Advanced quantum mechanics Reading instructions

Transcription

1 Advanced quantum mechanics Reading instructions All parts of the book are included in the course and are assumed to be read. But of course some concepts are more important than others. The main purpose of these instructions is to identify the central parts of each chapter. In addition some comments on important issues not treated in the book are made. A problematic aspect of the book is that it does not distinguish clearly enough between physical concepts and mathematical structures. For instance, large parts of Chapter 1 are essentially a rephrasing of concepts from Linear Algebra in terms that are used in quantum mechanics. Chapter 1: Fundamental concepts To a large extent this chapter is a repetition of material from previous courses. It describes the space of states in quantum mechanics, which is a Hilbert space, and various related notions and techniques, such as properties of discrete and continuous bases and linear operators acting on a Hilbert space. Unlike in many other texts, the space of states is described as an abstract Hilbert space, i.e. without any a priori concrete realization (such as in terms of square integrable functions on configuration space). Thereby the notion of a wave function becomes a derived concept. A central notion is the one of unitary operators, as well as their properties. An operator that might be unfamiliar is the one that generates (infinitesimal) translations; so make sure that this concept is well understood. To be able to follow the rest of the book it is indispensible to understand the concepts and techniques described in this chapter. No calculation or derivation should remain a mystery. Read thoroughly enough to make sure this goal is met. 1.2 vector space over the complex numbers (the definition should essentially be known from linear algebra), Hilbert space (unfortunately, no precise definition is given), ray; state vector ( ket, ket vector), null vector ( null ket ); superposition principle; observable, operator, eigenvector ( eigenket, eigenstate), eigenvalue; dual vector space ( bra space, is contravariant with respect to complex conjugation), duality ( dual correspondence ); inner product (hermitian inner product, bra(c)ket, unfortunately the given definition is not quite precise), positive-definiteness; orthogonal vectors, normalized vector, norm; null operator, adjoint (hermitian adjoint, hermitian conjugate) operator, multiplication of operators, outer product (= tensor product) of a ket vector and a bra vector, compatibility of outer and inner products ( associative law ). avqm HT

2 Warning: A product is a binary operation with suitable properties; just putting two symbols next to one another will, in general, not result in any sensible product structure. Various aspects of finite-dimensional linear algebra do not have a naive generalization to infinite-dimensional vector spaces. 1.3 reality of eigenvalues and orthogonality of eigenvectors of a hermitian operator; basis of a vector space, completeness in the form of (1.3.11); matrix realization of an operator, hermitian conjugate of a matrix; projection operator; spin- 1 2 system. 1.4 pure ensemble (collection of identically prepared systems); measurement, probaility of measuring an eigenvalue, expectation value; selective measurement, projection operator; commutation and anticommutation relations, e.g. of spin operators; (in)compatible observables, simultaneous eigenvectors of compatible observables; maximal set of commuting observables; mean square deviation (dispersion, variance), Schwarz inequality; general uncertainty relation. 1.5 unitary operator, change of basis as a unitary transformation, similarity transformation; trace, determinant; diagonalization; unitarily equivalent observables, having identical spectra of eigenvalues. 1.6 continuous spectrum, delta function ; position eigenstates (1.6.10); infinitesimal translation (1.6.13), infinitesimal translation operator; momentum as generator of infinitesimal translation; position-momentum uncertainty relation; canonical commutation relations; Poisson bracket, prescription (1.6.47) for the transition from classical to quantum. 1.7 orthogonality (1.7.2) for position eigenstates; position space wave function (1.7.5), with inner product (1.7.6); eigenfunction; avqm HT

