficulties. In fact, this is the case because although we cannot use the matrix S n (2) to remove the second-order off-diagonal terms which have appeared, we can make use of the first order matrix S n (1) (which so far has not been used and simply set to zero for convenience) to remove the off-diagonal elements of E n (2). That this is possible can be seen by rewriting Eq. (30) to show explicitly these so far unconstrained elements of S n (1) : E n (2) = n j + V ni n i S n (1) S n (1) V nj n j. (31) S deg V ni S(1) 6
7 Since the matrix element S n (1) has not previously been contrained and was set to zero only for simplicity, it can now be chosen to make this off-diagonal term E n (2) =0: S (1) n = 1 V nj n j V ni n i S deg V ni S(1) n j (32) V n nj 1 = V ni n V nj n j V. (33) ni n i S deg E n (0) j This is indeed first-order in V as required but an interesting ratio of a second-order divided by a first-order piece. Thus, if unique, nondegenerate states are resolved at first order in λ then no further difficulties arise at order λ 2. The equations which must be obeyed to make diagonal can be solved and end up determining the previous un- n specified coefficients S (1) n. 3. The N deg degenerate states φ ni remain degenerate at first order. Under these circumstances both the matrix elements S n (1) and S n (2) cancel from the expression giving E n (2) which implies that no choices remain to remove the off-diagonal terms from E n (2). As a result to second order in perturbation theory the matrix ( ) H n = δ n E (0) (2) n + λv nn + λe n, (34) where n is diagonal except for the N deg N deg block: n = V ni S(1) n j = V ni S deg S deg V n nj n j (35) which must be diagonalized explicilty, without the further use of perturbation theory. Finally we should observe that the second order energy shift E n (2) given in Eq. (27) for states φ n where n is not in the degenerate set S deg are actually independent of the basis used for the degenerate set of states { φ ni } ni S deg so these energies are unaffected by any final diagonalization steps taken in this subspace to deal with the second order degeneracy: n = n V n V n n (36) 7
8 = = n S deg n S deg V n V n n + n S deg V n V n V n V n + φ n V P degv φ n E n (0) n n deg n (37). (38) 8
Degenerate 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 informationPerturbation Theory 1
Perturbation Theory 1 1 Expansion of Complete System Let s take a look of an expansion for the function in terms of the complete system : (1) In general, this expansion is possible for any complete set.
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 informationMATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v
More informationA PRIMER ON SESQUILINEAR FORMS
A PRIMER ON SESQUILINEAR FORMS BRIAN OSSERMAN This is an alternative presentation of most of the material from 8., 8.2, 8.3, 8.4, 8.5 and 8.8 of Artin s book. Any terminology (such as sesquilinear form
More informationQuantum Physics II (8.05) Fall 2002 Assignment 3
Quantum Physics II (8.05) Fall 00 Assignment Readings The readings below will take you through the material for Problem Sets and 4. Cohen-Tannoudji Ch. II, III. Shankar Ch. 1 continues to be helpful. Sakurai
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 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 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 informationDiagonalization by a unitary similarity transformation
Physics 116A Winter 2011 Diagonalization by a unitary similarity transformation In these notes, we will always assume that the vector space V is a complex n-dimensional space 1 Introduction A semi-simple
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 informationPhysics 221A Fall 1996 Notes 14 Coupling of Angular Momenta
Physics 1A Fall 1996 Notes 14 Coupling of Angular Momenta In these notes we will discuss the problem of the coupling or addition of angular momenta. It is assumed that you have all had experience with
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
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 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 informationSince G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =
Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for
More informationApproximation Methods in QM
Chapter 3 Approximation Methods in QM Contents 3.1 Time independent PT (nondegenerate)............... 5 3. Degenerate perturbation theory (PT)................. 59 3.3 Time dependent PT and Fermi s golden
More informationarxiv:hep-ph/ v1 15 Jul 1996
UG FT 67/96 hep ph/9607313 July 1996 Numerical diagonalization of fermion mass matrices arxiv:hep-ph/9607313v1 15 Jul 1996 J. A. Aguilar Saavedra Departamento de Física Teórica y del Cosmos Universidad
More informationNotes on basis changes and matrix diagonalization
Notes on basis changes and matrix diagonalization Howard E Haber Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064 April 17, 2017 1 Coordinates of vectors and matrix
More informationWigner 3-j Symbols. D j 2. m 2,m 2 (ˆn, θ)(j, m j 1,m 1; j 2,m 2). (7)
Physics G6037 Professor Christ 2/04/2007 Wigner 3-j Symbols Begin by considering states on which two angular momentum operators J and J 2 are defined:,m ;,m 2. As the labels suggest, these states are eigenstates
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 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 informationChapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1)
Tor Kjellsson Stockholm University Chapter 6 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: where α is a constant. H = αδ(x a/ ( a Find the first-order correction
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 informationVectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1. x 2. x =
Linear Algebra Review Vectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1 x x = 2. x n Vectors of up to three dimensions are easy to diagram.
