Quantum Mechanics: Fundamental Principles and Applications

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1 Quantum Mechanics: Fundamental Principles and Applications John F. Dawson Department of Physics, University of New Hampshire, Durham, NH 0384 October 14, 009, 9:08am EST c 007 John F. Dawson, all rights reserved.

2 c 009 John F. Dawson, all rights reserved. ii

3 Contents Preface xv I Fundamental Principles 1 1 Linear algebra Linear vector spaces Linear independence Inner product The dual space Non-orthogonal basis sets Operators Eigenvalues and eigenvectors: Non-orthogonal basis vectors Projection operators: Spectral representations: Basis transformations Commuting operators Maximal sets of commuting operators Infinite dimensional spaces Translation of the coordinate system Measurement The uncertainty relation Time in non-relativistic quantum mechanics Canonical quantization 9.1 Classical mechanics review Symmetries of the action Galilean transformations Canonical quantization postulates The Heisenberg picture The Schrödinger picture Canonical transformations Schwinger s transformation theory Path integrals Space-time paths Some path integrals Matrix elements of coordinate operators Generating functionals iii

4 CONTENTS CONTENTS 3.5 Closed time path integrals Initial value conditions Connected Green functions Classical expansion Some useful integrals In and Out states The interaction representation The time development operator Forced oscillator Density matrix formalism Classical theory Classical time development operator Classical averages Classical correlation and Green functions Classical generating functional Quantum theory Thermal densities The canonical ensemble Ensemble averages Imaginary time formalism Thermal Green functions Path integral representation Thermovariable methods Green functions 77 8 Identical particles Coordinate representation Occupation number representation Particle fields Hamiltonian Symmetries Galilean transformations The Galilean group Group structure Galilean transformations Phase factors for the Galilean group Unitary transformations of the generators Commutation relations of the generators Center of mass operator Casimir invariants Extension of the Galilean group Finite dimensional representations The massless case Time translations Space translations and boosts Rotations The rotation operator c 009 John F. Dawson, all rights reserved. iv

5 CONTENTS CONTENTS 9.5. Rotations of the basis sets General Galilean transformations Improper transformations Parity Time reversal Charge conjugation Scale and conformal transformations Scale transformations Conformal transformations The Schrödinger group Wave equations Scalars Spinors Spinor particles Spinor antiparticles Vectors Massless wave equations Massless scalers Massless vectors Supersymmetry Grassmann variables Superspace and the 1D-N supersymmetry group D-N supersymmetry transformations in quantum mechanics Supersymmetric generators R-symmetry Extension of the supersymmetry group Differential forms II Applications Finite quantum systems Diatomic molecules Periodic chains Linear chains Impurities Bound state Scattering One and two dimensional wave mechanics Introduction Schrödinger s equation in one dimension Transmission of a barrier Wave packet propagation Time delays for reflection by a potential step Schrödinger s equation in two dimensions c 009 John F. Dawson, all rights reserved. v

6 CONTENTS CONTENTS 14 The WKB approximation Introduction Theory Connection formulas Positive slope Negative slope Examples Bound states Tunneling Spin systems Magnetic moments Pauli matrices The eigenvalue problem Spin precession in a magnetic field Driven spin system Spin decay: T 1 and T The Ising model Heisenberg models The harmonic oscillator The Lagrangian Energy eigenvalue and eigenvectors Other forms of the Lagrangian Coherent states Completeness relations Generating function Squeezed states The forced oscillator The three-dimensional oscillator The Fermi oscillator Action for a Fermi oscillator Electrons and phonons Electron-phonon action Equations of motion Numerical classical results Electron modes Vibrational modes Electron-phonon interaction The action revisited Quantization Block wave functions A one-dimensional periodic potential A lattice of delta-functions Numerical methods Schrödinger perturbation theory Time-independent perturbation theory Time-dependent perturbation theory c 009 John F. Dawson, all rights reserved. vi

7 CONTENTS CONTENTS 19 Variational methods Introduction Time dependent variations The initial value problem The eigenvalue problem Examples The harmonic oscillator The anharmonic oscillator Time-dependent Hartree-Fock Exactly solvable potential problems Supersymmetric quantum mechanics The hierarchy of Hamiltonians Shape invariance Angular momentum Eigenvectors of angular momentum Spin Orbital angular momentum Kinetic energy operator Parity and Time reversal Rotation of coordinate frames Rotation matrices Axis and angle parameterization Euler angles Cayley-Klein parameters Rotations in quantum mechanics Rotations using Euler angles Properties of D-functions Rotation of orbital angular momentum Sequential rotations Addition of angular momentum Coupling of two angular momenta Coupling of three and four angular momenta Rotation of coupled vectors Tensor operators Tensor operators and the Wigner-Eckart theorem Reduced matrix elements Angular momentum matrix elements of tensor operators Selected problems Spin-orbit force in hydrogen Transition rates for photon emission in Hydrogen Hyperfine splitting in Hydrogen The Zeeman effect in hydrogen The Stark effect in hydrogen Matrix elements of two-body nucleon-nucleon potentials Density matrix for the Deuteron c 009 John F. Dawson, all rights reserved. vii