3 matrix elements (1.7.14) of multiplication operators (and their specialization to position eigenstates, as in (1.7.12)); momentum operator in position space; momentum space wave function (1.7.25); transformation function (1.7.32) & (1.7.50); Gaussian wave packet, minimum uncertainty. Some relevant aspects of Hilbert spaces and of operators acting on them are not taken up in the book. Some of these are discussed in a separate lecture, with its own summary file. Much more information is available e.g. on wikipedia; some relevant links are: definition of Hilbert space examples: Lebesgue spaces, Sobolev spaces, Hardy spaces, Bergman spaces complete space bounded and unbounded operators symmetric and self-adjoint operators self-adjoint extension of a symmetric operator example of a non-trivial extension spectral theory for self-adjoint operators Chapter 2: Quantum dynamics This chapter deals with the description of the time development of a system. It is again partially a repetition of previous courses (in particular section 2.3), though to a lesser extent than chapter the time evolution operator U(t,t ), unitarity of U as a consequence of probability conservation, compositionofu(t 1,t )andu(t 2,t 1 ), therelation(2.1.21)betweeninfinitesimal timetranslations and the Hamiltonian; Schrödinger equation for the time evolution operator, Dyson series; energy eigenstates, conserved quantities, complete set of mutually commuting observables; stationary and non-stationary states; application: spin precession; correlation amplitude, energy-time uncertainty relation. 2.2 the Schrödinger and Heisenberg pictures; the Heisenberg equation of motion; classical limit, Poisson brackets; avqm HT

4 Ehrenfest s theorem; time dependence of eigenstates ( base kets ) in the Heisenberg picture, transition amplitudes. 2.3 description of the harmonic oscillator, in particular energy eigenstates and eigenvalues, with the help of creation and annihilation operators; time development of the harmonic oscillator in the Heisenberg picture; behavior of the step operators in the Heisenberg picture, coherent states. The contents of this section plays a fundamental role in further applications to many-particle systems and to quantum field theory. 2.4 the Schrödinger wave equation; the time-independent Schrödinger equation; probabilistic interpretation of the wave function, probability current; classical limit, Hamilton-Jacobi equation; regions with slowly varying potential, the WKB approximation; the bouncing ball potential. 2.5 propagator function, Green s function for the Schrödinger wave operator; transition amplitude, composition property; sum over paths; relation between the transition amplitude and the classical action, Feynman path integral. The path integral formalism is one of the standard tools in modern many-particle theory and quantum field theory. 2.6 gauge transformation; gauge transformation in electromagnetism, canonical and kinematical momentum; Aharonov-Bohm effect, magnetic flux quantum; magnetic monopole. It is recommended to also read Supplement 1 (Adiabatic change and geometrical phase, p. 464 of the revised edition from 1994). Chapter 3: Theory of angular momentum This chapter surveys the quantum-mechanical treatment of angular momentum. Some parts, in particular much of the contents of sections 3.5 and 3.6, should already be familiar from previous courses. The chapter contains a description of the groups SO(3) and SU(2), which belong to the class of soavqm HT

5 called Lie groups, and of their Lie algebras and their finite-dimensional irreducible representations. These topics are discussed in much more detail in the course Symmetries: groups, algebras and tensor calculus (FYAD08). Correspondingly the focus here should be more on the physical relevance than on the mathematical tools, but nevertheless you must be able to handle these tools. Also introduced are, in section 3.4, density operators and mixed ensembles. This is done in the context of spin, but is an independent subject, of relevance far beyond the context in which it is discussed here; it is in particular a crucial ingredient of quantum statistical mechanics. 3.1 finite versus infinitesimal rotations; active versus passive transformations; the mathematical structure of a group ; the rotation operator and its relation with the angular momentum operator; the commutation relations of the angular momentum operators. 3.2 application to spin- 1 2 systems; spinning system in a magnetic field, spin precession; the non-trivial change (3.2.15) of a spinors under a 2π-rotation, the test of this change by neutron interferometry; Pauli s two-component notation and two-component spinors; various identities involving Pauli matrices, in particular (3.2.35) and (3.2.39); the 2 2-matrix representation of the rotation operator. 3.3 the groups SO(3) and SU(2), as well as O(3) and U(3) and more generally, O(m) and U(n); parametrization of SU(2), Cayley-Klein parameters; Euler angles; the expression (3.3.22) for the 2 2-matrix representation in terms of (a variant of) Euler angles. 3.4 description of general ensembles a new axiom of quantum mechanics; sub-ensembles, incoherent mixture, pure and mixed ensembles, random ensembles; ensemble average (3.4.10) and density operator/ density matrix; discrete vs continuous probability distributions; time evolution of ensembles; entropy (3.4.41) as a measure of randomness; quantum statistical mechanics; canonical ensemble, partition function, internal energy. avqm HT

