Math 450/550, Spring 2015

Topics Covered in Each Lecture

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 10	Introduction of Classical Mechanics: Particle moving in a straight line, Newton's equation and its solution, conservative forces and conservation energy; solution of the Newton's equation for a conservative form in 1-dim; pure states and observables	Pages 19-22 of Hall
2	Feb. 12	Particle moving in R^n, Newton's equation and its solution, conservative forces and conservation energy; pure state and observables, Hamiltonian formulation of Classical Mechanics; system of particles, generalized coordinates and momenta, phase space and the algebra of observables (associative algebras); evolving observables and the Poisson bracket	Pages 23-28 of Hall & Pages 1-3 of Faddeev-Yakubovskii
3	Feb. 17	Properties of the Poisson bracket, the set of observables as a Lie algebra, Hamiltonian flows, dynamics as a one-parameter family of bijections of the phase space or as a one-parameter family of automorphisms of both the associative and Lie algebras of observables; real experiments and their verifiability; states as functions assigning to each observable a probability measure	Pages 33-40 of Hall & Pages 3-7 of Faddeev-Yakubovskii
4	Feb. 24	The mean or expectation value of an observable in a state, state as a linear functional acting in the vector space of observables, integral measure and the weight function defined by a state, pure states as states corresponding to Dirac delta weight functions, weight function as a probability density, Variance of an observable in a state as a measure of deviation from pure states, the time-evolution of expectation value of observables, the Hamilton and Liouville pictures, the Liouville's equation	Pages 8-15 of Faddeev-Yakubovskii
5	Feb. 26	Commutativity of the algebra of observables and the origin of the discrepancy between predictions of classical mechanics and experimental evidence; Review of basic linear algebra: Complex vector spaces, subspace, linear combination, span, and linear independence, dimension, image and inverse image of subsets under function, range and domain, linear operators, null-space, associative and Lie algebras gl(V) of linear operators acting on a vector space V,  isomorphisms and automorphisms, the automorphism group of a vector space GL(V), matrix representations of vectors and linear operators for finite-dimensional vector spaces, associative and Lie algebras algebra of complex square matrices gl(n,C), the group of invertible complex square matrices GL(n,C).	Pages 11-18 of Prugovecki
6	Mar. 03	Inner product spaces, norm and distance, unit and orthogonal vectors, orthonormal subsets, Gram-Schmidt orthonormalization, orthonormal besesof finite-dimensional inner-product spaces, the matrix representation of vector and linear operators in an orthonormal basis, inner product on the space of n ×1 matrices, Hermitian conjugation (adjoint) of a matrix, Hermitian and anti-Hermitian matrices, symmetric operators, automorphisms mapping orthonormal bases to orthonormal bases, unitary matrices and unitary groups, isometries and unitary operators.	Pages 18-24 of Prugovecki
7	Mar. 05	Isomorphic inner-product spaces, classification of finite-dimsional inner-product spaces using their dimension, Riesz Lemma (in finite-dim.), adjoint of a linear operator and self-adjoint (or Hermitian) opeartors (in finite-dim.), an example of the application of Gram-Schmidt process and the idea of a projection operator.	-
8	Mar. 10	Projection operators, orthogonal projection operators, characterization of orthogonal projection operators with one-dim. range,  their self-adjointness and one-to-one correspondence with 1-dim subspaces (rays), complete orthonormal system of orthogonal projection operators, spectral theorem for self-adjoint operators acting in a finite-dim. inner-product space, generalization to normal operators, application to unitary operators.	-
9	Mar. 12	Different orthonornal bases corresponding to a given complete orthonormal system of orthogonal operators with 1-dim range, functions of a normal operator, relation between Hermitian and unitary operators in finite-dimensions, trace of a linear operator in finite-dimensions and its properties, point spectrum, positive and positive-definite operators	-
10	Mar. 17	The inner product defined by trace on gl(V), the Dirac's bra-ket notation, orthonormality of the complete orthonormal system of projection operators with one-dimensional range with respect to the trace-inner product; Mathematical structures entering QM: State vecctors, states, the phase or state space of QM, observablea; the phase space of a quantum system when the space of state vectors is the Euclidean space E^2 (C^2 with Euclidean inner product), its identification with a 2-dim. sphere	-
11	Mar. 19	Algebra of observables in QM, its Lie algebra and inner-product space structure; States as certain linear functionals on the algebra of observables, their characterizations in terms of unit-trace positive operators as well as convex combination of pure states	Pages 27-35 of Faddeev-Yakubovskii