8 CONTENTS CONTENTS Electrodynamics 97.1 The Lagrangian Probability conservation Gauge transformations Constant electric field Hydrogen atom Eigenvalues and eigenvectors Matrix elements of the Runge-Lenz vector Symmetry group Operator factorization Operators for the principle quantum number SO(4, ) algebra The fine structure of hydrogen The hyperfine structure of hydrogen The Zeeman effect The Stark effect Atomic radiation Atomic transitions The photoelectric effect Resonance fluorescence Flux quantization Quantized flux The Aharonov-Bohm effect Magnetic monopoles Scattering theory Propagator theory Free particle Green function in one dimension S-matrix theory Scattering from a fixed potential Two particle scattering Resonance and time delays Proton-Neutron scattering III Appendices 343 A Table of physical constants 345 B Operator Relations 347 B.1 Commutator identities B. Operator functions B.3 Operator theorems C Binomial coefficients 351 D Fourier transforms 353 D.1 Finite Fourier transforms D. Finite sine and cosine transforms c 009 John F. Dawson, all rights reserved. viii

9 CONTENTS CONTENTS E Classical mechanics 355 E.1 Lagrangian and Hamiltonian dynamics E. Differential geometry E.3 The calculus of forms E.3.1 Derivatives of forms E.3. Integration of forms E.4 Non-relativistic space-time E.4.1 Symplectic manifolds E.4. Integral invariants E.4.3 Gauge connections F Statistical mechanics review 381 F.1 Thermal ensembles F. Grand canonical ensemble F..1 The canonical ensemble F.3 Some examples F.4 MSR formalism F.4.1 Classical statistical averages F.4. Generating functions F.4.3 Schwinger-Dyson equations F.5 Anharmonic oscillator F.5.1 The partition function for the anharmonic oscillator G Boson calculus 397 G.1 Boson calculus G. Connection to quantum field theory G.3 Hyperbolic vectors G.4 Coherent states G.5 Rotation matrices G.6 Addition of angular momentum G.7 Generating function G.8 Bose tensor operators Index 419 c 009 John F. Dawson, all rights reserved. ix

10 CONTENTS CONTENTS c 009 John F. Dawson, all rights reserved. x

11 List of Figures 3.1 The closed time path contour The Galilean transformation for Eq. (9.1) R-symmetry We plot the potential energy for an electron in two atomic sites. We also sketch wave functions ψ 1, (x) for an electron in the isolated atomic sites and the symmetric and antisymmetric combinations ψ ± (x) A molecule containing four atoms A molecule containing six atomic sites, arranged in a circular chain Construction for finding the six eigenvalues for an electron on the six periodic sites of Fig. 1.3, for values of k = 0,..., 5. Note the degeneracies for values of k = 1, 5 and k =, A molecule containing six atomic sites, arranged in a linear chain Six eigenvalues for the six linear sites of Fig. 1.5, for values of k = 1,..., A long chain with an impurity atom at site Transmission and reflection coefficients for electron scattering from an impurity for the case when (ɛ 0 ɛ 1 )/Γ 0 = and Γ 1 /Γ 0 = Two long connected chains A junction with three legs Two turning point situations Potential well Potential barrier Spin precession in the rotating coordinate system Retarded and advanced contours for the Green function of Eq. (16.117) Feynman (F ) (red) and anti-feynman (F ) (green) contours Plot of V (x) for the first 10 sites Plot of V (x) and V (x) for site n with wave functions for sites n, n ± 1, showing the overlap integrals between nearest neighbor sites Plot of x n (t) for the first 10 sites as a function of time for ɛ 0 = ω 0 = Γ = 1, and K = 0.5, for 100 sites Plot of y n (t) for the first 10 sites as a function of time for ɛ 0 = ω 0 = Γ = 1, and K = 0.5, for 100 sites Plot of φ n (t) for the first 10 sites as a function of time for ɛ 0 = ω 0 = Γ = 1, and K = 0.5, for 100 sites xi