More informationA Review of Perturbation Theory
A Review of Perturbation Theory April 17, 2002 Most quantum mechanics problems are not solvable in closed form with analytical techniques. To extend our repetoire beyond just particle-in-a-box, a number
More information(Again, this quantity is the correlation function of the two spins.) With z chosen along ˆn 1, this quantity is easily computed (exercise):
Lecture 30 Relevant sections in text: 3.9, 5.1 Bell s theorem (cont.) Assuming suitable hidden variables coupled with an assumption of locality to determine the spin observables with certainty we found
More informationTime Independent Perturbation Theory
apr_0-may_5.nb: 5/4/04::9:56:8 Time Independent Perturbation Theory Note: In producing a "final" vrsion of these notes I decided to change my notation from that used in class and by Sakurai. In class,
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
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 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 informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More information1 = I I I II 1 1 II 2 = normalization constant III 1 1 III 2 2 III 3 = normalization constant...
Here is a review of some (but not all) of the topics you should know for the midterm. These are things I think are important to know. I haven t seen the test, so there are probably some things on it that
More informationHilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.
Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,
More informationMATH 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
More informationBasic Calculus Review
Basic Calculus Review Lorenzo Rosasco ISML Mod. 2 - Machine Learning Vector Spaces Functionals and Operators (Matrices) Vector Space A vector space is a set V with binary operations +: V V V and : R V
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 informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More information«Random Vectors» Lecture #2: Introduction Andreas Polydoros
«Random Vectors» Lecture #2: Introduction Andreas Polydoros Introduction Contents: Definitions: Correlation and Covariance matrix Linear transformations: Spectral shaping and factorization he whitening
More informationSpectral Theorem for Self-adjoint Linear Operators
Notes for the undergraduate lecture by David Adams. (These are the notes I would write if I was teaching a course on this topic. I have included more material than I will cover in the 45 minute lecture;
More informationTime-Independent Perturbation Theory
4 Phys46.nb Time-Independent Perturbation Theory.. Overview... General question Assuming that we have a Hamiltonian, H = H + λ H (.) where λ is a very small real number. The eigenstates of the Hamiltonian
More informationPrincipal Component Analysis
Principal Component Analysis Laurenz Wiskott Institute for Theoretical Biology Humboldt-University Berlin Invalidenstraße 43 D-10115 Berlin, Germany 11 March 2004 1 Intuition Problem Statement Experimental
More informationPHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 8: Solutions. Topics covered: hydrogen fine structure
PHYS85 Quantum Mechanics II, Spring HOMEWORK ASSIGNMENT 8: Solutions Topics covered: hydrogen fine structure. [ pts] Let the Hamiltonian H depend on the parameter λ, so that H = H(λ). The eigenstates and
More informationWe will discuss matrix diagonalization algorithms in Numerical Recipes in the context of the eigenvalue problem in quantum mechanics, m A n = λ m
Eigensystems We will discuss matrix diagonalization algorithms in umerical Recipes in the context of the eigenvalue problem in quantum mechanics, A n = λ n n, (1) where A is a real, symmetric Hamiltonian
More informationLinear Algebra using Dirac Notation: Pt. 2
Linear Algebra using Dirac Notation: Pt. 2 PHYS 476Q - Southern Illinois University February 6, 2018 PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018
More informationOn common eigenbases of commuting operators
On common eigenbases of commuting operators Paolo Glorioso In this note we try to answer the question: Given two commuting Hermitian operators A and B, is each eigenbasis of A also an eigenbasis of B?