12	Mar. 26	Pureness of pure states (One cannot write a pure state as a convex combination of two different states.), characterization of the set of states for a quantum system with the two-dimensional complex Euclidean space as the space of state vectors with a ball of radius 1/2, Parallelism between the notion of state in classical and quantum mechanics and their difference (in QM, variance of an observable is not generally zero even for pure states); Theorem: Variance of a pure state is zero iff it is an eigenstate.	Pages 35-36 of Faddeev-Yakubovskii
Exam 1	Mar. 28
13	Mar. 31	Spectral representation of Hermitian operators with degenerate eigenvalues (finite-dimensional case), observables with a common orthonormal basis of eigenvectors, Heisenberg's uncertainty principle	Pages 36-39 of Faddeev-Yakubovskii
14	Apr. 02	Spectral function of an observable and the probability density; the Copenhagen interpretation of QM and the projection axiom; Dynamics in QM, Heisenberg and Schrödinger pictures, the Heisenberg and Liouville-von-Neumann equations and their solution; the time-evolution operator, the evolution of the pure states and the time-dependent Schrödinger equation; stationary states and the time-independent  Schrödinger equation	Pages 39-48 of Faddeev-Yakubovskii
Spring Break
15	Apr. 14	Cauchy-Schwarz inequality, The vector space C^∞of complex sequences, the generalization of the Euclidean inner product and the standard basis of E^n to C^∞; the space of square summable sequences, convergent and Cauchy sequences in an inner-product space, Hilbert spaces, closed, open, and dense subsets of an inner-product space.	Pages 25-44 of Prugovecki
16	Apr. 16	Schauder and orthonormal bases of an infinite-dimensional inner-product space, closure of a subset, Cauchy completion of inner-product spaces, the space L^2([0,1]) as completion of C^0([0,1]) endowed with the L^2 inner product, the need for Lebesgue integration,  the spaces L^2(R) and L^2(R^n).	-
17	Apr. 17	Extra Class: Continuous functions, topological dual of an inner-product space, Riesz lemma for Hilbert spaces, bounded linear operators, unboundedness of the X-operator ( (Xf)(x):=x f(x) ) in L^2(R); equivalence of continuity and boundedness for linear operators, boundedness of the point spectrum of bounded operators; bounded extension of densely-defined bounded operators; separable Hilbert spaces	Pages 537-539 of Hall
18	Apr. 21	E^n , l^2, L^2(R), and  L^2(R^n) as examples of separable Hilbert spaces; Closed operators and their graph, closed bounded operators, statements of Closed-Graph and Hellinger-Toeplitz theorems, closable operators and their closure; adjoint of a general densely defined linear operator mapping an inner-product space to another, its linearity, uniqueness, and closedness; self-adjoint and essentially self-adjoint operators, adjoint of the adjoint operator	Pages 169-174 of Hall
19	Apr. 24	Extra Class: Numerical range of a linear operator, characterization of symmetric operators in terms of the reality of the their numerical range, statement of various characterization theorem for the symmetric operators that are essentially self-adjoint or self-adjoint, the deficiency indices, positive operators, operators that are bounded from below, statement of the inverse mapping theorem, resolvent set and the spectrum of a linear operator, the point, continuous and residual spectra, spectrum of self-adjoint operators	Pages 174-180 of Hall
20	Apr. 28	Review of the construction of the spectral function in finite dimensions, the spectral theorem for self-adjoint operators A acting in an infinite-dimensional separable Hilbert space, the restrictions of A whose spectrum coincides with the point and continuous spectra of A; Hilbert-Schmidt operators	Pages 49-50 of Faddeev-Yakubovskii
21	Apr. 30	An inner-product on the vector space of Hilbert-Schmidt operators, The adjoint of a bounded operator is bounded; The adjoint of a Hilbert-Schmidt operator is a Hilbert-Schmidt operator; Product of bounded and Hilbert-Schmidt operators are Hilbert-Schmidt operators; trace-class operators, trace-class operators are Hilbert-Schmidt, properties of the trace of trace-class operators	Pages 421-423 of Hall
22	May 12	The choice of Hilbert space in QM and assignment of physical meaning to self-adjoint operators: Uniqueness of infinite-dim. separable Hilbert spaces up to unitary  equivalence, quantization of a classical system with phase space R^2n and Heisenberg-Weyl algebra hn, unitary, faithful, and irreducible representations of a Lie algebra, non-existence of a finite-dimensional faithful representation of hn, uniqueness of the unitary irreducible representation of hn up to unitary  equivalence	Pages 51-54 of Faddeev-Yakubovskii
23	May 14	The standard canonical quantization of a classical system with phase space R^2n, the standard position and momentum operators, the non-standard unitary equivalent quantizations, the factor-ordering ambiguity	Pages 62-64 & 82-83 of Hall
24	May 15	Extra Class: The position and momentum operators in the standard description of the quantum mechanics of a particle with classical phase space R^2n, emptiness of their point spectrum, generalized eigenfunctions of the position operator for n=1 and the concept of a distribution, definition and role of the Dirac delta function, the generalized eigenfunctions of the momentum operator P, the relation to Fourier transform and the Parseval's identity, the spectral representation of X and P operators in Dirac's bra-ket notation.	Pages 58-64 of Hall
Exam 2	May 18
Make-up Exam	May 25

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.