12 LIST OF FIGURES LIST OF FIGURES 17.6 Plot of dφ n (t)/dt for the first 10 sites as a function of time for ɛ 0 = ω 0 = Γ = 1, and K = 0.5, for 100 sites Plot of the electron and phonon energy spectra ɛ k and ω k on the periodic chain, as a function of k. Energies and k values have been normalized to unity. Note that near k = 0, the electron spectra is quadratic whereas the phonon spectrum is linear Construction for finding the oscillation frequencies for the six periodic sites of Fig. 1.3, for values of k = 0, ±1, ±, Plot of the right-hand side of Eqs. (17.107) and (17.108), for β = Plot of the right-hand side of Eqs. (17.107) and (17.108), for β = Plot of the right-hand side of Eqs. (17.107) and (17.108), for β = Plot of the energy, in units of m/, as a function of Ka/π for β = Euler angles for the rotations Σ Σ Σ Σ. The final axis is labeled (X, Y, Z) Mapping of points on a unit sphere to points on the equatorial plane, for x 3 > 0 (red lines) and x 3 < 0 (blue lines) The fine structure of hydrogen (not to scale). Levels with the same value of j are degenerate The hyperfine structure of the n = 1 and n = levels of hydrogen (not to scale) Zeeman splitting of the n = 1 hyperfine levels of hydrogen as a function of µ B B (not to scale) Stark splitting of the n = fine structure levels of hydrogen as a function of β = e a E 0. is the fine structure splitting energy. (not to scale) c 009 John F. Dawson, all rights reserved. xii

13 List of Tables 1.1 Relation between Nature and Quantum Theory The first five energies of the anharmonic oscillator computed using the time-dependent variational method compared to the exact results [?] and a SUSY-based variational method [?] Table of Clebsch-Gordan coefficients, spherical harmonics, and d-functions Algebric formulas for some 3j-symbols Algebric formulas for some 6j-symbols The first few radial wave functions for hydrogen A.1 Table of physical constants from the particle data group A. Table of physical constants from the particle data group xiii

14 LIST OF TABLES LIST OF TABLES c 009 John F. Dawson, all rights reserved. xiv

15 Preface In this book, I have tried to bridge the gap between material learned in an undergraduate course in quantum mechanics and an advanced relativistic field theory course. The book is a compilation of notes for a first year graduate course in non-relativistic quantum mechanics which I taught at the University of New Hampshire for a number of years. These notes assume an undergraduate knowledge of wave equation based quantum mechanics, on the level of Griffiths[1] or Liboff [], and undergraduate mathematical skills on the level of Boas [3]. This book places emphasis on learning new theoretical methods applied to old non-relativistic ideas, with a eye to what will be required in relativistic field theory and particle physics courses. The result provides an introduction to quantum mechanics which is, I believe, unique. The book is divided into two sections: Fundamental Principles and Applications. The fundamental principles section starts out in the usual way by reviewing linear algebra, vector spaces, and notation in the first chapter, and then in the second chapter, we discuss canonical quantization of classical systems. In the next two chapters, Path integrals and in- and out-states are discussed. Next is a chapter on the density matrix and Green functions in quantum mechanics where we also discuss thermal density matrices and Green functions. This is followed by a chapter on identical particles and second quantized non-relativistic fields. Next the Galilean group is discussed in detail and wave equations for massless and massive nonrelativistic particles explored. Finally, the last chapter of the fundamental principles section is devoted to supersymmetry in non-relativistic quantum mechanics. In the application section, I start by discussing finite quantum systems: the motion of electrons on molecules and on linear and circular chains. This is followed by chapters one and two dimensional wave mechanics and the WKB approximation. Then I discuss spin systems, the harmonic oscillator, and electrons and phonons on linear lattices. Approximation methods are discussed next, with chapters on perturbative and variational approximations. This is followed by a chapters on exactly solvable potential problems in nonrelativistic quantum mechanics, and a detailed chapter on angular momentum theory in quantum mechanics. In the next chapter, we discuss several problems concerning the interactions of non-relativistic electrons with a classical electromagnetic fields, including the Hydrogen atom, and lastly, we include a chapter on scattering theory. There are appendices giving operator identities, binomial coefficients, fourier transforms, and sections reviewing classical physics, differential geometry, classical statistical mechanics, and Schwinger s angular momentum theory. Much of the material for these notes come from the many excellent books on the subject, and in many cases, I have only rearranged them in my own way. I have tried to give references to original material when this was done. Of course, any misunderstandings are my own. I would like to thank... John Dawson September, 007 Durham, NH xv

16 REFERENCES REFERENCES References [1] D. J. Griffiths, Introduction to Quantum Mechanics (Pearson, Prentice Hall, Upper Saddle River, NJ, 005), second edition. [] R. L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, awp:adr, 1997), third edition. [3] M. L. Boaz, Mathematical Methods in the Physical Sciences (John Wiley & Sons, New York, NY, 1983). c 009 John F. Dawson, all rights reserved. xvi