6 3.5 ladder operators; derivation of the eigenvalue spectrum of J 2 and of J z ; matrix elements of J 2, J z and J ± ; the 3 3-matrix representation of the rotation operator; the matrices d (j) (β) with entries (3.5.51), explicit form of d (1/2) and d (1). 3.6 orbital angular momentum as a generator of rotations; spherical harmonics, Legendre polynomials; spherical harmonics as special rotation matrices. 3.7 addition of angular momenta as a tensor product (in proper mathematical terms: as the tensor product of finite-dimensional irreducible representations of the Lie algebra su(2)); Clebsch-Gordan coefficients, defined via the basis change (3.7.33); orthogonality properties of the Clebsch-Gordan coefficients; recursion relations for the Clebsch-Gordan coefficients; Wigner s 3j-symbols; Clebsch-Gordan series. 3.8 expressing the angular momentum operators through two uncoupled oscillators(formulas(3.8.8)); Wigner s formula (3.8.33) for d (j) (β). 3.9 measurements on composite systems; EPR paradox, hidden variable theories; Bell s inequality and its consequences for the conceptual foundation of quantum mechanics. Warning: quantum mechanics is inherently non-relativistic, but in the present context this is irrelevant because no information is exchanged between different observers vector operator, defined by the commutation relations (3.10.8) with angular momentum; Cartesian tensors versus spherical tensors; spherical tensor operator, defined by the commutation relations ( ) with angular momentum; product ( ) of irreducible spherical tensors; the selection rule ( ) for matrix elements of tensor operators; the Wigner-Eckart theorem ( ), reduced matrix elements; the projection theorem ( ) for vector operators. avqm HT

7 Chapter 4. Symmetry in Quantum Mechanics This chapter discusses various aspects of symmetries. In the context of time reversal symmetry, the notion of antilinear and anti-unitary operators is introduced. 4.1 symmetries in the Lagrangian and Hamiltonian formulation of classical mechanics; symmetries in quantum mechanics as unitary transformations generated by Hermitian operators; the relation between symmetries and conserved quantities. 4.2 parity (space inversion); pseudoscalars and pseudovectors; the behaviour of wave functions under the parity operation; the symmetric double-well potential as an example; parity selection rules; violation of parity symmetry in the weak interactions. 4.3 periodic potentials, Bloch s theorem; Brillouin zone. 4.4 time reversal / reversal of motion; antilinear and anti-unitary operators; Wigner s theorem; the operator K B acting as in (4.4.15), with chosen basis B, and its use in relating unitary and anti-unitary operators; the time-reversal operator T; commutation (4.4.31) of T with the Hamiltonian; T-even and T-odd hermitian operators; commutation relations (4.4.45), (4.4.47) and (4.4.53) of T with the operators p, x and J; reality of eigenfunctions for nondegenerate energy eigenvalues; action (4.4.72) of T 2 on a spin-j system; expression (4.4.73) for T with respect to an eigenbasis of J z ; T-even and T-odd spherical tensor operators; Kramers degeneracy and its lifting by a magnetic field. avqm HT

8 Chapter 5. Approximation Methods Again parts of this chapter should be known from previous courses. 5.1 perturbations of non-degenerate systems and the perturbation expansion (5.1.36); 1st and 2nd order energy corrections (5.1.42); 1st and 2nd order wave function corrections (5.1.44); wave-function renormalization; the quadratic Stark effect. 5.2 perturbation expansion for degenerate systems, summarized on p. 302; the linear Stark effect. 5.3 fine structure of the spectrum of a hydrogenic atom due to spin-orbit interaction; Thomas precession; Landé s interval rule; the (anomalous) Zeeman effect; van der Waals potential. 5.4 the variational method, giving upper bounds on energy eigenvalues. 5.5 the interaction picture, summarized in Table 5.2; resonance condition on the frequency of an external field in a two-state system; absorption-emission cycle; nuclear magnetic resonance. 5.6 Dyson series for the time-evolution operator in the interaction picture; the transition probability (5.6.19); the interpretation of its quadratic time-dependence (5.6.26) with the help of the density of final states; the transition rate and Fermi s golden rule; detailed balancing. 5.7 classical radiation field; absorption and stimulated emission; avqm HT