More information4.8 Arnoldi Iteration, Krylov Subspaces and GMRES
48 Arnoldi Iteration, Krylov Subspaces and GMRES We start with the problem of using a similarity transformation to convert an n n matrix A to upper Hessenberg form H, ie, A = QHQ, (30) with an appropriate
More informationSymmetric and self-adjoint matrices
Symmetric and self-adjoint matrices A matrix A in M n (F) is called symmetric if A T = A, ie A ij = A ji for each i, j; and self-adjoint if A = A, ie A ij = A ji or each i, j Note for A in M n (R) that
More informationPHYSICS 210A : STATISTICAL PHYSICS HW ASSIGNMENT
PHYSICS 210A : STATISTICAL PHYSICS HW ASSIGNMENT #1 (1) Consider a system with K possible states i, with i {1,,K}, where the transition rate W ij between any two states is the same, with W ij = γ > 0 (a)
More informationRepresentation theory & the Hubbard model
Representation theory & the Hubbard model Simon Mayer March 17, 2015 Outline 1. The Hubbard model 2. Representation theory of the symmetric group S n 3. Representation theory of the special unitary group
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationSignatures of GL n Multiplicity Spaces
Signatures of GL n Multiplicity Spaces UROP+ Final Paper, Summer 2016 Mrudul Thatte Mentor: Siddharth Venkatesh Project suggested by Pavel Etingof September 1, 2016 Abstract A stable sequence of GL n representations
More informationThen x 1,..., x n is a basis as desired. Indeed, it suffices to verify that it spans V, since n = dim(v ). We may write any v V as r
Practice final solutions. I did not include definitions which you can find in Axler or in the course notes. These solutions are on the terse side, but would be acceptable in the final. However, if you
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 informationReview of similarity transformation and Singular Value Decomposition
Review of similarity transformation and Singular Value Decomposition Nasser M Abbasi Applied Mathematics Department, California State University, Fullerton July 8 7 page compiled on June 9, 5 at 9:5pm
More informationQUANTUM MECHANICS I PHYS 516. Solutions to Problem Set # 5
QUANTUM MECHANICS I PHYS 56 Solutions to Problem Set # 5. Crossed E and B fields: A hydrogen atom in the N 2 level is subject to crossed electric and magnetic fields. Choose your coordinate axes to make
More informationFall TMA4145 Linear Methods. Exercise set Given the matrix 1 2
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an
More informationKet space as a vector space over the complex numbers
Ket space as a vector space over the complex numbers kets ϕ> and complex numbers α with two operations Addition of two kets ϕ 1 >+ ϕ 2 > is also a ket ϕ 3 > Multiplication with complex numbers α ϕ 1 >
More informationNOTES ON BILINEAR FORMS
NOTES ON BILINEAR FORMS PARAMESWARAN SANKARAN These notes are intended as a supplement to the talk given by the author at the IMSc Outreach Programme Enriching Collegiate Education-2015. Symmetric bilinear
More information2. Review of Linear Algebra
2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear
More information6.2 Unitary and Hermitian operators
6.2 Unitary and Hermitian operators Slides: Video 6.2.1 Using unitary operators Text reference: Quantum Mechanics for Scientists and Engineers Section 4.10 (starting from Changing the representation of
More informationLinear Operators, Eigenvalues, and Green s Operator
Chapter 10 Linear Operators, Eigenvalues, and Green s Operator We begin with a reminder of facts which should be known from previous courses. 10.1 Inner Product Space A vector space is a collection of
More informationCoordinate and Momentum Representation. Commuting Observables and Simultaneous Measurements. January 30, 2012
Coordinate and Momentum Representation. Commuting Observables and Simultaneous Measurements. January 30, 2012 1 Coordinate and Momentum Representations Let us consider an eigenvalue problem for a Hermitian
More informationVector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.
Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +
More informationMathematical Foundations of Quantum Mechanics
Mathematical Foundations of Quantum Mechanics 2016-17 Dr Judith A. McGovern Maths of Vector Spaces This section is designed to be read in conjunction with chapter 1 of Shankar s Principles of Quantum Mechanics,
More informationName: Final Exam MATH 3320
Name: Final Exam MATH 3320 Directions: Make sure to show all necessary work to receive full credit. If you need extra space please use the back of the sheet with appropriate labeling. (1) State the following
More informationIntroduction to Quantum Mechanics Physics Thursday February 21, Problem # 1 (10pts) We are given the operator U(m, n) defined by
Department of Physics Introduction to Quantum Mechanics Physics 5701 Temple University Z.-E. Meziani Thursday February 1, 017 Problem # 1 10pts We are given the operator Um, n defined by Ûm, n φ m >< φ
More informationProblem # Max points possible Actual score Total 120
FINAL EXAMINATION - MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to
More informationPhysics 129B, Winter 2010 Problem Set 4 Solution
Physics 9B, Winter Problem Set 4 Solution H-J Chung March 8, Problem a Show that the SUN Lie algebra has an SUN subalgebra b The SUN Lie group consists of N N unitary matrices with unit determinant Thus,
More informationLecture 10: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11)
Lecture 1: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11) The eigenvalue problem, Ax= λ x, occurs in many, many contexts: classical mechanics, quantum mechanics, optics 22 Eigenvectors and
More informationNotes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T.
Notes on singular value decomposition for Math 54 Recall that if A is a symmetric n n matrix, then A has real eigenvalues λ 1,, λ n (possibly repeated), and R n has an orthonormal basis v 1,, v n, where
More informationLinear Algebra and Dirac Notation, Pt. 2
Linear Algebra and Dirac Notation, Pt. 2 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 2 February 1, 2017 1 / 14
More informationLecture 5 : Projections
Lecture 5 : Projections EE227C. Lecturer: Professor Martin Wainwright. Scribe: Alvin Wan Up until now, we have seen convergence rates of unconstrained gradient descent. Now, we consider a constrained minimization
More informationAnnouncements Monday, November 20
Announcements Monday, November 20 You already have your midterms! Course grades will be curved at the end of the semester. The percentage of A s, B s, and C s to be awarded depends on many factors, and
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More informationGROUP THEORY PRIMER. New terms: so(2n), so(2n+1), symplectic algebra sp(2n)
GROUP THEORY PRIMER New terms: so(2n), so(2n+1), symplectic algebra sp(2n) 1. Some examples of semi-simple Lie algebras In the previous chapter, we developed the idea of understanding semi-simple Lie algebras
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW JC Stuff you should know for the exam. 1. Basics on vector spaces (1) F n is the set of all n-tuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication
More information18.06 Problem Set 8 - Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8 - Solutions Due Wednesday, 4 November 2007 at 4 pm in 2-06 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
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 informationQuantum Mechanics I Physics 5701
Quantum Mechanics I Physics 5701 Z. E. Meziani 02/23//2017 Physics 5701 Lecture Outline 1 General Formulation of Quantum Mechanics 2 Measurement of physical quantities and observables 3 Representations
More informationEIGENVALUES AND SINGULAR VALUE DECOMPOSITION
APPENDIX B EIGENVALUES AND SINGULAR VALUE DECOMPOSITION B.1 LINEAR EQUATIONS AND INVERSES Problems of linear estimation can be written in terms of a linear matrix equation whose solution provides the required