17 Part I Fundamental Principles 1

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19 Chapter 1 Basic principles of quantum theory Physical systems are represented in quantum theory by a complex vector space V with an inner product. The state of the system is described by a particular vector Ψ in this space. All the possible states of the system are represented by basis vectors in this space. Observables are represented by Hermitian operators acting in this vector space. The possible values of these observables are the eigenvalues of these operators. Probability amplitudes for observing these values are inner products. Symmetries of the physical system are represented by unitary transformations of the basis vectors in V. All this is summarized in Table 1.1 below. Thus it will be important for us to study linear vector spaces in detail, which is the subject of this chapter. Nature The physical system The state of the system All possible states of the system Observables Possible values of observables Probability amplitudes for events Symmetries Quantum Theory a vector space V a vector Ψ in V a set of basis vectors Hermitian operators eigenvalues of operators inner products unitary transformations Table 1.1: Relation between Nature and Quantum Theory 1.1 Linear vector spaces In the following 1, we will denote scalars (complex numbers) by a, b, c,..., and vectors by α, β, γ,.... Definition 1 (linear vector space). A linear vector space V is a set of objects called vectors ( α, β, γ,...) which are closed under addition and scalar multiplication. That is, if α and β are in V, then a α + b β is in V. Vectors addition and scalar multiplication have commutative, associative, and distributive properties: 1. α + β = β + α. commutative law 1 Much of the material in this chapter was taken from Serot [1, chapter 1] 3

20 1.. LINEAR INDEPENDENCE CHAPTER 1. LINEAR ALGEBRA. ( α + β ) + γ = α + ( β + γ ). associative law 3. a(b α ) = (ab) α. associative law 4. (a + b) α = a α + b α. distributive law 5. a( α + β ) = a α + a β. distributive law 6. There is a unique vector 0 in V, called the null vector, with the properties, α + 0 = α and 0 α = 0, for all α. Example 1 (C N ). The set of N complex numbers (c 1, c,..., c N ), where c i C. Addition of vectors is defined by addition of the components, and scalar multiplication by the multiplication of each element by the scalar. We usually write vectors as column matrices: c 1 c c =. Example (P N ). The set of all real polynomials c(t) = c 0 + c 1 t + c t + + c N t N of degree less than N in an independent real variable t, 1 t 1. A vector is defined by c = c(t). Addition and multiplication by a scalar are the ordinary ones for polynomials. Note that in this example, we define a secondary variable t R, which is not in the vector space. Example 3 (C[a, b]). The set of all continuous complex functions of a real variable on the closed interval [a, b]. Thus f = f(x), a x b. Again, in this example, we have a secondary base variable consisting of a real variable x R. 1. Linear independence A set of vectors e 1, e,..., e N, are linearly independent if the relation, c N (1.1) N c n e n = 0, (1.) n=1 can only be true if: c n = 0, n = 1,..., N. Otherwise, the set of vectors are linearly dependent, which means that one of them can be expressed as a linear combination of the others. The maximum number N of linearly independent vectors in a vector space V is called the dimension of the space, in which case the set of vectors provides a basis set for V. Any vector in the space can be written as a linear combination of the basis vectors. We can easily prove this: Theorem 1. Let e n, n = 1,..., N, be a basis in V. Then any vector α in V can be represented by: where a n are complex numbers. α = N a n e n, n=1 Proof. Since α and e n, n = 1,..., N are N + 1 vectors in V, they must be linearly dependent. So there must exist complex numbers c, c n, n = 1,..., N, not all zero, such that c α + N c n e n = 0. n=1 c 009 John F. Dawson, all rights reserved. 4

21 CHAPTER 1. LINEAR ALGEBRA 1.3. INNER PRODUCT But c 0, otherwise the set e n, n = 1,..., N, would be linearly dependent, which they are not. Therefore α = N n=1 c n c e n = N a n e n. where a n = c n /c. If a different set of coefficients b n, n = 1,..., N existed, we would then have by subtraction, N (a n b n ) e n = 0, n=1 which can be true only if b n = a n, n = 1,..., N, since the set e n is linearly dependent. Thus the components a n are unique for the basis set e n. In this chapter, we mostly consider linear vector spaces which have finite dimensions. Our examples 1 and above have dimension N, whereas example 3 has infinite dimensions. n=1 1.3 Inner product An inner product maps pairs of vectors to complex numbers. It is written: g( α, β ). That is, g is a function with two slots for vectors which map each pair of vectors in V to a complex number. The inner product must be defined so that it is anti-linear with respect to the first argument and linear with respect to the second argument. Because of this linearity and anti-linearity property, it is useful to write the inner product simply as: g( α, β ) α β. (1.3) The inner product must have the properties: 1. α β = β α = a complex number.. a α + b β γ = a α γ + b β γ. 3. γ a α + b β = a γ α + b γ β, 4. α α 0, with the equality holding only if α = 0. The norm, or length, of a vector is defined by α α α > 0. A Hilbert space is a linear vector space with an inner product for each pair of vectors in the space. Using our examples of linear vector spaces, one possible definition of the inner products is: Example 4 (C N ). For example: Example 5 (P N ). We can take: where a(t) and b(t) are members of the set. a b = a 1b 1 + a b + + a N b N, (1.4) a b = +1 1 a (t) b(t) dt, (1.5) Example 6 (C[a, b]). An inner product can be defined as an integral over the range with respect to a weight function w(x): f g = b c 009 John F. Dawson, all rights reserved. 5 a f (x) g(x) w(x) dx. (1.6)