9 absorption cross section, defined in (5.7.10) and expressed as in (5.7.14); electric dipole (E1) approximation, oscillator strength, Thomas-Reiche-Kuhn sum rule; differential cross section (5.7.36) in the photoelectric effect. 5.8 level shift (5.8.11), with first and second order contributions (5.8.13) and (5.8.15); decay width Γ, defined in (5.8.18); justification of the interpretation of Γ as a width. Chapter 6. Identical Particles This chapter studies consequences of the fact that in quantum theory identical particles are indistinguishable. A crucial ingredient is the distinction between bosons and fermions, satisfying Bose-Einstein and Fermi-Dirac statistics, respectively. This spin-statistics connection can actually be proven in suitable axiomatic approaches to quantum field theory, but this is much beyond the scope of the book. Another effect that is not treated in the book, but is crucial for several applications of quantum mechanics, such as the fractional quantum Hall effect, is the appearance of (pseudo)particles that behave neither as bosons nor as fermions, but in a sense interpolate between them; such particles are usually called anyons. Such a behavior, also referred to as fractional statistics or braid group statistics, is possible in systems in which particles are constrained to move in either one or two dimensions only. (The original article on the subject, by Leinaas & Myrheim, which can be accessed here, explains in particular how to interpret the exchange of two particles as a motion around a closed loop in the configuration space.) 6.1 identical particles, exchange degeneracy; permutation symmetry; tensor product of vector spaces and -notation for its elements; the permutation operator, with properties (6.1.5), (6.1.8) and (6.1.10) and eigenstates (6.1.11); the (anti)symmetrizer (6.1.12). 6.2 bosons and fermions, Bose-Einstein / Fermi-Dirac statistics; the symmetrization postulate (6.2.1); the Pauli principle; classical Maxwell-Boltzmann statistics; Bose-Einstein condensation. 6.3 factorization of permutation operators in a space part and a spin part; states of a two-electron system as products of a symmetric/ antisymmetric space part and an avqm HT

10 antisymmetric/ symmetric spin part; exchange density for a two-electron system. 6.4 the Hamiltonian (6.4.1) for the Helium atom; the successive approximations (6.4.4), (6.4.11) and (6.4.15) for the ground state energy of the Helium atom; effective (screened) charge; orthohelium and parahelium. 6.5 Young tableaux for the description of representations of SU(2) and of the decomposition of their tensor products (in particular, of the addition of angular momenta); extension of this method to the case of SU(3) and its relevance for the classification of hadrons as bound states of quarks; the description (6.5.21) od baryons as bound states of three quarks; the color degree of freedom of quarks, introduced for resolving the statistics paradox for baryons. The use of Young tableaux naturally extends to SU(N) for any N 2. Chapter 7. Scattering Theory Besides the description of bound systems, the study of collision processes is the second large area of applications of quantum mechanics. This is the topic of the present chapter. In scattering is studied in a time-independent description, while in 7.11 it is shown how to obtain the most important results with the help of the Green s function for the time-dependent Schrödinger equation. An important technical aspect is the relation between plane and spherical waves; this is actually relevant to other wave phenomena as well, already in classical physics. 7.1 the Lippmann-Schwinger equation, in which an infinitesimal imaginary part is added to the energy; different realizations (7.1.6), (7.1.6) and (7.1.10) of the Lippmann-Schwinger equation; the Green s function (7.1.12) for the Lippmann-Schwinger kernel; local potentials, see (7.1.20); interpretation of the process in terms of a plane wave plus an outgoing (respectively incoming) spherical wave, with amplitude (7.1.34); the formula (7.1.36) for the differential cross section. 7.2 avqm HT