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 information(VII.B) Bilinear Forms
(VII.B) Bilinear Forms There are two standard generalizations of the dot product on R n to arbitrary vector spaces. The idea is to have a product that takes two vectors to a scalar. Bilinear forms are
More informationLecture 10: Solving the Time-Independent Schrödinger Equation. 1 Stationary States 1. 2 Solving for Energy Eigenstates 3
Contents Lecture 1: Solving the Time-Independent Schrödinger Equation B. Zwiebach March 14, 16 1 Stationary States 1 Solving for Energy Eigenstates 3 3 Free particle on a circle. 6 1 Stationary States
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2006 Christopher J. Cramer. Lecture 5, January 27, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2006 Christopher J. Cramer Lecture 5, January 27, 2006 Solved Homework (Homework for grading is also due today) We are told
More information$ +! j. % i PERTURBATION THEORY AND SUBGROUPS (REVISED 11/15/08)
PERTURBATION THEORY AND SUBGROUPS REVISED 11/15/08) The use of groups and their subgroups is of much importance when perturbation theory is employed in understanding molecular orbital theory and spectroscopy
More informationLec 2: Mathematical Economics
Lec 2: Mathematical Economics to Spectral Theory Sugata Bag Delhi School of Economics 24th August 2012 [SB] (Delhi School of Economics) Introductory Math Econ 24th August 2012 1 / 17 Definition: Eigen
More informationParticles I, Tutorial notes Sessions I-III: Roots & Weights
Particles I, Tutorial notes Sessions I-III: Roots & Weights Kfir Blum June, 008 Comments/corrections regarding these notes will be appreciated. My Email address is: kf ir.blum@weizmann.ac.il Contents 1
More informationMathematical foundations - linear algebra
Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationLINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationGROUP THEORY PRIMER. New terms: tensor, rank k tensor, Young tableau, Young diagram, hook, hook length, factors over hooks rule
GROUP THEORY PRIMER New terms: tensor, rank k tensor, Young tableau, Young diagram, hook, hook length, factors over hooks rule 1. Tensor methods for su(n) To study some aspects of representations of a
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 informationMatrix Representation
Matrix Representation Matrix Rep. Same basics as introduced already. Convenient method of working with vectors. Superposition Complete set of vectors can be used to express any other vector. Complete set
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 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 informationSymmetric and anti symmetric matrices
Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal
More informationACM 104. Homework Set 4 Solutions February 14, 2001
ACM 04 Homework Set 4 Solutions February 4, 00 Franklin Chapter, Problem 4, page 55 Suppose that we feel that some observations are more important or reliable than others Redefine the function to be minimized
More informationIsotropic harmonic oscillator
Isotropic harmonic oscillator 1 Isotropic harmonic oscillator The hamiltonian of the isotropic harmonic oscillator is H = h m + 1 mω r (1) = [ h d m dρ + 1 ] m ω ρ, () ρ=x,y,z a sum of three one-dimensional
More informationLecture 6: Lies, Inner Product Spaces, and Symmetric Matrices
Math 108B Professor: Padraic Bartlett Lecture 6: Lies, Inner Product Spaces, and Symmetric Matrices Week 6 UCSB 2014 1 Lies Fun fact: I have deceived 1 you somewhat with these last few lectures! Let me
More informationH&M Chapter 5 Review of Dirac Equation
HM Chapter 5 Review of Dirac Equation Dirac s Quandary Notation Reminder Dirac Equation for free particle Mostly an exercise in notation Define currents Make a complete list of all possible currents Aside
More information