22 1.3. INNER PRODUCT CHAPTER 1. LINEAR ALGEBRA Definition. A basis set e n, n = 1,..., N is orthonormal if e i e j = δ ij. We first turn to a property of the inner product: the Schwartz inequality. Theorem (The Schwartz inequality). The Schwartz, or triangle, inequality states that for any two vectors in V, ψ φ ψ φ. Proof. We let Then the length of χ is positive definite: χ = ψ + λ φ. χ = χ χ = ψ ψ + λ ψ φ + λ φ ψ + λ φ φ 0. (1.7) This expression, as a function of λ and λ, will be a minimum when Thus χ χ = ψ φ + λ φ φ = 0, λ χ χ λ = φ ψ + λ φ φ = 0. λ = φ ψ / φ, λ = ψ φ / φ. Substituting this into (1.7) and taking the square root, we find: ψ φ ψ φ. The Schwartz inequality allows us to generalize the idea of an angle between two vectors. If we let then the inequality states that 0 cos γ The dual space cos γ = ψ φ /( ψ φ ), The dual vector is not a vector at all, but a function which operates on vectors to produce complex numbers defined by the inner product. The dual α is written with a slot ( ) for the vectors: α ( ) = g( α, ), (1.8) for all vectors α in V. That is, the dual only makes sense if it is acting on an arbitrary vector in V to produce a number: α ( β ) = g( α, β ) α β, (1.9) in agreement with our notation for inner product. The anti-linear property of the first slot of the inner product means that the set of dual functions form an anti-linear vector space also, called V D. So if we regard the dual α as right acting, we can just omit the parenthesis and the slot when writing the dual function. So if the set e i, i = 1,..., N are a basis in V, then the duals of the basis vectors are defined by: c 009 John F. Dawson, all rights reserved. 6 e i = g( e i, ), (1.10)

23 CHAPTER 1. LINEAR ALGEBRA 1.3. INNER PRODUCT with the property that e i e j = g ij. Then if α is a vector in V with the expansion: α = i a i e i. (1.11) Because of the anti-linear properties of the first slot in the definition of the inner product, the dual α is given uniquely by: α = i a i e i. (1.1) 1.3. Non-orthogonal basis sets If the basis set we have found for a linear vector space is not orthogonal, we have two choices: we can either construct an orthogonal set from the linearly independent basis set, or introduce contra- and covariant vectors. We first turn to the Gram-Schmidt orthogonalization method. Gram-Schmidt orthogonalization Given an arbitrary basis set, x n, n = 1,..., N, we can construct an orthonormal basis e n, n = 1,..., N as follows: 1. Start with e 1 = x 1 / x 1.. Next construct a vector orthogonal to e 1 from x and e 1, and normalize it: 3. Generalize this formula to the remaining vectors: for n =,..., N. e = x e 1 e 1 x x e 1 e 1 x. x n n 1 m=1 n 1 e m e m x n e n =. x n e m e m x n m=1 Contra- and co-variant vectors Another method of dealing with non-orthogonal basis vectors is to introduce contra- and co-variant vectors. We do that in this section. We first introduce a metric tensor g ij by the definition: g ij g( e i, e j ) = e i e j, (1.13) and assume that det{g} 0, so that the inverse metric, which we write with upper indices g 1 ij We sometimes write: g i k δ ik. g ij g jk = j j c 009 John F. Dawson, all rights reserved. 7 g ij exists: g ij g jk = δ ik, (1.14)