11 Born approximation; first-order Born amplitude (7.2.2), being essentially the Fourier transform of the potential; angular integration in the case of spherical potentials, see (7.2.4); the Rutherford cross section as the first-order Born approximation (7.2.9) to the Yukawa cross section; general properties of the first-order Born approximation, as listed after (7.2.11); the condition (7.2.12) for applicability of the first-order Born approximation; the transition operator T, defined by (7.2.16); higher-order approximations through the iterative solution (7.2.20) of (7.2.18). 7.3 the optical theorem (7.3.1), relating the imaginary part of forward scattering to the total cross section. 7.4 the eikonal approximation, applicable when the potential is approximately constant at the scale of the wavelength; the approximate result (7.4.6) for the semicalssical wave function, giving the formula (7.4.7) for the amplitude of the spherical wave. 7.5 spherical-wave state; the expression (7.5.15) of a momentum-space plane-wave basis state in terms of spherical-wave basis states; spherical Bessel function; the expression (7.5.21b) for the position-space wave function of a spherical-wave basis state. 7.6 the expansion (7.6.6) of the scattering amplitude in terms of the partial-wave amplitudes (7.6.5); the asymptotic behavior (7.6.7) of spherical Bessel functions; the change (7.6.9) of the outgoing spherical wave as a consequence of scattering; the unitarity relation (7.6.13) for partial waves; the partial-wave phase shifts, defined in (7.6.14); the expressions (7.6.17) and (7.6.18) for the scattering amplitude and the total cross section; the maximal partial-wave cross section (7.6.22); the expressions (7.6.35) for (the tangent of) partial-wave phase shifts; example: hard-sphere scattering, giving the formulas (7.6.44) and (7.6.47) for phase shifts and the unexpected approximate expressions (7.6.49) and (7.6.53) for the total cross section. 7.7 the effective potential (7.7.1) for a partial wave; the (approximate) threshold behavior (7.7.3) of partial-wave phase shifts; the Ramsauer-Townsend effect; avqm HT

12 the scattering length, defined in (7.7.13) and related to the low-energy limit of the total cross section as in (7.7.14); the geometric interpretation of the scattering length, see figure 7.9; sign change of the scattering length as indication of the formation of a bound state, and the relation (7.7.17) between the bound state wave function and the scattering length; bound states as poles of partial-wave scattering amplitudes; 7.8 quasi-bound states; resonance behavior of partial-wave cross sections at quasi-bound state energies; the Breit-Wigner formula (7.8.9) for the partial-wave cross section in the vicinity of a quasibound state energy, with width defined in (7.8.8). 7.9 constructive and destructive interference in a scattering process involving identical particles, exemplified by the formulas (7.9.2) for spin-0 particles and (7.9.3) for unpolarized spin-1/2 particles symmetry relations for matrix elements of the transition operator: (7.10.4) for unitary symmetry operators, ( ) for time reversal; detailed balance, following from invariance under both parity and time reversal the Green s operator (7.11.5) for the time-dependent Schrödinger equation, satisfying (7.11.3); the retarded boundary condition (7.11.4), accounting for causality; adiabatic switch-on of the potential, justified by considering wave packets; the relation ( ) between δ-function normalization and box normalization inelastic scattering; the form factor, defined in ( ); stopping power; the correction ( ) to Rutherford scattering resulting from the finite size of the nucleus; the approximate expression ( ) for the form factor of the nucleus the approximate expression ( ) for scattering solutions to the time-independent Schrödinger equation of the Coulomb problem; the expression ( ) for the partial-wave contributions, with phases satisfying (7.3.31); the difference between the result ( ), valid for the Coulomb force, and the behavior ( ) of the wave function, valid for short-range forces. The book does not mention interesting aspects of scattering theory that play an important role in recent developments and may be summarized under the header inverse scattering. avqm HT

13 Some of these are discussed separately; they are collected in their own summary file. For more information see e.g. the following links: The inverse scattering problem (Wikipedia) (Wikipedia) The inverse scattering transform Inverse scattering on the line (article, by Deift and Trubowitz) Introduction to inverse scattering theory (overview, by Devaney) avqm HT