24 1.3. INNER PRODUCT CHAPTER 1. LINEAR ALGEBRA Definition 3 (covariant vectors). We call the basis vectors e i with lower indices co-variant vectors, and then define basis vectors e i with upper indices by: e i = j e j g ji. (1.15) We call these vectors with upper indices contra-variant vectors. The duals of the contra-variant vectors are then given by: e i = [ g ji ] e j = g ij e j. (1.16) j j Remark 1. It is easy to show that the contra- and co-variant vectors obey the relations: e i e j = e i e j = δ ij, e i e i = e i e i = 1. (1.17) i i The set of dual vectors e i are not orthogonal with each other and are not normalized to one even if the set e i is normalized to one. If the basis vectors e i are orthonormal, then the contra- and co-variant basis vectors are identical, which was the case that Dirac had in mind when he invented the bra and ket notation. Remark. Since the sets e i and e i are both complete linearly independent basis sets, although not orthogonal, we can write a vector in one of two ways: v = i v i e i = i v i e i, (1.18) from which we find: e i v = v i, and e i v = v i, (1.19) which provides an easy methods to find the contra- and co-variant expansion coefficients of vectors. This was the reason for introducing contra- and co-variant base vectors in the first place. The two components of the vector v i and v i are related by the inner product matrix: v i = j g ij v j, and v i = j g ij v j. (1.0) That is, g ij and g ij lower and raise indices respectively. We now turn to a few examples. Example 7 (C ). Let us consider example 1 with N =, a two dimensional vector space of complex numbers. Vectors in this space are called spinors. A vector a is written as a two-component column matrix: ( ) a1 a =. (1.1) a The inner product of two vectors a and b is given by definition (1.4): a b = a 1b 1 + a b. (1.) Let us now take two linearly independent non-orthogonal basis vectors given by: ( ) 1 e 1 =, e 0 = 1 ( ) i. (1.3) 1 The basis bra s e 1 and e are then given by: e 1 = ( 1, 0 ), e = 1 ( i, 1 ). (1.4) Sometimes the co-variant vectors are called dual vectors. We do not use that terminology here because of the confusion with our definition of dual operators and the dual space of bra s. c 009 John F. Dawson, all rights reserved. 8

25 CHAPTER 1. LINEAR ALGEBRA 1.4. OPERATORS So the g ij matrix is given by: with det{g} = 1/. The inverse g ij is then: So the contra-variant vectors are given by: with the duals: e 1 = g ij = g ij = ( e i g i1 1 =, e i) = i=1 ( ) 1 i/ i/, (1.5) 1 ( ) i i. (1.6) e i g i = i=1 e 1 = ( 1, i ), e = ( 0, It is easy to see that these contra- and co-variant vectors satisfy the relations: ( ) e i e j = δ ij, and e 1 e 1 + e 1 0 e =, 0 1 ( ) e i e j = δ ij, and e 1 e 1 + e e 1 0 =, 0 1 ( ) 0, (1.7) ). (1.8) (1.9) Notice that the dual vectors are not normalized nor are they orthogonal with each other. They are orthonormal with the base vectors, however, which is the important requirement. Example 8 (P ). In this example, we take the non-orthogonal basis set to be powers of x. So we define co-variant vectors e i by: e i = x i, for i = 0, 1,,...,, (1.30) with an inner product rule given by integration over the range [ 1, 1]: { +1 g ij = e i e j = x i x j 0 for i + j odd, dx = 1 /(i + j + 1) for i + j even. 0 /3 0 0 /3 0 /5 = /3 0 / /5 0 / (1.31) We would have to invert this matrix to find g ij. This is not easy to do, and is left to the reader. 1.4 Operators An operator S maps a vector α V to another vector β V. We write: S( α ) = β, which is defined for some domain D and range R of vectors in the space. We usually consider cases where the domain and range is the full space V. Observables in quantum theory are represented by Hermitian operators and symmetry transformations by unitary or anti-unitary operators. These important operators are defined in this section for finite vector spaces. We start with a number of useful definitions. c 009 John F. Dawson, all rights reserved. 9

26 1.4. OPERATORS CHAPTER 1. LINEAR ALGEBRA Definition 4 (linear and anti-linear operators). A linear operator S = L has the properties: L(a α + b β ) = a L( α ) + b L( β ), (1.3) for any α, β V and a, b C. Similarly, an anti-linear operator S = A has the properties: A(a α + b β ) = a A( α ) + b A( β ). (1.33) So, for linear and anti-linear operators, we can just write: L α = L α = γ, and A α = A α = γ, without the parenthesis. Definition 5 (inverse operators). If the mapping S α = β is such that each β comes from a unique α, the mapping is called injective. In addition, if every vector β V is of the form β = S α, then the mapping is called surjective. If the mapping is both injective and surjective it is called bijective, and the inverse exists. (Physicists usually don t use such fancy names.) We write the inverse operation thus: S 1 β = α. Clearly, S S 1 = S 1 S = 1, the unit operator. A bijective, linear mapping is called an isomorphism. Remark 3. We note the following two theorems, which we state without proof: 3 1. If L is a linear operator, then so is L 1.. A linear operator has an inverse if and only if L α = 0 implies that α = 0. Definition 6 (adjoint operators). For linear operators, the adjoint operator L is defined by: For anti-linear operators, the adjoint A is defined by: α L β = L α β. (1.34) α A β = A α β = β A α. (1.35) Remark 4. For an orthonormal basis, the adjoint matrix of a linear operator is L ij = e i L e j = L e i e j = e j L e i = L ji, which is the complex conjugate of the transpose matrix. For an anti-linear operator, the adjoint matrix is given by: A ij = e i A e j = A e i e j = e j A e i = A ji, which is the transpose matrix. Definition 7 (unitary operators). A linear operator U is unitary if An anti-linear and anti-unitary operator U is defined by: U α U β = α U U β = α β. (1.36) U α U β = α U U β = α β = β α. (1.37) Thus for both unitary and anti-unitary operators U 1 = U. This was the reason for our differing definitions of the adjoint for linear and anti-linear operators. Definition 8 (Hermitian operators). A linear operator H is Hermitian if That is H = H. 3 the proofs can be found in Serot[1] α H β = H α β = α H β. (1.38) c 009 John F. Dawson, all rights reserved. 10

27 CHAPTER 1. LINEAR ALGEBRA 1.4. OPERATORS Remark 5. By convention, operators are right acting on kets: S α = β. The product of two linear operators is defined by: (AB) α = A (B α ) = A β = γ, (1.39) which is called the composition law, whereas Thus in general AB BA. Linear operators obey: (BA) α = B (A α ) = B δ = ɛ. (1.40) 1. A(BC) = (AB)C; associative law. (A + B)C = AC + BC; distributive law A(B + C) = AB + AC. The difference between the order of operations are important in quantum mechanics we call this difference, the commutator, and write: [A, B] = AB BA. In appendix B, we list some operator identities for commutators. Remark 6. left acting operators act in the space of linear functionals, or dual space V D of bra vectors, and are defined by the relation: ( α S ) β α ( S β ) = α S β { S α β for linear operators, = S α β for anti-linear operators. Thus, if for linear operators in V we have the relation then the corresponding mapping in V D is given by: (1.41) L α = L α = β, (1.4) α L = L α = β. (1.43) Dirac invented a notation for this. For an arbitrary operator S, he defined: { S α β for linear operators, α S β α S β = S α β for anti-linear operators. (1.44) We often call α S β a matrix element, the reasons for which will become apparent in the next section. In Dirac s notation, we can think of S as right acting on a ket vector or left acting on a bra vector. Definition 9. A Normal operator is one that commutes with it s adjoint: [A, A ] = 0. Remark 7. We shall learn below that normal operators are the most general kind of operators that can be diagonalized by a unitary transformation. Both unitary and Hermitian operators are examples of normal operators Eigenvalues and eigenvectors: For any operator A, if we can find a complex number a and a ket a such that A a = a a, (1.45) then a is called the eigenvalue and a the eigenvector. We assume in this section that the domain and range of the operator A is the full set V. Note that we have simplified our notation here by labeling the eigenvector by the eigenvalue a, rather than using Greek characters for labeling vectors. Vectors are distinguished by Dirac s ket notation. c 009 John F. Dawson, all rights reserved. 11

28 1.4. OPERATORS CHAPTER 1. LINEAR ALGEBRA Theorem 3 (Hermitian operators). The eigenvalues of Hermitian operators are real and the eigenvectors can be made orthonormal. Proof. The eigenvalue equation for Hermitian operators is written: Since H is Hermitian, we have the following relations: H h = h h (1.46) h H h = h h h, h H h = H h h = h h h. But for hermitian operators H = H, so subtracting these two equations, we find: So setting h = h, we have (h h ) h h = 0. (1.47) (h h ) h = 0. Since h 0, h = h. Thus h is real. Since all the eigenvalues are real, we have from (1.47), (h h ) h h = 0. (1.48) Thus, if h h, then h h = 0, and is orthogonal. The proof of orthogonality of the eigenvectors fails if there is more that one eigenvector with the same eigenvalue (we call this degenerate eigenvalues). However, by the Gram-Schmidt construction, that we can always find orthogonal eigenvectors among the eigenvectors with degenerate eigenvalues. Since h = 0, it is always possible to normalize them. Thus we can assume that hermitian operators have real and orthonormal eigenvectors, h h = δ h h. Theorem 4 (Unitary operators). The eigenvalues of unitary operators have unit magnitude and the eigenvectors can be made orthogonal. Proof. The eigenvalue equation for Unitary operators is written: Unitary operators obey U U = 1, so we have: So we find: U u = u u (1.49) u U U u = U u U u = u u u u = u u. (1 u u) u u = 0. Therefore, if u = u, (1 u ) u = 0, and we must have u = 1. This means we can write u = e iθ, where θ is real. In addition if u u, then the eigenvectors are orthogonal, u u = 0. Degenerate eigenvalues can again be orthonormalized by the Gram-Schmidt method. Remark 8 (finding eigenvalues and eigenvectors). For finite systems, we can find eigenvalues and eigenvectors of an operator A by solving a set of linear equations. Let e i, i = 1,..., N be an orthonormal basis in V. Then we can write: N a = e i c i (a), c i (α) = e i a. (1.50) Then Eq. (1.45), becomes: i=1 N A ij c j (a) = a c i (a), for i = 1,..., N, (1.51) j=1 c 009 John F. Dawson, all rights reserved. 1

29 CHAPTER 1. LINEAR ALGEBRA 1.4. OPERATORS where A ij = e i A e j. We write A as the matrix with elements A ij. By Cramer s rule[][p. 9], Eq. (1.51) has nontrivial solutions if f(a) = det[ A ai ] = 0. (1.5) f(a) is a polynomial in a of degree N, and is called the characteristic polynomial. In general, it has N complex roots, some of which may be the same. We call the number of multiple roots the degeneracy of the root. If we call all these roots a n then we can write formally: f(a) = N c n a n = n=1 N (a a n ) = 0, (1.53) For Hermitian operators, these roots are all real. For unitary operators, they are complex with unit magnitude. The coefficients c i (a) can be found for each eigenvalue from the linear set of equations (1.51). If there are no multiple roots, the N eigenvectors so found are orthogonal and span V. For the case of multiple roots, it is possible to still find N linearly independent eigenvectors and orthorgonalize them by the Schmidt procedure. Thus we can assume that we can construct, by these methods, a complete set of orthonormal vectors that span the vector space Non-orthogonal basis vectors Let us examine in this section how to write matrix elements of operators using non-orthogonal basis sets. First let A be a linear operator satisfying: A v = u (1.54) Expanding the vectors in terms of the co-variant basis set e i, we have: n=1 A v = j v j A e j = u = i u i e i, (1.55) Right operating on this by the bra e i gives: A i j v j = u i, where A i j = e i A e j e i A e j. (1.56) j This can be interpreted as matrix multiplication of a square matrix A i j with the column matrix v j to give the column matrix u i. Similarly, expanding the vectors in terms of the contra-variant basis vectors e i gives a corresponding expression: A v = v j A e j = u = u i e i, (1.57) j i Right operating again on this expression by e i gives: A j i v j = u i, where A j i = e i A e j e i A e j. (1.58) j This can also be interpreted as matrix multiplication of a square matrix A i j with the column matrix v j to give the column matrix u i. One can easily check that A i j = i j g ii A i j gj j. (1.59) A i j and A i j are not the same matrix. We can also define matrices A ij and A ij by: c 009 John F. Dawson, all rights reserved. 13 A ij = e i A e j, and A ij = e i A e j. (1.60)

30 1.4. OPERATORS CHAPTER 1. LINEAR ALGEBRA These matrices are related to the others by raising or lowering indices using g ii and g jj. For example, we have: A ij = g ii A i j = j A i g j j = g ii A i j g j j. (1.61) i j i j Example 9 (Adjoint). From definition (6) of the adjoint of linear operators, we find that matrix elements in non-orthogonal basis sets are given by: [L ] ij = [L ji ], [L ] i j = [L i j ] (1.6) [L ] i j = [L i j] [L ] ij = [L ji ]. Only the first and last of these matrix forms give a definition of adjoint matrix that relate the complex conjugate of a matrix to the transpose of the same matrix. The second and third form relate complex conjugates of matrices with upper and lower indices to the transpose of matrices with lower and upper indices, and do not even refer to the same matrix. Example 10 (Hermitian operators). For Hermitian operators such that Ĥ = Ĥ, we find: H ij = [H ji ], H i j = [H i j ] (1.63) H i j = [H i j] H ij = [H ji ]. Example 11 (P ). Returning to our example of continuous functions defined on the interval [ 1, 1] and using the non-orthogonal basis P, as defined in Example 8 above, we define an operator X which is multiplication of functions in the vector space by x. We take the co-variant basis vectors to be given by Eq. (1.30). On this basis set, the X operation is easily described as: X e i = e i+1, for i = 0, 1,...,. (1.64) So we easily construct the mixed tensor X i j: X i j = e i X e j = e i e j+1 = δ i,j+1 = (1.65) However, there are three other tensors we can write down. For example, the matrix X ij is given by: 0 /3 0 /5 /3 0 /5 0 X ij = e i X e j = e i e j+1 = g i,j+1 = 0 /5 0 /7. (1.66) /5 0 / This matrix obeys Xji = X ij, and is clearly Hermitian. Since we (now) know that X is a hermitian operator, we can deduce that: X j i = [ X i j ] = , (1.67) without knowing what the contra-variant vectors are. c 009 John F. Dawson, all rights reserved. 14

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